MFEMVectorFESpace

Overview

This is a convenience class for building finite element spaces to represent vector variables. The family of shape functions is selected from the fec_type parameter, and the order is controlled using the fec_order parameter. The number of vector components _in reference space_ is set by the range_dim parameter and defaults to the same as the dimension of the mesh. The number of vector components in physical space may differ, if a lower-dimensional mesh is embedded in a higher dimensional space.

The dimension of the resulting finite element collection/space will be the same as the dimension of the mesh. The value of vdim (the number of degrees of freedom per basis function) will depend on fec_type.

H1 and L2(Int) spaces will have vdim set to range_dim
ND and RT spaces will always have vdim set to 1

Note that Nédélec and Raviart-Thomas shape functions do not support vectors with fewer components than the dimension of the problem. Furthermore, MFEM does not currently support Nédélec and Raviart-Thomas finite elements in a 1D reference space but with 2D vectors.

If you need a finite element space that can't be constructed using the options available in this class, you can use MFEMGenericFESpace instead.

Example Input File Syntax

[FESpaces<<<{"href": "../../../syntax/FESpaces/index.html"}>>>]
  [H1FESpace]
    type = MFEMVectorFESpace<<<{"description": "Convenience class to construct vector finite element spaces, abstracting away some of the mathematical complexity of specifying the dimensions.", "href": "MFEMVectorFESpace.html"}>>>
    fec_type<<<{"description": "Specifies the family of FE shape functions."}>>> = H1
    fec_order<<<{"description": "Order of the FE shape function to use."}>>> = FIRST
    range_dim<<<{"description": "The number of components of the vectors in reference space. Zero (the default) means it will be the same as the problem dimension. Note that MFEM does not currently support 2D vectors in 1D space for ND and RT elements."}>>> = 3
    ordering<<<{"description": "Ordering style to use for vector DoFs."}>>> = "vdim"
  []
[]

[Variables<<<{"href": "../../../syntax/Variables/index.html"}>>>]
  [displacement]
    type = MFEMVariable<<<{"description": "Class for adding MFEM variables to the problem (`mfem::ParGridFunction`s).", "href": "../variables/MFEMVariable.html"}>>>
    fespace<<<{"description": "The finite element space this variable is defined on."}>>> = H1FESpace
  []
[]

Input Parameters

basisGaussLobattoSpecifies the basis used for H1 and L2(Int) vector elements. H1 spaces require a closed basis (GaussLobatto Positive ClosedUniform Serendipity ClosedGL)
Default:GaussLobatto
C++ Type:MooseEnum
Options:GaussLegendre, GaussLobatto, Positive, OpenUniform, ClosedUniform, OpenHalfUniform, Serendipity, ClosedGL, IntegratedGLL
Controllable:No
Description:Specifies the basis used for H1 and L2(Int) vector elements. H1 spaces require a closed basis (GaussLobatto Positive ClosedUniform Serendipity ClosedGL)
closed_basisGaussLobattoSpecifies the closed basis used for ND and RT elements.
Default:GaussLobatto
C++ Type:MooseEnum
Options:GaussLobatto, ClosedUniform, Serendipity, ClosedGL
Controllable:No
Description:Specifies the closed basis used for ND and RT elements.
fec_orderFIRSTOrder of the FE shape function to use.
Default:FIRST
C++ Type:MooseEnum
Options:CONSTANT, FIRST, SECOND, THIRD, FOURTH, FIFTH, SIXTH, SEVENTH, EIGHTH, NINTH, TENTH, ELEVENTH, TWELFTH, THIRTEENTH, FOURTEENTH, FIFTEENTH, SIXTEENTH, SEVENTEENTH, EIGHTTEENTH, NINETEENTH, TWENTIETH, TWENTYFIRST, TWENTYSECOND, TWENTYTHIRD, TWENTYFOURTH, TWENTYFIFTH, TWENTYSIXTH, TWENTYSEVENTH, TWENTYEIGHTH, TWENTYNINTH, THIRTIETH, THIRTYFIRST, THIRTYSECOND, THIRTYTHIRD, THIRTYFOURTH, THIRTYFIFTH, THIRTYSIXTH, THIRTYSEVENTH, THIRTYEIGHTH, THIRTYNINTH, FORTIETH, FORTYFIRST, FORTYSECOND, FORTYTHIRD
Controllable:No
Description:Order of the FE shape function to use.
fec_typeH1Specifies the family of FE shape functions.
Default:H1
C++ Type:MooseEnum
Options:H1, ND, RT, L2, L2Int
Controllable:No
Description:Specifies the family of FE shape functions.
open_basisGaussLegendreSpecifies the open basis used for ND and RT elements.
Default:GaussLegendre
C++ Type:MooseEnum
Options:GaussLegendre, GaussLobatto, Positive, OpenUniform, ClosedUniform, OpenHalfUniform, Serendipity, ClosedGL, IntegratedGLL
Controllable:No
Description:Specifies the open basis used for ND and RT elements.
orderingVDIMOrdering style to use for vector DoFs.
Default:VDIM
C++ Type:MooseEnum
Options:NODES, VDIM
Controllable:No
Description:Ordering style to use for vector DoFs.
range_dim0The number of components of the vectors in reference space. Zero (the default) means it will be the same as the problem dimension. Note that MFEM does not currently support 2D vectors in 1D space for ND and RT elements.
Default:0
C++ Type:int
Controllable:No
Description:The number of components of the vectors in reference space. Zero (the default) means it will be the same as the problem dimension. Note that MFEM does not currently support 2D vectors in 1D space for ND and RT elements.
submeshSubmesh to define the FESpace on. Leave blank to use base mesh.
C++ Type:std::string
Controllable:No
Description:Submesh to define the FESpace on. Leave blank to use base mesh.
vdim1The number of degrees of freedom per basis function.
Default:1
C++ Type:int
Controllable:No
Description:The number of degrees of freedom per basis function.

Optional Parameters

control_tagsAdds user-defined labels for accessing object parameters via control logic.
C++ Type:std::vector<std::string>
Controllable:No
Description:Adds user-defined labels for accessing object parameters via control logic.
enableTrueSet the enabled status of the MooseObject.
Default:True
C++ Type:bool
Controllable:No
Description:Set the enabled status of the MooseObject.

Advanced Parameters

Input Files

(test/tests/mfem/kernels/graddiv.i)
(test/tests/mfem/transfers/mfem_parent_mfem_sub/mfem_parent_vector.i)
(test/tests/mfem/auxkernels/projection.i)
(test/tests/mfem/ics/vector_ic.i)
(test/tests/mfem/vectorpostprocessors/point_value_sampler/point_value_sampler_diffusion.i)
(test/tests/mfem/auxkernels/2Dmagnetostatic.i)
(test/tests/mfem/kernels/gravity.i)
(test/tests/mfem/kernels/irrotational.i)
(test/tests/mfem/complex/complex_waveguide.i)
(test/tests/mfem/kernels/darcy.i)
(test/tests/mfem/transfers/mfem_parent_mfem_sub/mfem_sub_vector.i)
(test/tests/mfem/kernels/diffusion_amr.i)
(test/tests/mfem/vectorpostprocessors/point_value_sampler/point_value_sampler_curlcurl.i)
(test/tests/mfem/kernels/mixed_heattransfer.i)
(test/tests/mfem/kernels/linearelasticity.i)
(test/tests/mfem/submeshes/magnetostatic.i)
(test/tests/mfem/vectorpostprocessors/line_value_sampler/line_value_sampler_curlcurl.i)
(test/tests/mfem/kernels/maxwell_eigenproblem.i)
(test/tests/mfem/submeshes/av_magnetostatic.i)
(test/tests/mfem/vectorpostprocessors/line_value_sampler/line_value_sampler_diffusion.i)
(test/tests/mfem/submeshes/hphi_magnetostatic.i)
(test/tests/mfem/submeshes/hphi_magnetodynamic.i)
(test/tests/mfem/kernels/curlcurl.i)
(test/tests/mfem/kernels/diffusion.i)
(test/tests/mfem/submeshes/cut_closed_coil.i)
(test/tests/mfem/auxkernels/crossproduct.i)

(test/tests/mfem/kernels/gravity.i)

[Mesh]
  type = MFEMMesh
  file = ../mesh/beam-tet.mesh
  dim = 3
  uniform_refine = 2
  displacement = "displacement"
[]

[Problem]
  type = MFEMProblem
[]

[FESpaces]
  [H1FESpace]
    type = MFEMVectorFESpace
    fec_type = H1
    fec_order = FIRST
    range_dim = 3
    ordering = "vdim"
  []
[]

[Variables]
  [displacement]
    type = MFEMVariable
    fespace = H1FESpace
  []
[]

[BCs]
  [dirichlet]
    type = MFEMVectorDirichletBC
    variable = displacement
    boundary = '1'
  []
[]

[FunctorMaterials]
  [Rigidium]
    type = MFEMGenericFunctorMaterial
    prop_names = 'lambda mu'
    prop_values = '50.0 50.0'
    block = 1
  []
  [Bendium]
    type = MFEMGenericFunctorMaterial
    prop_names = 'lambda mu'
    prop_values = '1.0 1.0'
    block = 2
  []
  [RigidiumWeightDensity]
    type = MFEMGenericFunctorVectorMaterial
    prop_names = 'gravitational_force_density'
    prop_values = '{0.0 0.0 -1e-2}'
    block = 1
  []
  [BendiumWeightDensity]
    type = MFEMGenericFunctorVectorMaterial
    prop_names = 'gravitational_force_density'
    prop_values = '{0.0 0.0 -5e-3}'
    block = 2
  []
[]

[Kernels]
  [diff]
    type = MFEMLinearElasticityKernel
    variable = displacement
    lambda = lambda
    mu = mu
  []
  [gravity]
    type = MFEMVectorDomainLFKernel
    variable = displacement
    vector_coefficient = gravitational_force_density
  []
[]

[Preconditioner]
  [boomeramg]
    type = MFEMHypreBoomerAMG
    fespace = H1FESpace
    l_max_its = 20
    l_tol = 1e-5
    print_level = 2
  []
[]

[Solver]
  type = MFEMHyprePCG
  preconditioner = boomeramg
  l_max_its = 100
  l_tol = 1e-4
  l_abs_tol = 0.0
  print_level = 2
[]

[Executioner]
  type = MFEMSteady
  device = "cpu"
[]

[Outputs]
  [ParaViewDataCollection]
    type = MFEMParaViewDataCollection
    file_base = OutputData/Gravity
    vtk_format = ASCII
  []
[]

(test/tests/mfem/kernels/graddiv.i)

# Grad-div problem using method of manufactured solutions,
# based on MFEM Example 4.

[Mesh]
  type = MFEMMesh
  file = ../mesh/beam-tet.mesh
  dim = 3
  uniform_refine = 1
[]

[Problem]
  type = MFEMProblem
[]

