Eigenvalue

Eigenvalue solves a standard/generalized linear or nonlinear eigenvalue problem

Input Parameters

auto_initializationTrueIf true, we will set an initial eigen vector in moose, otherwise EPS solver will initial eigen vector
Default:True
C++ Type:bool
Options:
Description:If true, we will set an initial eigen vector in moose, otherwise EPS solver will initial eigen vector
contact_line_search_allowed_lambda_cuts2The number of times lambda is allowed to be cut in half in the contact line search. We recommend this number be roughly bounded by 0 <= allowed_lambda_cuts <= 3
Default:2
C++ Type:unsigned int
Options:
Description:The number of times lambda is allowed to be cut in half in the contact line search. We recommend this number be roughly bounded by 0 <= allowed_lambda_cuts <= 3
contact_line_search_ltolThe linear relative tolerance to be used while the contact state is changing between non-linear iterations. We recommend that this tolerance be looser than the standard linear tolerance
C++ Type:double
Options:
Description:The linear relative tolerance to be used while the contact state is changing between non-linear iterations. We recommend that this tolerance be looser than the standard linear tolerance
custom_abs_tol1e-50The absolute nonlinear residual to shoot for during fixed point iterations. This check is performed based on postprocessor defined by the custom_pp residual.
Default:1e-50
C++ Type:double
Options:
Description:The absolute nonlinear residual to shoot for during fixed point iterations. This check is performed based on postprocessor defined by the custom_pp residual.
custom_rel_tol1e-08The relative nonlinear residual drop to shoot for during fixed point iterations. This check is performed based on the postprocessor defined by custom_pp residual.
Default:1e-08
C++ Type:double
Options:
Description:The relative nonlinear residual drop to shoot for during fixed point iterations. This check is performed based on the postprocessor defined by custom_pp residual.
eigen_max_its10000Max Iterations for Eigen Solver
Default:10000
C++ Type:unsigned int
Options:
Description:Max Iterations for Eigen Solver
eigen_problem_typeGEN_NON_HERMITIANType of the eigenvalue problem we are solving HERMITIAN: Hermitian NON_HERMITIAN: Non-Hermitian GEN_HERMITIAN: Generalized Hermitian GEN_NON_HERMITIAN: Generalized Non-Hermitian GEN_INDEFINITE: Generalized indefinite Hermitian POS_GEN_NON_HERMITIAN: Generalized Non-Hermitian with positive (semi-)definite B SLEPC_DEFAULT: Use whatever SLEPC has by default
Default:GEN_NON_HERMITIAN
C++ Type:MooseEnum
Options:HERMITIAN, NON_HERMITIAN, GEN_HERMITIAN, GEN_NON_HERMITIAN, GEN_INDEFINITE, POS_GEN_NON_HERMITIAN, SLEPC_DEFAULT
Description:Type of the eigenvalue problem we are solving HERMITIAN: Hermitian NON_HERMITIAN: Non-Hermitian GEN_HERMITIAN: Generalized Hermitian GEN_NON_HERMITIAN: Generalized Non-Hermitian GEN_INDEFINITE: Generalized indefinite Hermitian POS_GEN_NON_HERMITIAN: Generalized Non-Hermitian with positive (semi-)definite B SLEPC_DEFAULT: Use whatever SLEPC has by default
eigen_tol0.0001Relative Tolerance for Eigen Solver
Default:0.0001
C++ Type:double
Options:
Description:Relative Tolerance for Eigen Solver
extra_power_iterations0The number of extra free power iterations
Default:0
C++ Type:unsigned int
Options:
Description:The number of extra free power iterations
fixed_point_algorithmpicardThe fixed point algorithm to converge the sequence of problems.
Default:picard
C++ Type:MooseEnum
Options:picard, secant, steffensen
Description:The fixed point algorithm to converge the sequence of problems.
free_power_iterations4The number of free power iterations
Default:4
C++ Type:unsigned int
Options:
Description:The number of free power iterations
l_abs_tol1e-50Absolute Tolerances for Linear Solver
Default:1e-50
C++ Type:double
Options:
Description:Absolute Tolerances for Linear Solver
line_searchdefaultSpecifies the line search type (Note: none = basic)
Default:default
C++ Type:MooseEnum
Options:basic, bt, contact, cp, default, l2, none, project, shell
Description:Specifies the line search type (Note: none = basic)
line_search_packagepetscThe solver package to use to conduct the line-search
Default:petsc
C++ Type:MooseEnum
Options:petsc, moose
Description:The solver package to use to conduct the line-search
matrix_freeFalseWhether or not to use a matrix free fashion to form operators. If true, shell matrices will be used and meanwhile a preconditioning matrixmay be formed as well.
Default:False
C++ Type:bool
Options:
Description:Whether or not to use a matrix free fashion to form operators. If true, shell matrices will be used and meanwhile a preconditioning matrixmay be formed as well.
mffd_typewpSpecifies the finite differencing type for Jacobian-free solve types. Note that the default is wp (for Walker and Pernice).
Default:wp
C++ Type:MooseEnum
Options:wp, ds
Description:Specifies the finite differencing type for Jacobian-free solve types. Note that the default is wp (for Walker and Pernice).
n_basis_vectors3The dimension of eigen subspaces
Default:3
C++ Type:unsigned int
Options:
Description:The dimension of eigen subspaces
n_eigen_pairs1The number of eigen pairs
Default:1
C++ Type:unsigned int
Options:
Description:The number of eigen pairs
n_max_nonlinear_pingpong100The maximum number of times the non linear residual can ping pong before requesting halting the current evaluation and requesting timestep cut
Default:100
C++ Type:unsigned int
Options:
Description:The maximum number of times the non linear residual can ping pong before requesting halting the current evaluation and requesting timestep cut
nl_abs_div_tol1e+50Nonlinear Absolute Divergence Tolerance. A negative value disables this check.
Default:1e+50
C++ Type:double
Options:
Description:Nonlinear Absolute Divergence Tolerance. A negative value disables this check.
nl_div_tol1e+10Nonlinear Relative Divergence Tolerance. A negative value disables this check.
Default:1e+10
C++ Type:double
Options:
Description:Nonlinear Relative Divergence Tolerance. A negative value disables this check.
nl_forced_its0The Number of Forced Nonlinear Iterations
Default:0
C++ Type:unsigned int
Options:
Description:The Number of Forced Nonlinear Iterations
normal_factorNormalize eigenvector to make a defined norm equal to this factor
C++ Type:double
Options:
Description:Normalize eigenvector to make a defined norm equal to this factor
normalizationPostprocessor evaluating norm of eigenvector for normalization
C++ Type:PostprocessorName
Options:
Description:Postprocessor evaluating norm of eigenvector for normalization
petsc_optionsSingleton PETSc options
C++ Type:MultiMooseEnum
Options:-dm_moose_print_embedding, -dm_view, -ksp_converged_reason, -ksp_gmres_modifiedgramschmidt, -ksp_monitor, -ksp_monitor_snes_lg-snes_ksp_ew, -ksp_snes_ew, -snes_converged_reason, -snes_ksp, -snes_ksp_ew, -snes_linesearch_monitor, -snes_mf, -snes_mf_operator, -snes_monitor, -snes_test_display, -snes_view
Description:Singleton PETSc options
petsc_options_inameNames of PETSc name/value pairs
C++ Type:MultiMooseEnum
Options:-ksp_atol, -ksp_gmres_restart, -ksp_max_it, -ksp_pc_side, -ksp_rtol, -ksp_type, -mat_fd_coloring_err, -mat_fd_type, -mat_mffd_type, -pc_asm_overlap, -pc_factor_levels, -pc_factor_mat_ordering_type, -pc_hypre_boomeramg_grid_sweeps_all, -pc_hypre_boomeramg_max_iter, -pc_hypre_boomeramg_strong_threshold, -pc_hypre_type, -pc_type, -snes_atol, -snes_linesearch_type, -snes_ls, -snes_max_it, -snes_rtol, -snes_divergence_tolerance, -snes_type, -sub_ksp_type, -sub_pc_type