[FESpaces]
  [HDivFESpace]
    type = MFEMVectorFESpace
    fec_type = RT
    fec_order = CONSTANT
    ordering = "vdim"
  []
  [L2FESpace]
    type = MFEMScalarFESpace
    fec_type = L2
    fec_order = CONSTANT
    basis = GaussLegendre
  []
[]

[Variables]
  [F]
    type = MFEMVariable
    fespace = HDivFESpace
  []
[]

[AuxVariables]
  [divF]
    type = MFEMVariable
    fespace = L2FESpace
  []
[]

[AuxKernels]
  [div]
    type = MFEMDivAux
    variable = divF
    source = F
    execute_on = TIMESTEP_END
  []
[]

[Functions]
  [f]
    type = ParsedVectorFunction
    expression_x = '(1. + 2*kappa * kappa) * cos(kappa * x) * sin(kappa * y)'
    expression_y = '(1. + 2*kappa * kappa) * cos(kappa * y) * sin(kappa * x)'
    expression_z = '0'

    symbol_names = kappa
    symbol_values = 3.1415926535
  []
  [F_exact]
    type = ParsedVectorFunction
    expression_x = 'cos(kappa * x) * sin(kappa * y)'
    expression_y = 'cos(kappa * y) * sin(kappa * x)'
    expression_z = '0'

    symbol_names = kappa
    symbol_values = 3.1415926535
  []
[]

[BCs]
  [dirichlet]
    type = MFEMVectorNormalDirichletBC
    variable = F
    boundary = '1 2 3'
    vector_coefficient = F_exact
  []
[]

[Kernels]
  [divdiv]
    type = MFEMDivDivKernel
    variable = F
  []
  [mass]
    type = MFEMVectorFEMassKernel
    variable = F
  []
  [source]
    type = MFEMVectorFEDomainLFKernel
    variable = F
    vector_coefficient = f
  []
[]

[Preconditioner]
  [ADS]
    type = MFEMHypreADS
    fespace = HDivFESpace
  []
[]

[Solver]
  type = MFEMCGSolver
  preconditioner = ADS
  l_tol = 1e-16
  l_max_its = 1000
  print_level = 2
[]

[Executioner]
  type = MFEMSteady
  device = "cpu"
[]

[Outputs]
  [ParaViewDataCollection]
    type = MFEMParaViewDataCollection
    file_base = OutputData/GradDiv
    vtk_format = ASCII
  []
[]

(test/tests/mfem/transfers/mfem_parent_mfem_sub/mfem_parent_vector.i)

[Mesh]
  type = MFEMMesh
  file = ../../mesh/cube.e
[]

[Problem]
  type = MFEMProblem
  solve = false
[]

[FESpaces]
  [H1FESpace]
    type = MFEMVectorFESpace
    fec_type = H1
    fec_order = FIRST
  []
  [HCurlFESpace]
    type = MFEMVectorFESpace
    fec_type = ND
    fec_order = FIRST
  []
  [HDivFESpace]
    type = MFEMVectorFESpace
    fec_type = RT
    fec_order = CONSTANT
  []
  [L2FESpace]
    type = MFEMVectorFESpace
    fec_type = L2
    fec_order = CONSTANT
  []
[]

[Variables]
  [mfem_parent_h1_vector_var]
    type = MFEMVariable
    fespace = H1FESpace
  []
  [mfem_parent_hcurl_vector_var]
    type = MFEMVariable
    fespace = HCurlFESpace
  []
  [mfem_parent_hdiv_vector_var]
    type = MFEMVariable
    fespace = HDivFESpace
  []
  [mfem_parent_l2_vector_var]
    type = MFEMVariable
    fespace = L2FESpace
  []
[]

[AuxVariables]
  [mfem_sub_h1_vector_var]
    type = MFEMVariable
    fespace = H1FESpace
  []
  [mfem_sub_hcurl_vector_var]
    type = MFEMVariable
    fespace = HCurlFESpace
  []
  [mfem_sub_hdiv_vector_var]
    type = MFEMVariable
    fespace = HDivFESpace
  []
  [mfem_sub_l2_vector_var]
    type = MFEMVariable
    fespace = L2FESpace
  []
[]

[Functions]
  [vector_field]
    type = ParsedVectorFunction
    expression_x = 'sin(pi * y)'
    expression_y = 'sin(pi * z)'
    expression_z = 'sin(pi * x)'
  []
[]

[ICs]
  [h1_vector_ic]
    type = MFEMVectorIC
    variable = mfem_parent_h1_vector_var
    vector_coefficient = vector_field
  []
  [hcurl_vector_ic]
    type = MFEMVectorIC
    variable = mfem_parent_hcurl_vector_var
    vector_coefficient = vector_field
  []
  [hdiv_vector_ic]
    type = MFEMVectorIC
    variable = mfem_parent_hdiv_vector_var
    vector_coefficient = vector_field
  []
  [l2_vector_ic]
    type = MFEMVectorIC
    variable = mfem_parent_l2_vector_var
    vector_coefficient = vector_field
  []
[]

[Executioner]
  type = MFEMSteady
  device = cpu
[]

[MultiApps]
  [mfem_app]
    type = FullSolveMultiApp
    input_files = mfem_sub_vector.i
    execute_on = 'INITIAL'
  []
[]

[Transfers]
  [transfer_from_subapp]
    type = MultiAppMFEMShapeEvaluationTransfer
    source_variables = 'mfem_sub_h1_vector_var mfem_sub_hcurl_vector_var mfem_sub_hdiv_vector_var mfem_sub_l2_vector_var'
    variables = 'mfem_sub_h1_vector_var mfem_sub_hcurl_vector_var mfem_sub_hdiv_vector_var mfem_sub_l2_vector_var'
    from_multi_app = mfem_app
  []
[]

[Postprocessors]
  [H1_Var_L2_Error]
    type = MFEMVectorL2Error
    variable = mfem_parent_h1_vector_var
    function = mfem_sub_h1_vector_var
    execute_on = TIMESTEP_END
  []
  [HCurl_Var_L2_Error]
    type = MFEMVectorL2Error
    variable = mfem_parent_hcurl_vector_var
    function = mfem_sub_hcurl_vector_var
    execute_on = TIMESTEP_END
  []
  [HDiv_Var_L2_Error]
    type = MFEMVectorL2Error
    variable = mfem_parent_hdiv_vector_var
    function = mfem_sub_hdiv_vector_var
    execute_on = TIMESTEP_END
  []
  [L2_Var_L2_Error]
    type = MFEMVectorL2Error
    variable = mfem_parent_l2_vector_var
    function = mfem_sub_l2_vector_var
    execute_on = TIMESTEP_END
  []
[]

[Outputs]
  file_base = 'mfem_parent_mfem_sub_vector_h1_hcurl_hdiv_l2_hex'
  csv = true
[]

(test/tests/mfem/auxkernels/projection.i)

[Mesh]
  type = MFEMMesh
  file = ../mesh/hinomaru.e
  dim = 2
[]

[Problem]
  type = MFEMProblem
[]

[FESpaces]
  [H1FESpace]
    type = MFEMScalarFESpace
    fec_type = H1
    fec_order = FIRST
  []
  [HCurlFESpace]
    type = MFEMVectorFESpace
    fec_type = ND
    fec_order = FIRST
  []
  [L2FESpace]
    type = MFEMScalarFESpace
    fec_type = L2
    fec_order = CONSTANT
    basis = GaussLegendre
  []
[]

[Variables]
  [Az]
    type = MFEMVariable
    fespace = H1FESpace
  []
[]

[AuxVariables]
  [J]
    type = MFEMVariable
    fespace = L2FESpace
  []
  [GAz]
    type = MFEMVariable
    fespace = HCurlFESpace
  []
  [GAz(copy)]
    type = MFEMVariable
    fespace = HCurlFESpace
  []
[]

[Kernels]
  [diffusion]
    type = MFEMDiffusionKernel
    variable = Az
  []
  [source]
    type = MFEMDomainLFKernel
    variable = Az
    coefficient = J_source
  []
[]

[AuxKernels]
  [J]
    type = MFEMScalarProjectionAux
    variable = J
    coefficient = J_source
  []
  [GAz]
    type = MFEMGradAux
    variable = GAz
    source = Az
  []
  [GAz(copy)]
    type = MFEMVectorProjectionAux
    variable = GAz(copy)
    vector_coefficient = GAz
  []
[]

[BCs]
  [essential]
    type = MFEMScalarDirichletBC
    variable = Az
    boundary = outer
    coefficient = 1
  []
[]

[FunctorMaterials]
  [J_wire]
    type = MFEMGenericFunctorMaterial
    prop_names = J_source
    prop_values = 8.0
    block = wire
  []
[]

[Preconditioner]
  [boomeramg]
    type = MFEMHypreBoomerAMG
  []
[]

[Solver]
  type = MFEMHyprePCG
  preconditioner = boomeramg
  l_tol = 1e-16
[]

[Executioner]
  type = MFEMSteady
[]

[Outputs]
  [ParaViewDataCollection]
    type = MFEMParaViewDataCollection
    file_base = OutputData/Projection
    vtk_format = ASCII
  []
[]

(test/tests/mfem/ics/vector_ic.i)

[Mesh]
  type = MFEMMesh
  file = ../mesh/cylinder-hex-q2.gen
[]

[Problem]
  type = MFEMProblem
  solve = false
[]

[FESpaces]
  [H1VectorFESpace]
    type = MFEMVectorFESpace
    fec_type = H1
    fec_order = FIRST
  []
  [HCurlFESpace]
    type = MFEMVectorFESpace
    fec_type = ND
    fec_order = FIRST
  []
  [HDivFESpace]
    type = MFEMVectorFESpace
    fec_type = RT
    fec_order = CONSTANT
  []
  [L2VectorFESpace]
    type = MFEMVectorFESpace
    fec_type = L2
    fec_order = CONSTANT
    basis = GaussLegendre
  []
[]

[Variables]
  [h1_vector]
    type = MFEMVariable
    fespace = H1VectorFESpace
  []
  [nd_vector]
    type = MFEMVariable
    fespace = HCurlFESpace
  []
  [rt_vector]
    type = MFEMVariable
    fespace = HDivFESpace
  []
  [l2_vector]
    type = MFEMVariable
    fespace = L2VectorFESpace
  []
[]

[Functions]
  [external_vector_field]
    type = ParsedVectorFunction
    expression_x = 'sin(kappa * y)'
    expression_y = 'sin(kappa * z)'
    expression_z = 'sin(kappa * x)'

    symbol_names = kappa
    symbol_values = 3.1415926535
  []
[]

[ICs]
  [h1_vector_ic]
    type = MFEMVectorIC
    variable = h1_vector
    vector_coefficient = external_vector_field
  []
  [l2_vector_ic]
    type = MFEMVectorIC
    variable = l2_vector
    vector_coefficient = external_vector_field
  []
  [nd_vector_ic]
    type = MFEMVectorIC
    variable = nd_vector
    vector_coefficient = external_vector_field
  []
  [rt_vector_ic]
    type = MFEMVectorIC
    variable = rt_vector
    vector_coefficient = external_vector_field
  []
[]

[Executioner]
  type = MFEMSteady
  device = cpu
[]

[Outputs]
  [ParaViewDataCollection]
    type = MFEMParaViewDataCollection
    file_base = OutputData/VectorIC
    vtk_format = ASCII
  []
[]

(test/tests/mfem/vectorpostprocessors/point_value_sampler/point_value_sampler_diffusion.i)

# MFEM diffusion problem sampled with MFEMPointValueSampler.

[Mesh]
  type = MFEMMesh
  file = ../../mesh/mug.e
[]

[Problem]
  type = MFEMProblem
[]

[FESpaces]
  [H1FESpace]
    type = MFEMScalarFESpace
    fec_type = H1
    fec_order = FIRST
  []
  [HCurlFESpace]
    type = MFEMVectorFESpace
    fec_type = ND
    fec_order = FIRST
  []
[]

[Variables]
  [concentration]
    type = MFEMVariable
    fespace = H1FESpace
  []
[]

[AuxVariables]
  [concentration_gradient]
    type = MFEMVariable
    fespace = HCurlFESpace
  []
[]

[AuxKernels]
  [grad]
    type = MFEMGradAux
    variable = concentration_gradient
    source = concentration
    execute_on = TIMESTEP_END
  []
[]

[ICs]
  [diffused_ic]
    type = MFEMScalarIC
    coefficient = one
    variable = concentration
  []
[]

[Functions]
  [one]
    type = ParsedFunction
    expression = 1.0
  []
[]

[BCs]
  [bottom]
    type = MFEMScalarDirichletBC
    variable = concentration
    boundary = 'bottom'
    coefficient = 1.0
  []
  [top]
    type = MFEMScalarDirichletBC
    variable = concentration
    boundary = 'top'
  []
[]

[FunctorMaterials]
  [Substance]
    type = MFEMGenericFunctorMaterial
    prop_names = diffusivity
    prop_values = 1.0
    block = 'the_domain'
  []
[]

[Kernels]
  [diff]
    type = MFEMDiffusionKernel
    variable = concentration
    coefficient = diffusivity
  []
[]

[Preconditioner]
  [boomeramg]
    type = MFEMHypreBoomerAMG
  []
  [jacobi]
    type = MFEMOperatorJacobiSmoother
  []
[]

[Solver]
  type = MFEMHypreGMRES
  preconditioner = boomeramg
  l_tol = 1e-16
  l_max_its = 1000
[]

[VectorPostprocessors]
  [point_sample]
    type = MFEMPointValueSampler
    variable = 'concentration'
    points = '2.125 0 -1.375  2.125 0 1.125'
  []
[]

[Executioner]
  type = MFEMSteady
  device = cpu
[]

[Outputs]
  execute_on = 'timestep_end'
  csv = true
[]

(test/tests/mfem/auxkernels/2Dmagnetostatic.i)

[Mesh]
    type = MFEMMesh
    file = ../mesh/hinomaru.e
[]

[Problem]
    type = MFEMProblem
[]

[FESpaces]
 [H1FESpace]
   type = MFEMScalarFESpace
   fec_type = H1
   fec_order = FIRST
 []
 [HCurlFESpace]
    type = MFEMVectorFESpace
    fec_type = ND
    fec_order = FIRST
  []
  #For compatible pairing H1 order p -> ND order p -> RT order p-1
  [RTFESpace]
    type = MFEMVectorFESpace
    fec_type = RT
    fec_order = CONSTANT
  []
   [L2FESpace]
    type = MFEMScalarFESpace
    fec_type = L2
    fec_order = CONSTANT
    basis = GaussLegendre
  []
[]

[Variables]
   [Az]
    type = MFEMVariable
    fespace = H1FESpace
  []
[]

[FunctorMaterials]
  [J_wire]
    type = MFEMGenericFunctorMaterial
    prop_names = J_source
    prop_values = 8.0
    block = wire
  []
[]

[AuxVariables]
  [J]
    type = MFEMVariable
    fespace = L2FESpace
  []
  [gradAz]
    type = MFEMVariable
    fespace = HCurlFESpace
  []
  [B]
    type = MFEMVariable
    fespace = RTFESpace
  []
[]

[Kernels]
  [diffusion]
    type = MFEMDiffusionKernel
    variable = Az
  []
  [source]
    type = MFEMDomainLFKernel
    variable = Az
    coefficient = J_source
  []
[]

[AuxKernels]
  [J]
    type = MFEMScalarProjectionAux
    variable = J
    coefficient = J_source
  []
  [gradAz]
    type = MFEMGradAux
    variable = gradAz
    source = Az
  []
  [B_from_gradAz]
    type = MFEMNDtoRTAux
    variable = B
    source = gradAz
    scale_factor = 1.0
  []
[]

[BCs]
  [essential]
    type = MFEMScalarDirichletBC
    variable = Az
    boundary = outer
    coefficient = 1
  []
[]

[Preconditioner]
  [boomeramg]
    type = MFEMHypreBoomerAMG
  []
[]

[Solver]
  type = MFEMHyprePCG
  preconditioner = boomeramg
  l_tol = 1e-8
[]

[VectorPostprocessors]
  [line_sample]
    type = MFEMLineValueSampler
    variable = 'B'
    start_point = '0 2 0'
    end_point = '0 -2 0'
    num_points = 10
  []
[]

[Executioner]
  type = MFEMSteady
[]

[Outputs]
  [ReportedPostprocessors]
    type = CSV
    file_base = 2DMagnetostatic
  []
[]

(test/tests/mfem/kernels/gravity.i)

[Mesh]
  type = MFEMMesh
  file = ../mesh/beam-tet.mesh
  dim = 3
  uniform_refine = 2
  displacement = "displacement"
[]

[Problem]
  type = MFEMProblem
[]

[FESpaces]
  [H1FESpace]
    type = MFEMVectorFESpace
    fec_type = H1
    fec_order = FIRST
    range_dim = 3
    ordering = "vdim"
  []
[]

[Variables]
  [displacement]
    type = MFEMVariable
    fespace = H1FESpace
  []
[]

[BCs]
  [dirichlet]
    type = MFEMVectorDirichletBC
    variable = displacement
    boundary = '1'
  []
[]

[FunctorMaterials]
  [Rigidium]
    type = MFEMGenericFunctorMaterial
    prop_names = 'lambda mu'
    prop_values = '50.0 50.0'
    block = 1
  []
  [Bendium]
    type = MFEMGenericFunctorMaterial
    prop_names = 'lambda mu'
    prop_values = '1.0 1.0'
    block = 2
  []
  [RigidiumWeightDensity]
    type = MFEMGenericFunctorVectorMaterial
    prop_names = 'gravitational_force_density'
    prop_values = '{0.0 0.0 -1e-2}'
    block = 1
  []
  [BendiumWeightDensity]
    type = MFEMGenericFunctorVectorMaterial
    prop_names = 'gravitational_force_density'
    prop_values = '{0.0 0.0 -5e-3}'
    block = 2
  []
[]

[Kernels]
  [diff]
    type = MFEMLinearElasticityKernel
    variable = displacement
    lambda = lambda
    mu = mu
  []
  [gravity]
    type = MFEMVectorDomainLFKernel
    variable = displacement
    vector_coefficient = gravitational_force_density
  []
[]

[Preconditioner]
  [boomeramg]
    type = MFEMHypreBoomerAMG
    fespace = H1FESpace
    l_max_its = 20
    l_tol = 1e-5
    print_level = 2
  []
[]

[Solver]
  type = MFEMHyprePCG
  preconditioner = boomeramg
  l_max_its = 100
  l_tol = 1e-4
  l_abs_tol = 0.0
  print_level = 2
[]

[Executioner]
  type = MFEMSteady
  device = "cpu"
[]

[Outputs]
  [ParaViewDataCollection]
    type = MFEMParaViewDataCollection
    file_base = OutputData/Gravity
    vtk_format = ASCII
  []
[]

(test/tests/mfem/kernels/irrotational.i)

# 2D irrotational vortex with Nedelec elements of the first kind.

centre_x = -0.75
centre_y = 0.1

[Mesh]
  type = MFEMMesh
  file = ../mesh/vortex.msh
  dim = 2
[]

[Problem]
  type = MFEMProblem
[]

[FESpaces]
  [H1FESpace]
    type = MFEMScalarFESpace
    fec_type = H1
    fec_order = SEVENTH
  []
  [HCurlFESpace]
    type = MFEMVectorFESpace
    fec_type = ND
    fec_order = SEVENTH
  []
[]

[Variables]
  [velocity_potential]
    type = MFEMVariable
    fespace = H1FESpace
  []
[]

[AuxVariables]
  [velocity]
    type = MFEMVariable
    fespace = HCurlFESpace
  []
[]

[Functions]
  [speed]
    type = ParsedFunction
    expression = '1 / sqrt((x-x0)^2 + (y-y0)^2)'
    symbol_names = 'x0 y0'
  symbol_values = '${centre_x} ${centre_y}'
  []
  [theta]
    type = ParsedFunction
    expression = 'atan2(y-y0, x-x0)'
    symbol_names = 'x0 y0'
    symbol_values = '${centre_x} ${centre_y}'
  []
  [exact_velocity]
    type = ParsedVectorFunction
    expression_x = '-v * sin(th)'
    expression_y = 'v * cos(th)'
    symbol_names = 'v th'
    symbol_values = 'speed theta'
  []
[]

[BCs]
  [potential_velocity_boundary]
    type = MFEMScalarDirichletBC
    variable = velocity_potential
    boundary = '1'
    coefficient = theta
  []
[]

[Kernels]
  [laplacian]
    type = MFEMDiffusionKernel
    variable = velocity_potential
  []
[]

[AuxKernels]
  [grad]
    type = MFEMGradAux
    variable = velocity
    source = velocity_potential
    execute_on = TIMESTEP_END
  []
[]

[Preconditioner]
  [boomeramg]
    type = MFEMHypreBoomerAMG
  []
[]

[Solver]
  type = MFEMHypreGMRES
  preconditioner = boomeramg
  l_tol = 1e-16
  l_max_its = 1000
[]

[Executioner]
  type = MFEMSteady
  device = cpu
[]

[Postprocessors]
  [potential_error]
    type = MFEML2Error
    variable = velocity_potential
    function = theta
  []
  [velocity_error]
    type = MFEMVectorL2Error
    variable = velocity
    function = exact_velocity
  []
[]

[Outputs]
  [ParaViewDataCollection]
    type = MFEMParaViewDataCollection
    file_base = OutputData/Irrotational
    vtk_format = ASCII
  []
  [L2CSV]
    type = CSV
    file_base = OutputData/Irrotational
  []
[]

(test/tests/mfem/complex/complex_waveguide.i)

freq = 900e6
angfreq = ${fparse 2*pi*freq}
epsilon0 = 8.8541878176e-12
mu0 = ${fparse 4e-7*pi}
magnetic_reluctivity = ${fparse 1/mu0}
elec_cond_mouse = 0.97
elec_cond_air = 1e-323