Description:Names of PETSc name/value pairs
petsc_options_valueValues of PETSc name/value pairs (must correspond with "petsc_options_iname"
C++ Type:std::vector<std::string>
Options:
Description:Values of PETSc name/value pairs (must correspond with "petsc_options_iname"
precond_matrix_freeFalseWhether or not to use a matrix free fashion for forming the preconditioning matrix. If true, a shell matrix will be used for preconditioner.
Default:False
C++ Type:bool
Options:
Description:Whether or not to use a matrix free fashion for forming the preconditioning matrix. If true, a shell matrix will be used for preconditioner.
precond_matrix_includes_eigenFalseWhether or not to include eigen kernels in the preconditioning matrix. If true, the preconditioning matrix will include eigen kernels.
Default:False
C++ Type:bool
Options:
Description:Whether or not to include eigen kernels in the preconditioning matrix. If true, the preconditioning matrix will include eigen kernels.
resid_vs_jac_scaling_param0A parameter that indicates the weighting of the residual vs the Jacobian in determining variable scaling parameters. A value of 1 indicates pure residual-based scaling. A value of 0 indicates pure Jacobian-based scaling
Default:0
C++ Type:double
Options:
Description:A parameter that indicates the weighting of the residual vs the Jacobian in determining variable scaling parameters. A value of 1 indicates pure residual-based scaling. A value of 0 indicates pure Jacobian-based scaling
scaling_group_variablesName of variables that are grouped together for determining scale factors. (Multiple groups can be provided, separated by semicolon)
C++ Type:std::vector<std::vector<std::string>>
Options:
Description:Name of variables that are grouped together for determining scale factors. (Multiple groups can be provided, separated by semicolon)
skip_exception_checkFalseSpecifies whether or not to skip exception check
Default:False
C++ Type:bool
Options:
Description:Specifies whether or not to skip exception check
solve_typePJFNKPOWER: Power / Inverse / RQI ARNOLDI: Arnoldi KRYLOVSCHUR: Krylov-Schur JACOBI_DAVIDSON: Jacobi-Davidson NONLINEAR_POWER: Nonlinear Power NEWTON: Newton PJFNK: Preconditioned Jacobian-free Newton-KyrlovJFNK: Jacobian-free Newton-Kyrlov
Default:PJFNK
C++ Type:MooseEnum
Options:POWER, ARNOLDI, KRYLOVSCHUR, JACOBI_DAVIDSON, NONLINEAR_POWER, NEWTON, PJFNK, JFNK
Description:POWER: Power / Inverse / RQI ARNOLDI: Arnoldi KRYLOVSCHUR: Krylov-Schur JACOBI_DAVIDSON: Jacobi-Davidson NONLINEAR_POWER: Nonlinear Power NEWTON: Newton PJFNK: Preconditioned Jacobian-free Newton-KyrlovJFNK: Jacobian-free Newton-Kyrlov
splittingTop-level splitting defining a hierarchical decomposition into subsystems to help the solver.
C++ Type:std::vector<std::string>
Options:
Description:Top-level splitting defining a hierarchical decomposition into subsystems to help the solver.
time0System time
Default:0
C++ Type:double
Options:
Description:System time
verboseFalseSet to true to print additional information
Default:False
C++ Type:bool
Options:
Description:Set to true to print additional information
which_eigen_pairsWhich eigenvalue pairs to obtain from the solution LARGEST_MAGNITUDE SMALLEST_MAGNITUDE LARGEST_REAL SMALLEST_REAL LARGEST_IMAGINARY SMALLEST_IMAGINARY TARGET_MAGNITUDE TARGET_REAL TARGET_IMAGINARY ALL_EIGENVALUES SLEPC_DEFAULT
C++ Type:MooseEnum
Options:LARGEST_MAGNITUDE, SMALLEST_MAGNITUDE, LARGEST_REAL, SMALLEST_REAL, LARGEST_IMAGINARY, SMALLEST_IMAGINARY, TARGET_MAGNITUDE, TARGET_REAL, TARGET_IMAGINARY, ALL_EIGENVALUES, SLEPC_DEFAULT
Description:Which eigenvalue pairs to obtain from the solution LARGEST_MAGNITUDE SMALLEST_MAGNITUDE LARGEST_REAL SMALLEST_REAL LARGEST_IMAGINARY SMALLEST_IMAGINARY TARGET_MAGNITUDE TARGET_REAL TARGET_IMAGINARY ALL_EIGENVALUES SLEPC_DEFAULT