[Mesh]
  type = MFEMMesh
  file = ../mesh/waveguide.g
  dim = 3
[]

[Problem]
  type = MFEMProblem
  numeric_type = complex
[]

[FESpaces]
  [HCurlFESpace]
    type = MFEMVectorFESpace
    fec_type = ND
    fec_order = FIRST
  []
  [HDivFESpace]
    type = MFEMVectorFESpace
    fec_type = RT
    fec_order = CONSTANT
  []
[]

[Variables]
  [E]
    type = MFEMComplexVariable
    fespace = HCurlFESpace
  []
[]

[Functions]
  [mass_coef_mouse]
    type = ParsedFunction
    expression = -43*${epsilon0}*${angfreq}^2
  []
  [loss_coef_mouse]
    type = ParsedFunction
    expression = ${angfreq}*${elec_cond_mouse}
  []
  [mass_coef_air]
    type = ParsedFunction
    expression = -${epsilon0}*${angfreq}^2
  []
  [loss_coef_air]
    type = ParsedFunction
    expression = ${angfreq}*${elec_cond_air}
  []
[]

[FunctorMaterials]
  [Mouse]
    type = MFEMGenericFunctorMaterial
    prop_names = 'massCoef lossCoef MagReluctivity'
    prop_values = 'mass_coef_mouse loss_coef_mouse ${magnetic_reluctivity}'
    block = 1
  []
  [Air]
    type = MFEMGenericFunctorMaterial
    prop_names = 'massCoef lossCoef MagReluctivity'
    prop_values = 'mass_coef_air loss_coef_air ${magnetic_reluctivity}'
    block = 2
  []
[]

[BCs]
  [tangential_E]
    type = MFEMComplexVectorTangentialDirichletBC
    variable = E
    boundary = '2 3 4'
  []
  [WaveguidePortIn]
    type = MFEMRWTE10IntegratedBC
    variable = E
    boundary = '5'
    input_port = true
    port_length_vector = "24.76e-2 0.0 0.0"
    port_width_vector = "0.0 12.38e-2 0.0"
    frequency = ${freq}
    epsilon = ${epsilon0}
    mu = ${mu0}
  []
  [WaveguidePortOut]
    type = MFEMRWTE10IntegratedBC
    variable = E
    boundary = '6'
    input_port = false
    port_length_vector = "24.76e-2 0.0 0.0"
    port_width_vector = "0.0 12.38e-2 0.0"
    frequency = ${freq}
    epsilon = ${epsilon0}
    mu = ${mu0}
  []
[]

[Kernels]
  [curlcurl]
    type = MFEMComplexKernel
    variable = E
    [RealComponent]
      type = MFEMCurlCurlKernel
      coefficient = MagReluctivity
    []
  []
  [mass_loss]
    type = MFEMComplexKernel
    variable = E
    [RealComponent]
      type = MFEMVectorFEMassKernel
      coefficient = massCoef
    []
    [ImagComponent]
      type = MFEMVectorFEMassKernel
      coefficient = lossCoef
    []
  []
[]

[Solver]
  type = MFEMMUMPS
[]

[Executioner]
  type = MFEMSteady
  assembly_level = legacy
[]

[Postprocessors]
  [ObstructionAbsorption]
    type = MFEMComplexVectorPeriodAveragedPostprocessor
    coefficient = ${elec_cond_mouse}
    dual_variable = E
    primal_variable = E
    block = 1
  []
[]


[Outputs]
  [ReportedPostprocessors]
    type = CSV
    file_base = OutputData/ComplexWaveguide
  []
[]

(test/tests/mfem/kernels/darcy.i)

[Mesh]
  type = MFEMMesh
  file = ../mesh/star.mesh
  uniform_refine = 2
[]

[Problem]
  type = MFEMProblem
[]

[Functions]
  [exact_velocity]
    type = ParsedVectorFunction
    expression_x = '-exp(x) * sin(y)'
    expression_y = '-exp(x) * cos(y)'
  []
  [exact_pressure]
    type = ParsedFunction
    expression = 'exp(x) * sin(y)'
  []
  [exact_pressure_rhs]
    type = ParsedFunction
    expression = '-exp(x) * sin(y)'
  []
[]

[FESpaces]
  [HDivFESpace]
    type = MFEMVectorFESpace
    fec_type = RT
    fec_order = SECOND
  []
  [L2FESpace]
    type = MFEMScalarFESpace
    fec_type = L2
    fec_order = SECOND
    basis = GaussLegendre
  []
[]

[Variables]
  [velocity]
    type = MFEMVariable
    fespace = HDivFESpace
  []
  [pressure]
    type = MFEMVariable
    fespace = L2FESpace
  []
[]

[BCs]
  [flux_boundaries]
    type = MFEMVectorFEBoundaryFluxIntegratedBC
    variable = velocity
    coefficient = exact_pressure_rhs
  []
[]

[Kernels]
  [VelocityMass]
    type = MFEMVectorFEMassKernel
    variable = velocity
  []
  [PressureGrad]
    type = MFEMVectorFEDivergenceKernel
    trial_variable = pressure
    variable = velocity
    coefficient = -1
    transpose = true
  []
  [VelocityDiv]
    type = MFEMVectorFEDivergenceKernel
    trial_variable = velocity
    variable = pressure
    coefficient = -1
  []
[]

[Solver]
  type = MFEMMUMPS
[]

[Executioner]
  type = MFEMSteady
  device = cpu
[]

[Postprocessors]
  [velocity_error]
    type = MFEMVectorL2Error
    variable = velocity
    function = exact_velocity
  []
  [pressure_error]
    type = MFEML2Error
    variable = pressure
    function = exact_pressure
  []
[]

[Outputs]
  [ParaViewDataCollection]
    type = MFEMParaViewDataCollection
    file_base = OutputData/Darcy
    vtk_format = ASCII
  []
  [DarcyErrorCSV]
    type = CSV
    file_base = OutputData/Darcy
  []
[]

(test/tests/mfem/transfers/mfem_parent_mfem_sub/mfem_sub_vector.i)

[Mesh]
  type = MFEMMesh
  file = ../../mesh/cube.e
[]

[Problem]
  type = MFEMProblem
  solve = false
[]

[FESpaces]
  [H1FESpace]
    type = MFEMVectorFESpace
    fec_type = H1
    fec_order = FIRST
  []
  [HCurlFESpace]
    type = MFEMVectorFESpace
    fec_type = ND
    fec_order = FIRST
  []
  [HDivFESpace]
    type = MFEMVectorFESpace
    fec_type = RT
    fec_order = CONSTANT
  []
  [L2FESpace]
    type = MFEMVectorFESpace
    fec_type = L2
    fec_order = CONSTANT
  []
[]

[Variables]
  [mfem_sub_h1_vector_var]
    type = MFEMVariable
    fespace = H1FESpace
  []
  [mfem_sub_hcurl_vector_var]
    type = MFEMVariable
    fespace = HCurlFESpace
  []
  [mfem_sub_hdiv_vector_var]
    type = MFEMVariable
    fespace = HDivFESpace
  []
  [mfem_sub_l2_vector_var]
    type = MFEMVariable
    fespace = L2FESpace
  []
[]

[Functions]
  [vector_field]
    type = ParsedVectorFunction
    expression_x = 'sin(pi * y)'
    expression_y = 'sin(pi * z)'
    expression_z = 'sin(pi * x)'
  []
[]

[ICs]
  [h1_vector_ic]
    type = MFEMVectorIC
    variable = mfem_sub_h1_vector_var
    vector_coefficient = vector_field
  []
  [hcurl_vector_ic]
    type = MFEMVectorIC
    variable = mfem_sub_hcurl_vector_var
    vector_coefficient = vector_field
  []
  [hdiv_vector_ic]
    type = MFEMVectorIC
    variable = mfem_sub_hdiv_vector_var
    vector_coefficient = vector_field
  []
  [l2_vector_ic]
    type = MFEMVectorIC
    variable = mfem_sub_l2_vector_var
    vector_coefficient = vector_field
  []
[]

[Executioner]
  type = Steady
[]

(test/tests/mfem/kernels/diffusion_amr.i)

[Mesh]
  type = MFEMMesh
  file = ../mesh/square.msh
  nonconforming = true
[]

[Adaptivity]
  [Indicators]
    [l2zz]
      type = MFEML2ZienkiewiczZhuIndicator
      variable = concentration
      kernel = diff
    []
  []
  [Markers]
    [ref]
      type = MFEMRefinementMarker
      threshold = 0.7
      indicator = l2zz
      max_h_level = 1
    []
  []
[]

[Problem]
  type = MFEMProblem
[]

[FESpaces]
  [H1FESpace]
    type = MFEMScalarFESpace
    fec_type = H1
    fec_order = FIRST
  []
  [HCurlFESpace]
    type = MFEMVectorFESpace
    fec_type = ND
    fec_order = FIRST
  []
[]

[Variables]
  [concentration]
    type = MFEMVariable
    fespace = H1FESpace
  []
[]

[AuxVariables]
  [concentration_gradient]
    type = MFEMVariable
    fespace = HCurlFESpace
  []
[]

[AuxKernels]
  [grad]
    type = MFEMGradAux
    variable = concentration_gradient
    source = concentration
    execute_on = TIMESTEP_END
  []
[]

[BCs]
  [top]
    type = MFEMScalarDirichletBC
    variable = concentration
    boundary = 4
    coefficient = 1
  []
  [bottom]
    type = MFEMScalarDirichletBC
    variable = concentration
    boundary = 2
  []
[]

[Functions]
  [D]
    type = ParsedFunction
    expression = 1+1/(1+exp(20*y-10))
  []
  [solution]
    type = ParsedFunction
    expression = (20*y+log(exp(20*y)+2*exp(10))-log(1+2*exp(10)))/(30+log(2+exp(10))-log(1+2*exp(10)))
  []
[]

[Kernels]
  [diff]
    type = MFEMDiffusionKernel
    variable = concentration
    coefficient = D
  []
[]

[Preconditioner]
  [boomeramg]
    type = MFEMHypreBoomerAMG
  []
[]

[Solver]
  type = MFEMHypreGMRES
  preconditioner = boomeramg
  l_tol = 1e-16
[]

[Executioner]
  type = MFEMSteady
  device = cpu
[]

[Postprocessors]
  [error]
    type = MFEML2Error
    variable = concentration
    function = solution
  []
[]

[Outputs]
  csv = true
  file_base = OutputData/DiffusionHRefinement
[]

(test/tests/mfem/vectorpostprocessors/point_value_sampler/point_value_sampler_curlcurl.i)

# Definite Maxwell problem solved with Nedelec elements of the first kind
# based on MFEM Example 3. Sampled with MFEMPointValueSampler.

[Mesh]
  type = MFEMMesh
  file = ../../mesh/small_fichera.mesh
[]

[Problem]
  type = MFEMProblem
[]

[FESpaces]
  [HCurlFESpace]
    type = MFEMVectorFESpace
    fec_type = ND
    fec_order = FIRST
  []
  [HDivFESpace]
    type = MFEMVectorFESpace
    fec_type = RT
    fec_order = CONSTANT
  []
[]

[Variables]
  [e_field]
    type = MFEMVariable
    fespace = HCurlFESpace
  []
[]

[AuxVariables]
  [db_dt_field]
    type = MFEMVariable
    fespace = HDivFESpace
  []
[]

[AuxKernels]
  [curl]
    type = MFEMCurlAux
    variable = db_dt_field
    source = e_field
    scale_factor = -1.0
    execute_on = TIMESTEP_END
  []
[]

[Functions]
  [exact_e_field]
    type = ParsedVectorFunction
    expression_x = 'sin(kappa * y)'
    expression_y = 'sin(kappa * z)'
    expression_z = 'sin(kappa * x)'

    symbol_names = kappa
    symbol_values = 3.1415926535
  []

  [forcing_field]
    type = ParsedVectorFunction
    expression_x = '(1. + kappa * kappa) * sin(kappa * y)'
    expression_y = '(1. + kappa * kappa) * sin(kappa * z)'
    expression_z = '(1. + kappa * kappa) * sin(kappa * x)'

    symbol_names = kappa
    symbol_values = 3.1415926535
  []
[]

[BCs]
  [tangential_E_bdr]
    type = MFEMVectorTangentialDirichletBC
    variable = e_field
    vector_coefficient = exact_e_field
  []
[]

[Kernels]
  [curlcurl]
    type = MFEMCurlCurlKernel
    variable = e_field
  []
  [mass]
    type = MFEMVectorFEMassKernel
    variable = e_field
  []
  [source]
    type = MFEMVectorFEDomainLFKernel
    variable = e_field
    vector_coefficient = forcing_field
  []
[]

[Preconditioner]
  [ams]
    type = MFEMHypreAMS
    fespace = HCurlFESpace
  []
[]

[Solver]
  type = MFEMHypreGMRES
  preconditioner = ams
  l_tol = 1e-6
[]

[VectorPostprocessors]
  [point_sample]
    type = MFEMPointValueSampler
    variable = 'e_field'
    points = '1 1 -0.5  1 1 0.5'
  []
[]


[Executioner]
  type = MFEMSteady
  device = cpu
[]

[Outputs]
  execute_on = 'timestep_end'
  csv = true
[]

(test/tests/mfem/kernels/mixed_heattransfer.i)

# Mixed heat transfer problem.
# Based on Firedrake Irksome demo_mixed_heat example:
# https://www.firedrakeproject.org/Irksome/demos/demo_mixed_heat.py.html

[Mesh]
  type = MFEMMesh
  file = ../mesh/square.e
[]

[Problem]
  type = MFEMProblem
[]

[FESpaces]
  [HDivFESpace]
    type = MFEMVectorFESpace
    fec_type = RT
    fec_order = FIRST
  []
  [L2FESpace]
    type = MFEMScalarFESpace
    fec_type = L2
    fec_order = FIRST
  []
[]

[Variables]
  [time_integrated_heat_flux]
    type = MFEMVariable
    fespace = HDivFESpace
    time_derivative = heat_flux
  []
  [temperature]
    type = MFEMVariable
    fespace = L2FESpace
  []
[]

[Kernels]
  [dT_dt,T']
    type = MFEMTimeDerivativeMassKernel
    variable = temperature
  []
  [divh,T']
    type = MFEMVectorFEDivergenceKernel
    trial_variable = heat_flux
    variable = temperature
  []
  [h,h']
    type = MFEMTimeDerivativeVectorFEMassKernel
    variable = time_integrated_heat_flux
  []
  [-T,div.h']
    type = MFEMVectorFEDivergenceKernel
    trial_variable = temperature
    variable = time_integrated_heat_flux
    coefficient = -1.0
    transpose = true
  []
[]

[BCs]
  [gamma_T_right]
    type = MFEMVectorFEBoundaryFluxIntegratedBC
    variable = time_integrated_heat_flux
    coefficient = 0.0
    boundary = 2
  []
  [gamma_T_left]
    type = MFEMVectorFEBoundaryFluxIntegratedBC
    variable = time_integrated_heat_flux
    coefficient = -1.0
    boundary = 4
  []
  [gamma_h_topbottom]
    type = MFEMVectorNormalDirichletBC
    variable = time_integrated_heat_flux
    vector_coefficient = '0.0 0.0'
    boundary = '1 3'
  []
[]

[Solver]
  type = MFEMSuperLU
[]

[Executioner]
  type = MFEMTransient
  device = cpu
  assembly_level = legacy
  dt = 0.03
  start_time = 0.0
  end_time = 0.09
[]

[Outputs]
  [ParaViewDataCollection]
    type = MFEMParaViewDataCollection
    file_base = OutputData/MixedHeatTransfer
    vtk_format = ASCII
  []
[]

(test/tests/mfem/kernels/linearelasticity.i)

[Mesh]
  type = MFEMMesh
  file = ../mesh/beam-tet.mesh
  dim = 3
  uniform_refine = 2
  displacement = "displacement"
[]

[Problem]
  type = MFEMProblem
[]

[FESpaces]
  [H1FESpace]
    type = MFEMVectorFESpace
    fec_type = H1
    fec_order = FIRST
    range_dim = 3
    ordering = "vdim"
  []
[]

[Variables]
  [displacement]
    type = MFEMVariable
    fespace = H1FESpace
  []
[]

[BCs]
  [dirichlet]
    type = MFEMVectorDirichletBC
    variable = displacement
    boundary = '1'
  []
  [pull_down]
    type = MFEMVectorBoundaryIntegratedBC
    variable = displacement
    boundary = '2'
    vector_coefficient = '0.0 0.0 -0.01'
  []
[]

[FunctorMaterials]
  [Rigidium]
    type = MFEMGenericFunctorMaterial
    prop_names = 'lambda mu'
    prop_values = '50.0 50.0'
    block = 1
  []
  [Bendium]
    type = MFEMGenericFunctorMaterial
    prop_names = 'lambda mu'
    prop_values = '1.0 1.0'
    block = 2
  []
[]

[Kernels]
  [diff]
    type = MFEMLinearElasticityKernel
    variable = displacement
    lambda = lambda
    mu = mu
  []
[]

[Preconditioner]
  [boomeramg]
    type = MFEMHypreBoomerAMG
    l_max_its = 500
    l_tol = 1e-8
    print_level = 2
  []
[]

[Solver]
  type = MFEMHyprePCG
  #preconditioner = boomeramg
  l_max_its = 5000
  l_tol = 1e-8
  l_abs_tol = 0.0
  print_level = 2
[]

[Executioner]
  type = MFEMSteady
  device = "cpu"
[]

[Outputs]
  [ParaViewDataCollection]
    type = MFEMParaViewDataCollection
    file_base = OutputData/LinearElasticity
    vtk_format = ASCII
  []
[]

(test/tests/mfem/submeshes/magnetostatic.i)

# Definite Maxwell problem solved with Nedelec elements of the first kind
# based on MFEM Example 3.

[Mesh]
  type = MFEMMesh
  file = ../mesh/cylinder-hex-q2.gen
[]

[Problem]
  type = MFEMProblem
[]

[FESpaces]
  [H1FESpace]
    type = MFEMScalarFESpace
    fec_type = H1
    fec_order = FIRST
  []
  [HCurlFESpace]
    type = MFEMVectorFESpace
    fec_type = ND
    fec_order = FIRST
  []
  [HDivFESpace]
    type = MFEMVectorFESpace
    fec_type = RT
    fec_order = CONSTANT
  []
[]

[Variables]
  [a_field]
    type = MFEMVariable
    fespace = HCurlFESpace
  []
  [electric_potential]
    type = MFEMVariable
    fespace = H1FESpace
  []
[]

[AuxVariables]
  [b_field]
    type = MFEMVariable
    fespace = HDivFESpace
  []
[]

[AuxKernels]
  [curl]
    type = MFEMCurlAux
    variable = b_field
    source = a_field
    execute_on = TIMESTEP_END
  []
[]

[BCs]
  [tangential_a_bdr]
    type = MFEMVectorTangentialDirichletBC
    variable = a_field
    boundary = '1 2 3'
  []
[]

[Kernels]
  inactive = coefficient_source
  [curlcurl]
    type = MFEMCurlCurlKernel
    variable = a_field
  []
  [auxvar_source]
    type = MFEMMixedVectorGradientKernel
    trial_variable = electric_potential
    variable = a_field
    block = 1
  []
  [coefficient_source]
    type = MFEMVectorFEDomainLFKernel
    vector_coefficient = electric_potential_grad
    variable = a_field
    block = 1
  []
[]

[Preconditioner]
  [ams]
    type = MFEMHypreAMS
    fespace = HCurlFESpace
    singular = true
  []
[]

[Solver]
  type = MFEMHypreGMRES
  preconditioner = ams
  l_tol = 1e-12
[]

[Executioner]
  type = MFEMSteady
  device = cpu
[]

[MultiApps]
  [subapp]
    type = FullSolveMultiApp
    input_files = open_coil_source.i
    execute_on = INITIAL
  []
[]

[Transfers]
  [from_sub]
    type = MultiAppMFEMCopyTransfer
    source_variables = electric_potential
    variables = electric_potential
    from_multi_app = subapp
  []
[]

[Outputs]
  [ParaViewDataCollection]
    type = MFEMParaViewDataCollection
    file_base = OutputData/Magnetostatic
    vtk_format = ASCII
  []
[]

(test/tests/mfem/vectorpostprocessors/line_value_sampler/line_value_sampler_curlcurl.i)

# Definite Maxwell problem solved with Nedelec elements of the first kind
# based on MFEM Example 3. Sampled with MFEMLineValueSampler.