Optional Parameters

accept_on_max_fixed_point_iterationFalseTrue to treat reaching the maximum number of fixed point iterations as converged.
Default:False
C++ Type:bool
Options:
Description:True to treat reaching the maximum number of fixed point iterations as converged.
auto_advanceFalseWhether to automatically advance sub-applications regardless of whether their solve converges, for transient executioners only.
Default:False
C++ Type:bool
Options:
Description:Whether to automatically advance sub-applications regardless of whether their solve converges, for transient executioners only.
custom_ppPostprocessor for custom fixed point convergence check.
C++ Type:PostprocessorName
Options:
Description:Postprocessor for custom fixed point convergence check.
direct_pp_valueFalseTrue to use direct postprocessor value (scaled by value on first iteration). False (default) to use difference in postprocessor value between fixed point iterations.
Default:False
C++ Type:bool
Options:
Description:True to use direct postprocessor value (scaled by value on first iteration). False (default) to use difference in postprocessor value between fixed point iterations.
disable_fixed_point_residual_norm_checkFalseDisable the residual norm evaluation thus the three parameters fixed_point_rel_tol, fixed_point_abs_tol and fixed_point_force_norms.
Default:False
C++ Type:bool
Options:
Description:Disable the residual norm evaluation thus the three parameters fixed_point_rel_tol, fixed_point_abs_tol and fixed_point_force_norms.
fixed_point_abs_tol1e-50The absolute nonlinear residual to shoot for during fixed point iterations. This check is performed based on the main app's nonlinear residual.
Default:1e-50
C++ Type:double
Options:
Description:The absolute nonlinear residual to shoot for during fixed point iterations. This check is performed based on the main app's nonlinear residual.
fixed_point_force_normsFalseForce the evaluation of both the TIMESTEP_BEGIN and TIMESTEP_END norms regardless of the existence of active MultiApps with those execute_on flags, default: false.
Default:False
C++ Type:bool
Options:
Description:Force the evaluation of both the TIMESTEP_BEGIN and TIMESTEP_END norms regardless of the existence of active MultiApps with those execute_on flags, default: false.
fixed_point_max_its1Specifies the maximum number of fixed point iterations.
Default:1
C++ Type:unsigned int
Options:
Description:Specifies the maximum number of fixed point iterations.
fixed_point_min_its1Specifies the minimum number of fixed point iterations.
Default:1
C++ Type:unsigned int
Options:
Description:Specifies the minimum number of fixed point iterations.
fixed_point_rel_tol1e-08The relative nonlinear residual drop to shoot for during fixed point iterations. This check is performed based on the main app's nonlinear residual.
Default:1e-08
C++ Type:double
Options:
Description:The relative nonlinear residual drop to shoot for during fixed point iterations. This check is performed based on the main app's nonlinear residual.
relaxation_factor1Fraction of newly computed value to keep.Set between 0 and 2.
Default:1
C++ Type:double
Options:
Description:Fraction of newly computed value to keep.Set between 0 and 2.
transformed_postprocessorsList of main app postprocessors to transform during fixed point iterations
C++ Type:std::vector<PostprocessorName>
Options:
Description:List of main app postprocessors to transform during fixed point iterations
transformed_variablesList of main app variables to transform during fixed point iterations
C++ Type:std::vector<std::string>
Options:
Description:List of main app variables to transform during fixed point iterations