[Mesh]
  type = MFEMMesh
  file = ../../mesh/small_fichera.mesh
[]

[Problem]
  type = MFEMProblem
[]

[FESpaces]
  [HCurlFESpace]
    type = MFEMVectorFESpace
    fec_type = ND
    fec_order = FIRST
  []
  [HDivFESpace]
    type = MFEMVectorFESpace
    fec_type = RT
    fec_order = CONSTANT
  []
[]

[Variables]
  [e_field]
    type = MFEMVariable
    fespace = HCurlFESpace
  []
[]

[AuxVariables]
  [db_dt_field]
    type = MFEMVariable
    fespace = HDivFESpace
  []
[]

[AuxKernels]
  [curl]
    type = MFEMCurlAux
    variable = db_dt_field
    source = e_field
    scale_factor = -1.0
    execute_on = TIMESTEP_END
  []
[]

[Functions]
  [exact_e_field]
    type = ParsedVectorFunction
    expression_x = 'sin(kappa * y)'
    expression_y = 'sin(kappa * z)'
    expression_z = 'sin(kappa * x)'

    symbol_names = kappa
    symbol_values = 3.1415926535
  []

  [forcing_field]
    type = ParsedVectorFunction
    expression_x = '(1. + kappa * kappa) * sin(kappa * y)'
    expression_y = '(1. + kappa * kappa) * sin(kappa * z)'
    expression_z = '(1. + kappa * kappa) * sin(kappa * x)'

    symbol_names = kappa
    symbol_values = 3.1415926535
  []
[]

[BCs]
  [tangential_E_bdr]
    type = MFEMVectorTangentialDirichletBC
    variable = e_field
    vector_coefficient = exact_e_field
  []
[]

[Kernels]
  [curlcurl]
    type = MFEMCurlCurlKernel
    variable = e_field
  []
  [mass]
    type = MFEMVectorFEMassKernel
    variable = e_field
  []
  [source]
    type = MFEMVectorFEDomainLFKernel
    variable = e_field
    vector_coefficient = forcing_field
  []
[]

[Preconditioner]
  [ams]
    type = MFEMHypreAMS
    fespace = HCurlFESpace
  []
[]

[Solver]
  type = MFEMHypreGMRES
  preconditioner = ams
  l_tol = 1e-6
[]

[VectorPostprocessors]
  [line_sample]
    type = MFEMLineValueSampler
    variable = 'e_field'
    start_point = '1 1 -1'
    end_point = '1 1 1'
    num_points = 11
  []
[]


[Executioner]
  type = MFEMSteady
  device = cpu
[]

[Outputs]
  execute_on = 'timestep_end'
  csv = true
[]

(test/tests/mfem/kernels/maxwell_eigenproblem.i)

[Mesh]
  type = MFEMMesh
  file = ../mesh/beam-tet.mesh
  dim = 3
[]

[Problem]
  type = MFEMEigenproblem
  num_modes = 5
[]

[FESpaces]
  [HCurlFESpace]
    type = MFEMVectorFESpace
    fec_type = ND
    fec_order = FIRST
  []
[]

[Variables]
  [E]
    type = MFEMVariable
    fespace = HCurlFESpace
  []
[]

[BCs]
  [all]
    type = MFEMVectorTangentialDirichletBC
    variable = E
    vector_coefficient = '0 0 0'
  []
[]

[Kernels]
  [diff]
    type = MFEMCurlCurlKernel
    variable = E
  []
[]

[Preconditioner]
  [ams]
    type = MFEMHypreAMS
    fespace = HCurlFESpace
    print_level = 0
    singular = true
  []
[]

[Solver]
  type = MFEMHypreAME
  preconditioner = ams
  print_level = 0
  l_tol = 1e-8
  l_max_its = 100
[]

[Executioner]
  type = MFEMSteady
  device = cpu
[]

[VectorPostprocessors]
  [eigenvalues]
    type = MFEMEigenvaluesPostprocessor
  []
[]

[Outputs]
  execute_on = 'timestep_end'
  csv = true
  file_base = OutputData/MaxwellEigenproblem
[]

(test/tests/mfem/submeshes/av_magnetostatic.i)

# Magnetostatic problem solved on a closed conductor subject to
# global loop voltage constraint.

[Mesh]
  type = MFEMMesh
  file = ../mesh/embedded_concentric_torus.e
[]

[Problem]
  type = MFEMProblem
[]

[SubMeshes]
  inactive = 'fluxcut'
  [fluxcut]
    type = MFEMCutTransitionSubMesh
    cut_boundary = 'MeasurementPlane'
    block = 'TorusCore TorusSheath'
    transition_subdomain = transition_dom
    transition_subdomain_boundary = transition_bdr
    closed_subdomain = coil_dom
  []
[]

[FESpaces]
  inactive = 'FluxFESpace'
  [HCurlFESpace]
    type = MFEMVectorFESpace
    fec_type = ND
    fec_order = FIRST
  []
  [HDivFESpace]
    type = MFEMVectorFESpace
    fec_type = RT
    fec_order = CONSTANT
  []
  [FluxFESpace]
    type = MFEMVectorFESpace
    fec_type = ND
    fec_order = FIRST
    submesh = fluxcut
  []
[]

[Variables]
  [a_field]
    type = MFEMVariable
    fespace = HCurlFESpace
  []
[]

[AuxVariables]
  inactive = 'flux_e_field'
  [b_field]
    type = MFEMVariable
    fespace = HDivFESpace
  []
  [e_field]
    type = MFEMVariable
    fespace = HCurlFESpace
  []
  [flux_e_field]
    type = MFEMVariable
    fespace = FluxFESpace
  []
[]

[AuxKernels]
  [curl]
    type = MFEMCurlAux
    variable = b_field
    source = a_field
    scale_factor = 1.0
    execute_on = TIMESTEP_END
  []
[]

[Functions]
  [exact_a_field]
    type = ParsedVectorFunction
    expression_x = '0'
    expression_y = '0'
    expression_z = '0'
  []
[]

[BCs]
  [tangential_a_bdr]
    type = MFEMVectorTangentialDirichletBC
    variable = a_field
    vector_coefficient = exact_a_field
    boundary = 'Exterior'
  []
[]

[FunctorMaterials]
  inactive = 'ConductorBoundary'
  [Vacuum]
    type = MFEMGenericFunctorMaterial
    prop_names = reluctivity
    prop_values = 1.0
  []
  [Conductor]
    type = MFEMGenericFunctorMaterial
    prop_names = conductivity
    prop_values = 1.0
    block = 'TorusCore TorusSheath'
  []
  [ConductorBoundary]
    type = MFEMGenericFunctorMaterial
    prop_names = conductivity_boundary
    prop_values = 1.0
    boundary = 'MeasurementPlane'
  []
[]

[Kernels]
  [mass]
    type = MFEMVectorFEMassKernel
    variable = a_field
    coefficient = 1e-10
  []
  [curlcurl]
    type = MFEMCurlCurlKernel
    variable = a_field
    coefficient = reluctivity
  []
  [source]
    type = MFEMMixedVectorMassKernel
    variable = a_field
    trial_variable = e_field
    coefficient = conductivity
    block = 'TorusCore TorusSheath'
  []
[]

[Preconditioner]
  [ams]
    type = MFEMHypreAMS
    fespace = HCurlFESpace
  []
[]

[Solver]
  type = MFEMHyprePCG
  preconditioner = ams
  l_tol = 1e-14
  l_max_its = 1000
[]

[Executioner]
  type = MFEMSteady
  device = cpu
[]

[MultiApps]
  [coil]
    type = FullSolveMultiApp
    input_files = cut_closed_coil.i
    execute_on = INITIAL
  []
[]

[Transfers]
  inactive = 'submesh_transfer_to_fluxsurface'
  [from_coil]
    type = MultiAppMFEMCopyTransfer
    source_variables = e_field
    variables = e_field
    from_multi_app = coil
  []
  [submesh_transfer_to_fluxsurface]
    type = MFEMSubMeshTransfer
    from_variable = e_field
    to_variable = flux_e_field
    execute_on = TIMESTEP_END
  []
[]

[Postprocessors]
  inactive = 'CoilCurrent'
  [CoilPower]
    type = MFEMVectorFEInnerProductIntegralPostprocessor
    coefficient = conductivity
    dual_variable = e_field
    primal_variable = e_field
    block = 'TorusCore TorusSheath'
  []
  [CoilCurrent]
    type = MFEMVectorBoundaryFluxIntegralPostprocessor
    coefficient = conductivity_boundary
    variable = flux_e_field
    boundary = 'MeasurementPlane'
  []
[]

[Outputs]
  [ParaViewDataCollection]
    type = MFEMParaViewDataCollection
    file_base = OutputData/MagnetostaticClosedCoil
    vtk_format = ASCII
  []
  [ReportedPostprocessors]
    type = CSV
    file_base = OutputData/AVMagnetostaticClosedCoilCSV
  []
[]

(test/tests/mfem/vectorpostprocessors/line_value_sampler/line_value_sampler_diffusion.i)

# MFEM diffusion problem sampled with MFEMLineValueSampler.

[Mesh]
  type = MFEMMesh
  file = ../../mesh/mug.e
[]

[Problem]
  type = MFEMProblem
[]

[FESpaces]
  [H1FESpace]
    type = MFEMScalarFESpace
    fec_type = H1
    fec_order = FIRST
  []
  [HCurlFESpace]
    type = MFEMVectorFESpace
    fec_type = ND
    fec_order = FIRST
  []
[]

[Variables]
  [concentration]
    type = MFEMVariable
    fespace = H1FESpace
  []
[]

[AuxVariables]
  [concentration_gradient]
    type = MFEMVariable
    fespace = HCurlFESpace
  []
[]

[AuxKernels]
  [grad]
    type = MFEMGradAux
    variable = concentration_gradient
    source = concentration
    execute_on = TIMESTEP_END
  []
[]

[ICs]
  [diffused_ic]
    type = MFEMScalarIC
    coefficient = one
    variable = concentration
  []
[]

[Functions]
  [one]
    type = ParsedFunction
    expression = 1.0
  []
[]

[BCs]
  [bottom]
    type = MFEMScalarDirichletBC
    variable = concentration
    boundary = 'bottom'
    coefficient = 1.0
  []
  [top]
    type = MFEMScalarDirichletBC
    variable = concentration
    boundary = 'top'
  []
[]

[FunctorMaterials]
  [Substance]
    type = MFEMGenericFunctorMaterial
    prop_names = diffusivity
    prop_values = 1.0
    block = 'the_domain'
  []
[]

[Kernels]
  [diff]
    type = MFEMDiffusionKernel
    variable = concentration
    coefficient = diffusivity
  []
[]

[Preconditioner]
  [boomeramg]
    type = MFEMHypreBoomerAMG
  []
  [jacobi]
    type = MFEMOperatorJacobiSmoother
  []
[]

[Solver]
  type = MFEMHypreGMRES
  preconditioner = boomeramg
  l_tol = 1e-16
  l_max_its = 1000
[]

[VectorPostprocessors]
  [line_sample]
    type = MFEMLineValueSampler
    variable = 'concentration'
    start_point = '2.125 0 -2.375'
    end_point = '2.125 0 2.625'
    num_points = 11
  []
[]

[Executioner]
  type = MFEMSteady
  device = cpu
[]

[Outputs]
  execute_on = 'timestep_end'
  csv = true
[]

(test/tests/mfem/submeshes/hphi_magnetostatic.i)

# Solve for the magnetic field around a closed conductor subject to
# global current constraint.

initial_vacuum_domains = 'Exterior'
vacuum_cut_surface = 'Cut'
conductor_current = 1.0
vacuum_permeability = 1.0

[Problem]
  type = MFEMProblem
[]

[Mesh]
    type = MFEMMesh
    file = ../mesh/split_embedded_concentric_torus.e
[]

[FunctorMaterials]
  [Conductor]
    type = MFEMGenericFunctorMaterial
    prop_names = permeability
    prop_values = ${vacuum_permeability}
  []
[]