Fixed Point Iterations Parameters

automatic_scalingFalseWhether to use automatic scaling for the variables.
Default:False
C++ Type:bool
Options:
Description:Whether to use automatic scaling for the variables.
compute_initial_residual_before_preset_bcsFalseUse the residual norm computed *before* preset BCs are imposed in relative convergence check
Default:False
C++ Type:bool
Options:
Description:Use the residual norm computed *before* preset BCs are imposed in relative convergence check
compute_scaling_onceTrueWhether the scaling factors should only be computed once at the beginning of the simulation through an extra Jacobian evaluation. If this is set to false, then the scaling factors will be computed during an extra Jacobian evaluation at the beginning of every time step.
Default:True
C++ Type:bool
Options:
Description:Whether the scaling factors should only be computed once at the beginning of the simulation through an extra Jacobian evaluation. If this is set to false, then the scaling factors will be computed during an extra Jacobian evaluation at the beginning of every time step.
l_max_its10000Max Linear Iterations
Default:10000
C++ Type:unsigned int
Options:
Description:Max Linear Iterations
l_tol0.01Linear Tolerance
Default:0.01
C++ Type:double
Options:
Description:Linear Tolerance
nl_abs_step_tol0Nonlinear Absolute step Tolerance
Default:0
C++ Type:double
Options:
Description:Nonlinear Absolute step Tolerance
nl_abs_tol1e-50Nonlinear Absolute Tolerance
Default:1e-50
C++ Type:double
Options:
Description:Nonlinear Absolute Tolerance
nl_max_funcs10000Max Nonlinear solver function evaluations
Default:10000
C++ Type:unsigned int
Options:
Description:Max Nonlinear solver function evaluations
nl_max_its50Max Nonlinear Iterations
Default:50
C++ Type:unsigned int
Options:
Description:Max Nonlinear Iterations
nl_rel_step_tol0Nonlinear Relative step Tolerance
Default:0
C++ Type:double
Options:
Description:Nonlinear Relative step Tolerance
nl_rel_tol1e-08Nonlinear Relative Tolerance
Default:1e-08
C++ Type:double
Options:
Description:Nonlinear Relative Tolerance
num_grids1The number of grids to use for a grid sequencing algorithm. This includes the final grid, so num_grids = 1 indicates just one solve in a time-step
Default:1
C++ Type:unsigned int
Options:
Description:The number of grids to use for a grid sequencing algorithm. This includes the final grid, so num_grids = 1 indicates just one solve in a time-step
off_diagonals_in_auto_scalingFalseWhether to consider off-diagonals when determining automatic scaling factors.
Default:False
C++ Type:bool
Options:
Description:Whether to consider off-diagonals when determining automatic scaling factors.
snesmf_reuse_baseTrueSpecifies whether or not to reuse the base vector for matrix-free calculation
Default:True
C++ Type:bool
Options:
Description:Specifies whether or not to reuse the base vector for matrix-free calculation

Solver Parameters

control_tagsAdds user-defined labels for accessing object parameters via control logic.
C++ Type:std::vector<std::string>
Options:
Description:Adds user-defined labels for accessing object parameters via control logic.
enableTrueSet the enabled status of the MooseObject.
Default:True
C++ Type:bool
Options:
Description:Set the enabled status of the MooseObject.
outputsVector of output names were you would like to restrict the output of variables(s) associated with this object
C++ Type:std::vector<OutputName>
Options:
Description:Vector of output names were you would like to restrict the output of variables(s) associated with this object

Advanced Parameters

max_xfem_update4294967295Maximum number of times to update XFEM crack topology in a step due to evolving cracks
Default:4294967295
C++ Type:unsigned int
Options:
Description:Maximum number of times to update XFEM crack topology in a step due to evolving cracks
update_xfem_at_timestep_beginFalseShould XFEM update the mesh at the beginning of the timestep
Default:False
C++ Type:bool
Options:
Description:Should XFEM update the mesh at the beginning of the timestep