[ICs]
  [vacuum_cut_potential_ic]
    type = MFEMScalarBoundaryIC
    variable = vacuum_cut_potential
    boundary = ${vacuum_cut_surface}
    coefficient = ${conductor_current}
  []
[]

[SubMeshes]
  [cut]
    type = MFEMCutTransitionSubMesh
    cut_boundary = ${vacuum_cut_surface}
    block = ${initial_vacuum_domains}
    transition_subdomain = transition_dom
    transition_subdomain_boundary = transition_bdr
    closed_subdomain = vacuum_dom
  []
  [vacuum]
    type = MFEMDomainSubMesh
    block = vacuum_dom
  []
[]

[FESpaces]
  [VacuumH1FESpace]
    type = MFEMScalarFESpace
    fec_type = H1
    fec_order = FIRST
    submesh = vacuum
  []
  [VacuumHCurlFESpace]
    type = MFEMVectorFESpace
    fec_type = ND
    fec_order = FIRST
    submesh = vacuum
  []
  [TransitionH1FESpace]
    type = MFEMScalarFESpace
    fec_type = H1
    fec_order = FIRST
    submesh = cut
  []
  [TransitionHCurlFESpace]
    type = MFEMVectorFESpace
    fec_type = ND
    fec_order = FIRST
    submesh = cut
  []
  [HCurlFESpace]
    type = MFEMVectorFESpace
    fec_type = ND
    fec_order = FIRST
  []
[]

[Variables]
  [vacuum_magnetic_potential]
    type = MFEMVariable
    fespace = VacuumH1FESpace
  []
[]

[AuxVariables]
  [vacuum_cut_potential]
    type = MFEMVariable
    fespace = VacuumH1FESpace
  []
  [transition_cut_potential]
    type = MFEMVariable
    fespace = TransitionH1FESpace
  []
  [transition_cut_function_field]
    type = MFEMVariable
    fespace = TransitionHCurlFESpace
  []
  [background_h_field]
    type = MFEMVariable
    fespace = VacuumHCurlFESpace
  []
  [cut_function_field]
    type = MFEMVariable
    fespace = VacuumHCurlFESpace
  []
  [vacuum_h_field]
    type = MFEMVariable
    fespace = VacuumHCurlFESpace
  []
  [h_field]
    type = MFEMVariable
    fespace = HCurlFESpace
  []
[]

[AuxKernels]
  [update_background_h_field]
    type = MFEMGradAux
    variable = background_h_field
    source = vacuum_magnetic_potential
    scale_factor = -1.0
    execute_on = TIMESTEP_END
  []
  [update_transition_cut_function_field]
    type = MFEMGradAux
    variable = transition_cut_function_field
    source = transition_cut_potential
    scale_factor = -1.0
    execute_on = TIMESTEP_END
  []
  [update_total_h_field]
    type = MFEMSumAux
    variable = vacuum_h_field
    source_variables = 'background_h_field cut_function_field'
    execute_on = TIMESTEP_END
  []
[]

[BCs]
  # Set zero of magnetic potential on symmetry plane
  [Exterior]
    type = MFEMScalarDirichletBC
    variable = vacuum_magnetic_potential
    boundary = 'Cut'
    coefficient = 0.0
  []
[]

[Kernels]
  [diff]
    type = MFEMDiffusionKernel
    variable = vacuum_magnetic_potential
    coefficient = permeability
  []
  [source]
    type = MFEMMixedGradGradKernel
    trial_variable = vacuum_cut_potential
    variable = vacuum_magnetic_potential
    coefficient = permeability
    block = 'transition_dom'
  []
[]

[Preconditioner]
  [boomeramg]
    type = MFEMHypreBoomerAMG
  []
[]

[Solver]
  type = MFEMHypreGMRES
  preconditioner = boomeramg
  l_tol = 1e-8
  l_max_its = 100
[]

[Executioner]
  type = MFEMSteady
[]

[Transfers]
  [submesh_transfer_to_transition]
    type = MFEMSubMeshTransfer
    from_variable = vacuum_cut_potential
    to_variable = transition_cut_potential
    execute_on = TIMESTEP_END
  []
  [submesh_transfer_from_transition]
    type = MFEMSubMeshTransfer
    from_variable = transition_cut_function_field
    to_variable = cut_function_field
    execute_on = TIMESTEP_END
  []
  [submesh_transfer_from_vacuum]
    type = MFEMSubMeshTransfer
    from_variable = vacuum_h_field
    to_variable = h_field
    execute_on = TIMESTEP_END
  []
[]

[Postprocessors]
  [MagneticEnergy]
    type = MFEMVectorFEInnerProductIntegralPostprocessor
    coefficient = ${fparse 0.5*vacuum_permeability}
    dual_variable = vacuum_h_field
    primal_variable = vacuum_h_field
    block = 'Exterior'
  []
[]

[Outputs]
  [ReportedPostprocessors]
    type = CSV
    file_base = OutputData/HPhiMagnetostaticClosedCoilCSV
  []
  [VacuumParaViewDataCollection]
    type = MFEMParaViewDataCollection
    file_base = OutputData/HPhiMagnetostaticClosedCoil
    vtk_format = ASCII
    submesh = vacuum
  []
[]

(test/tests/mfem/submeshes/hphi_magnetodynamic.i)

# Solve for the magnetic field around a closed conductor subject to
# global current constraint.

conductor_domains = 'TorusCore TorusSheath'
conductor_resistivity = 1.0
vacuum_permeability = 1.0

[Problem]
  type = MFEMProblem
[]

[Mesh]
    type = MFEMMesh
    file = ../mesh/split_embedded_concentric_torus.e
[]

[FunctorMaterials]
  [Conductor]
    type = MFEMGenericFunctorMaterial
    prop_names = 'resistivity'
    prop_values = ${conductor_resistivity}
    block = ${conductor_domains}
  []
  [Vacuum]
    type = MFEMGenericFunctorMaterial
    prop_names = 'permeability'
    prop_values = '${vacuum_permeability}'
  []
[]

[SubMeshes]
  [conductor]
    type = MFEMDomainSubMesh
    block = ${conductor_domains}
    submesh_boundary = conductor_surface
  []
[]

[FESpaces]
  [H1FESpace]
    type = MFEMScalarFESpace
    fec_type = H1
    fec_order = FIRST
  []
  [HCurlFESpace]
    type = MFEMVectorFESpace
    fec_type = ND
    fec_order = FIRST
  []
  [HDivFESpace]
    type = MFEMVectorFESpace
    fec_type = RT
    fec_order = CONSTANT
  []
  [CoilHCurlFESpace]
    type = MFEMVectorFESpace
    fec_type = ND
    fec_order = FIRST
    submesh = conductor
  []
[]

[Variables]
  [coil_induced_h_field]
    type = MFEMVariable
    fespace = CoilHCurlFESpace
  []
[]

[AuxVariables]
  [h_field]
    type = MFEMVariable
    fespace = HCurlFESpace
  []
  [coil_external_h_field]
    type = MFEMVariable
    fespace = CoilHCurlFESpace
  []
  [j_field]
    type = MFEMVariable
    fespace = HDivFESpace
  []
[]

[AuxKernels]
  [update_j_field]
    type = MFEMCurlAux
    variable = j_field
    source = h_field
    scale_factor = 1.0
    execute_on = TIMESTEP_END
  []
[]

[BCs]
  [conductor_bdr]
    type = MFEMVectorTangentialDirichletBC
    variable = coil_induced_h_field
    vector_coefficient = coil_external_h_field
    boundary = conductor_surface
  []
[]

[Kernels]
  [dBdt]
    type = MFEMTimeDerivativeVectorFEMassKernel
    variable = coil_induced_h_field
    coefficient = permeability
  []
  [curlE]
    type = MFEMCurlCurlKernel
    variable = coil_induced_h_field
    coefficient = resistivity
  []
[]

[Preconditioner]
  [ams]
    type = MFEMHypreAMS
    fespace = CoilHCurlFESpace
  []
[]

[Solver]
  type = MFEMHyprePCG
  preconditioner = ams
  l_tol = 1e-9
  l_max_its = 100
[]

[Executioner]
  type = MFEMTransient
  dt = 0.5
  start_time = 0.0
  end_time = 2.0
[]

[MultiApps]
  [hphi_magnetostatic]
    type = FullSolveMultiApp
    input_files = hphi_magnetostatic.i
    execute_on = INITIAL
  []
[]

[Transfers]
  [from_external_field]
    type = MultiAppMFEMCopyTransfer
    source_variables = h_field
    variables = h_field
    from_multi_app = hphi_magnetostatic
  []
  [submesh_transfer_to_coil]
    type = MFEMSubMeshTransfer
    from_variable = h_field
    to_variable = coil_external_h_field
    execute_on = TIMESTEP_BEGIN
  []
  [submesh_transfer_from_coil]
    type = MFEMSubMeshTransfer
    from_variable = coil_induced_h_field
    to_variable = h_field
    execute_on = TIMESTEP_END
  []
[]

[Postprocessors]
  [CoilPower]
    type = MFEMVectorFEInnerProductIntegralPostprocessor
    coefficient = resistivity
    dual_variable = j_field
    primal_variable = j_field
    block = 'TorusCore TorusSheath'
  []
[]

[Outputs]
  [ReportedPostprocessors]
    type = CSV
    file_base = OutputData/HPhiMagnetodynamicClosedCoilCSV
  []
  [VacuumParaViewDataCollection]
    type = MFEMParaViewDataCollection
    file_base = OutputData/HPhiMagnetodynamicClosedCoil
    vtk_format = ASCII
  []
[]

(test/tests/mfem/kernels/curlcurl.i)

# Definite Maxwell problem solved with Nedelec elements of the first kind
# based on MFEM Example 3.

[Mesh]
  type = MFEMMesh
  file = ../mesh/small_fichera.mesh
  dim = 3
[]

[Problem]
  type = MFEMProblem
[]

[FESpaces]
  inactive = "L2FESpace"
  [HCurlFESpace]
    type = MFEMVectorFESpace
    fec_type = ND
    fec_order = FIRST
  []
  [HDivFESpace]
    type = MFEMVectorFESpace
    fec_type = RT
    fec_order = CONSTANT
  []
  [L2FESpace]
    type = MFEMScalarFESpace
    fec_type = L2
    fec_order = CONSTANT
  []
[]

[Variables]
  [e_field]
    type = MFEMVariable
    fespace = HCurlFESpace
  []
[]

[AuxVariables]
  inactive = "joule_heating"
  [db_dt_field]
    type = MFEMVariable
    fespace = HDivFESpace
  []
  [joule_heating]
    type = MFEMVariable
    fespace = L2FESpace
  []
[]

[AuxKernels]
  inactive = "joule_Q_aux"
  [curl]
    type = MFEMCurlAux
    variable = db_dt_field
    source = e_field
    scale_factor = -1.0
    execute_on = TIMESTEP_END
  []
  [joule_Q_aux]
    type = MFEMInnerProductAux
    variable = joule_heating
    first_source_vec = e_field
    second_source_vec = e_field
    execute_on = TIMESTEP_END
  []
[]

[Functions]
  [exact_e_field]
    type = ParsedVectorFunction
    expression_x = 'sin(kappa * y)'
    expression_y = 'sin(kappa * z)'
    expression_z = 'sin(kappa * x)'

    symbol_names = kappa
    symbol_values = 3.1415926535
  []

  [forcing_field]
    type = ParsedVectorFunction
    expression_x = '(1. + kappa * kappa) * sin(kappa * y)'
    expression_y = '(1. + kappa * kappa) * sin(kappa * z)'
    expression_z = '(1. + kappa * kappa) * sin(kappa * x)'

    symbol_names = kappa
    symbol_values = 3.1415926535
  []
[]

[BCs]
  [tangential_E_bdr]
    type = MFEMVectorTangentialDirichletBC
    variable = e_field
    vector_coefficient = exact_e_field
  []
[]

[Kernels]
  [curlcurl]
    type = MFEMCurlCurlKernel
    variable = e_field
  []
  [mass]
    type = MFEMVectorFEMassKernel
    variable = e_field
  []
  [source]
    type = MFEMVectorFEDomainLFKernel
    variable = e_field
    vector_coefficient = forcing_field
  []
[]

[Preconditioner]
  [ams]
    type = MFEMHypreAMS
    fespace = HCurlFESpace
  []
[]

[Solver]
  type = MFEMHypreGMRES
  preconditioner = ams
  l_tol = 1e-12
[]

[Executioner]
  type = MFEMSteady
  device = cpu
[]

[Outputs]
  [ParaViewDataCollection]
    type = MFEMParaViewDataCollection
    file_base = OutputData/CurlCurl
    vtk_format = ASCII
  []
[]

(test/tests/mfem/kernels/diffusion.i)

[Mesh]
  type = MFEMMesh
  file = ../mesh/mug.e
  dim = 3
[]

[Problem]
  type = MFEMProblem
[]

[FESpaces]
  [H1FESpace]
    type = MFEMScalarFESpace
    fec_type = H1
    fec_order = FIRST
  []
  [HCurlFESpace]
    type = MFEMVectorFESpace
    fec_type = ND
    fec_order = FIRST
  []
[]

[Variables]
  [concentration]
    type = MFEMVariable
    fespace = H1FESpace
  []
[]

[AuxVariables]
  [concentration_gradient]
    type = MFEMVariable
    fespace = HCurlFESpace
  []
[]

[AuxKernels]
  [grad]
    type = MFEMGradAux
    variable = concentration_gradient
    source = concentration
    execute_on = TIMESTEP_END
  []
[]

[BCs]
  [bottom]
    type = MFEMScalarDirichletBC
    variable = concentration
    boundary = 'bottom'
    coefficient = 1.0
  []
  [top]
    type = MFEMScalarDirichletBC
    variable = concentration
    boundary = 'top'
  []
[]

[Kernels]
  [diff]
    type = MFEMDiffusionKernel
    variable = concentration
  []
[]

[Preconditioner]
  [boomeramg]
    type = MFEMHypreBoomerAMG
  []
  [jacobi]
    type = MFEMOperatorJacobiSmoother
  []
[]

[Solver]
  type = MFEMHypreGMRES
  preconditioner = boomeramg
  l_tol = 1e-16
  l_max_its = 1000
[]

[Executioner]
  type = MFEMSteady
  device = cpu
[]

[Outputs]
  active = ParaViewDataCollection
  [ParaViewDataCollection]
    type = MFEMParaViewDataCollection
    file_base = OutputData/Diffusion
    vtk_format = ASCII
  []
  [VisItDataCollection]
    type = MFEMVisItDataCollection
    file_base = OutputData/VisItDataCollection
  []
  [ConduitDataCollection]
    type = MFEMConduitDataCollection
    file_base = OutputData/ConduitDataCollection/Run
    protocol = conduit_bin
  []
[]

(test/tests/mfem/submeshes/cut_closed_coil.i)

# Solve for the electric field on a closed conductor subject to
# global loop voltage constraint.

initial_coil_domains = 'TorusCore TorusSheath'
coil_cut_surface = 'Cut'
coil_loop_voltage = -1.0
coil_conductivity = 1.0

[Problem]
  type = MFEMProblem
[]

[Mesh]
    type = MFEMMesh
    file = ../mesh/embedded_concentric_torus.e
[]

[FunctorMaterials]
  [Conductor]
    type = MFEMGenericFunctorMaterial
    prop_names = conductivity
    prop_values = ${coil_conductivity}
  []
[]

[ICs]
  [coil_external_potential_ic]
    type = MFEMScalarBoundaryIC
    variable = coil_external_potential
    boundary = ${coil_cut_surface}
    coefficient = ${coil_loop_voltage}
  []
[]

[SubMeshes]
  [cut]
    type = MFEMCutTransitionSubMesh
    cut_boundary = ${coil_cut_surface}
    block = ${initial_coil_domains}
    transition_subdomain = transition_dom
    transition_subdomain_boundary = transition_bdr
    closed_subdomain = coil_dom
  []
  [coil]
    type = MFEMDomainSubMesh
    block = coil_dom
  []
[]

[FESpaces]
  [H1FESpace]
    type = MFEMScalarFESpace
    fec_type = H1
    fec_order = FIRST
  []
  [HCurlFESpace]
    type = MFEMVectorFESpace
    fec_type = ND
    fec_order = FIRST
  []
  [CoilH1FESpace]
    type = MFEMScalarFESpace
    fec_type = H1
    fec_order = FIRST
    submesh = coil
  []
  [CoilHCurlFESpace]
    type = MFEMVectorFESpace
    fec_type = ND
    fec_order = FIRST
    submesh = coil
  []
  [TransitionH1FESpace]
    type = MFEMScalarFESpace
    fec_type = H1
    fec_order = FIRST
    submesh = cut
  []
  [TransitionHCurlFESpace]
    type = MFEMVectorFESpace
    fec_type = ND
    fec_order = FIRST
    submesh = cut
  []
[]

[Variables]
  [coil_induced_potential]
    type = MFEMVariable
    fespace = CoilH1FESpace
  []
[]

[AuxVariables]
  [coil_external_potential]
    type = MFEMVariable
    fespace = CoilH1FESpace
  []
  [transition_external_potential]
    type = MFEMVariable
    fespace = TransitionH1FESpace
  []
  [transition_external_e_field]
    type = MFEMVariable
    fespace = TransitionHCurlFESpace
  []
  [induced_potential]
    type = MFEMVariable
    fespace = H1FESpace
  []
  [induced_e_field]
    type = MFEMVariable
    fespace = HCurlFESpace
  []
  [external_e_field]
    type = MFEMVariable
    fespace = HCurlFESpace
  []
  [e_field]
    type = MFEMVariable
    fespace = HCurlFESpace
  []
[]

[AuxKernels]
  [update_induced_e_field]
    type = MFEMGradAux
    variable = induced_e_field
    source = induced_potential
    scale_factor = -1.0
    execute_on = TIMESTEP_END
  []
  [update_external_e_field]
    type = MFEMGradAux
    variable = transition_external_e_field
    source = transition_external_potential
    scale_factor = -1.0
    execute_on = TIMESTEP_END
  []
  [update_total_e_field]
    type = MFEMSumAux
    variable = e_field
    source_variables = 'induced_e_field external_e_field'
    execute_on = TIMESTEP_END
  []
[]

[Kernels]
  [diff]
    type = MFEMDiffusionKernel
    variable = coil_induced_potential
    coefficient = conductivity
  []
  [source]
    type = MFEMMixedGradGradKernel
    trial_variable = coil_external_potential
    variable = coil_induced_potential
    coefficient = conductivity
    block = 'transition_dom'
  []
[]

[Solver]
  type = MFEMSuperLU
[]

[Executioner]
  type = MFEMSteady
[]

[Transfers]
  [submesh_transfer_from_coil]
    type = MFEMSubMeshTransfer
    from_variable = coil_induced_potential
    to_variable = induced_potential
    execute_on = TIMESTEP_END
  []
  [submesh_transfer_to_transition]
    type = MFEMSubMeshTransfer
    from_variable = coil_external_potential
    to_variable = transition_external_potential
    execute_on = TIMESTEP_END
  []
  [submesh_transfer_from_transition]
    type = MFEMSubMeshTransfer
    from_variable = transition_external_e_field
    to_variable = external_e_field
    execute_on = TIMESTEP_END
  []
[]

[Outputs]
  [GlobalParaViewDataCollection]
    type = MFEMParaViewDataCollection
    file_base = OutputData/WholePotentialCoil
    vtk_format = ASCII
  []
  [TransitionParaViewDataCollection]
    type = MFEMParaViewDataCollection
    file_base = OutputData/CutPotentialCoil
    vtk_format = ASCII
    submesh = cut
  []
  [CoilParaViewDataCollection]
    type = MFEMParaViewDataCollection
    file_base = OutputData/Coil
    vtk_format = ASCII
    submesh = coil
  []
[]

(test/tests/mfem/auxkernels/crossproduct.i)

[Mesh]
  type = MFEMMesh
  file = ../mesh/ref-cube.mesh
[]

[Problem]
  type = MFEMProblem
  solve = false
[]

[FESpaces]
  [HCurlFESpace]
    type = MFEMVectorFESpace
    fec_type = ND
    fec_order = FIRST
  []
  [HDivFESpace]
    type = MFEMVectorFESpace
    fec_type = RT
    fec_order = CONSTANT
  []
  [L2VectorFESpace]
    type = MFEMVectorFESpace
    fec_type = L2
    fec_order = CONSTANT
  []
[]

[Variables]
  [e_field]
    type = MFEMVariable
    fespace = HCurlFESpace
  []
  [b_field]
    type = MFEMVariable
    fespace = HDivFESpace
  []
[]

[AuxVariables]
  [lorentz_force]
    type = MFEMVariable
    fespace = L2VectorFESpace
  []
[]

[Functions]
  [external_e_field]
    type = ParsedVectorFunction
    expression_x = '1'
    expression_y = '0'
    expression_z = '0'
  []
  [external_b_field]
    type = ParsedVectorFunction
    expression_x = '0'
    expression_y = '1'
    expression_z = '0'
  []
[]

[ICs]
  [e_field_ic]
    type = MFEMVectorIC
    variable = e_field
    vector_coefficient = external_e_field
  []
  [b_field_ic]
    type = MFEMVectorIC
    variable = b_field
    vector_coefficient = external_b_field
  []
[]

[AuxKernels]
  [cross]
    type = MFEMCrossProductAux
    variable = lorentz_force
    first_source_vec = e_field
    second_source_vec = b_field
    execute_on = TIMESTEP_END
  []
[]

[Executioner]
  type = MFEMSteady
  device = cpu
[]

[Outputs]
  [ParaViewDataCollection]
    type = MFEMParaViewDataCollection
    file_base = OutputData/CrossProduct
    vtk_format = ASCII
  []
[]

Overview
Example Input File Syntax
Input Parameters
Input Files