Xfem Fixed Point Iterations Parameters

Restart Parameters

Input Files

(test/tests/problems/eigen_problem/eigensolvers/ne_coupled_scaled.i)
(test/tests/problems/eigen_problem/eigensolvers/ne.i)
(test/tests/problems/eigen_problem/initial_condition/ne_ic_no_free.i)
(test/tests/tag/eigen_tag.i)
(test/tests/problems/eigen_problem/arraykernels/ne_array_kernels.i)
(test/tests/problems/eigen_problem/eigensolvers/gipm_ibc.i)
(test/tests/problems/eigen_problem/eigensolvers/gipm.i)
(test/tests/problems/eigen_problem/initial_condition/ne_ic.i)
(test/tests/problems/eigen_problem/eigensolvers/ane.i)
(test/tests/problems/eigen_problem/eigensolvers/scalar.i)
(test/tests/problems/eigen_problem/eigensolvers/ne_coupled_picard.i)
(test/tests/problems/eigen_problem/eigensolvers/ne_hmg.i)
(test/tests/problems/eigen_problem/eigensolvers/ipm.i)
(test/tests/problems/eigen_problem/eigensolvers/ne_deficient_b.i)
(test/tests/problems/eigen_problem/eigensolvers/dg_krylovschur.i)
(test/tests/problems/eigen_problem/eigensolvers/ne-coupled-resid-scaling.i)
(test/tests/problems/eigen_problem/eigensolvers/ne_coupled_picard_subT_sub.i)
(test/tests/problems/eigen_problem/eigensolvers/ne_coupled.i)
(test/tests/problems/eigen_problem/preconditioners/ne_pbp.i)
(test/tests/problems/eigen_problem/arraykernels/ne_two_variables.i)
(test/tests/problems/eigen_problem/eigensolvers/ne-coupled-scaling.i)

(test/tests/problems/eigen_problem/eigensolvers/ne_coupled_scaled.i)

[Mesh]
  type = GeneratedMesh
  dim = 2
  xmin = 0
  xmax = 10
  ymin = 0
  ymax = 10
  elem_type = QUAD4
  nx = 8
  ny = 8
[]

[Variables]
  [./u]
    order = FIRST
    family = LAGRANGE
  [../]

  [./T]
    order = FIRST
    family = LAGRANGE
  [../]
[]

[AuxVariables]
  [./power]
    order = FIRST
    family = LAGRANGE
  [../]
[]

[Kernels]
  [./diff]
    type = DiffMKernel
    variable = u
    mat_prop = diffusion
    offset = 0.0
  [../]

  [./rhs]
    type = CoefReaction
    variable = u
    coefficient = -1.0
    extra_vector_tags = 'eigen'
  [../]

  [./diff_T]
    type = Diffusion
    variable = T
  [../]
  [./src_T]
    type = CoupledForce
    variable = T
    v = power
  [../]
[]

[AuxKernels]
  [./power_ak]
    type = NormalizationAux
    variable = power
    source_variable = u
    normalization = unorm
    # this coefficient will affect the eigenvalue.
    normal_factor = 10
    execute_on = linear
  [../]
[]

[BCs]
  [./homogeneous]
    type = DirichletBC
    variable = u
    boundary = '0 1 2 3'
    value = 0
  [../]

  [./eigenU]
    type = EigenDirichletBC
    variable = u
    boundary = '0 1 2 3'
  [../]

  [./homogeneousT]
    type = DirichletBC
    variable = T
    boundary = '0 1 2 3'
    value = 0
  [../]
[]

[Materials]
  [./dc]
    type = VarCouplingMaterial
    var = T
    block = 0
    base = 1.0
    coef = 1.0
  [../]
[]

[Executioner]
  type = Eigenvalue
  solve_type = PJFNK

  # Postprocessor value to normalize
  normalization = unorm
  # Value to set normilization to
  normal_factor = 17
[]

[Postprocessors]
  [./unorm]
    type = ElementIntegralVariablePostprocessor
    variable = u
    execute_on = linear
  [../]
[]

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

(test/tests/problems/eigen_problem/eigensolvers/ne.i)

[Mesh]
  type = GeneratedMesh
  dim = 2
  xmin = 0
  xmax = 10
  ymin = 0
  ymax = 10
  elem_type = QUAD4
  nx = 8
  ny = 8
[]

# the minimum eigenvalue of this problem is 2*(PI/a)^2;
# Its inverse is 0.5*(a/PI)^2 = 5.0660591821169. Here a is equal to 10.

[Variables]
  [./u]
    order = FIRST
    family = LAGRANGE
  [../]
[]

[Kernels]
  [./diff]
    type = Diffusion
    variable = u
  [../]

  [./rhs]
    type = CoefReaction
    variable = u
    coefficient = -1.0
    extra_vector_tags = 'eigen'
  [../]
[]

[BCs]
  [./homogeneous]
    type = DirichletBC
    variable = u
    boundary = '0 1 2 3'
    value = 0
  [../]
  [./eigen]
    type = EigenDirichletBC
    variable = u
    boundary = '0 1 2 3'
  [../]
[]

[Executioner]
  type = Eigenvalue
  solve_type = PJFNK
[]

[VectorPostprocessors]
  [./eigenvalues]
    type = Eigenvalues
    execute_on = 'timestep_end'
  [../]
[]

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

(test/tests/problems/eigen_problem/initial_condition/ne_ic_no_free.i)

[Mesh]
  file = 'gold/ne_ic_out.e'
[]

# the minimum eigenvalue of this problem is 2*(PI/a)^2;
# Its inverse is 0.5*(a/PI)^2 = 5.0660591821169. Here a is equal to 10.

[Variables]
  [./u]
    order = FIRST
    family = LAGRANGE
    # Use a good initial so that Newton can converge when we do not use free power iterations
    initial_from_file_var = u
    initial_from_file_timestep = 1
  [../]
[]

[Kernels]
  [./diff]
    type = Diffusion
    variable = u
  [../]

  [./rhs]
    type = CoefReaction
    variable = u
    coefficient = -1.0
    extra_vector_tags = 'eigen'
  [../]
[]

[BCs]
  [./homogeneous]
    type = DirichletBC
    variable = u
    boundary = '0 1 2 3'
    value = 0
  [../]
  [./eigen]
    type = EigenDirichletBC
    variable = u
    boundary = '0 1 2 3'
  [../]
[]

[Executioner]
  type = Eigenvalue
  solve_type = PJFNK
  free_power_iterations = 0
  nl_abs_tol = 1e-8
  nl_rel_tol = 1e-6
[]

[VectorPostprocessors]
  [./eigenvalues]
    type = Eigenvalues
    execute_on = 'timestep_end'
  [../]
[]

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

(test/tests/tag/eigen_tag.i)

[Mesh/gmg]
  type = GeneratedMeshGenerator
  dim = 2
  nx = 5
  ny = 5
[]

[Variables/u]
[]

[AuxVariables]
  [vec_tag_diff]
    order = FIRST
    family = LAGRANGE
  []
  [vec_tag_rhs]
    order = FIRST
    family = LAGRANGE
  []
  [mat_tag_diff]
    order = FIRST
    family = LAGRANGE
  []
  [mat_tag_rhs]
    order = FIRST
    family = LAGRANGE
  []
[]

[Kernels]
  [diff]
    type = Diffusion
    variable = u
    extra_vector_tags = 'tag_diff'
    extra_matrix_tags = 'tag_diff'
  []
  [rhs]
    type = CoefReaction
    variable = u
    extra_vector_tags = 'eigen tag_rhs'
    extra_matrix_tags = 'tag_rhs'
  []
[]

[AuxKernels]
  [vec_tag_diff]
    type = TagVectorAux
    variable = vec_tag_diff
    v = u
    vector_tag = tag_diff
  []
  [vec_tag_rhs]
    type = TagVectorAux
    variable = vec_tag_rhs
    v = u
    vector_tag = tag_rhs
  []

  [mat_tag_diff]
    type = TagVectorAux
    variable = mat_tag_diff
    v = u
    vector_tag = tag_diff
  []
  [mat_tag_rhs]
    type = TagVectorAux
    variable = mat_tag_diff
    v = u
    vector_tag = tag_rhs
  []
[]

[BCs/homogeneous]
  type = DirichletBC
  boundary = 'top right bottom left'
  variable = u
  value = 0
[]

[Problem]
  extra_tag_vectors = 'tag_diff tag_rhs'
  extra_tag_matrices = 'tag_diff tag_rhs'
[]


[Executioner]
  type = Eigenvalue
  solve_type = NEWTON
  eigen_problem_type = GEN_NON_HERMITIAN
[]

[Outputs]
  exodus = true
[]

(test/tests/problems/eigen_problem/arraykernels/ne_array_kernels.i)

[Mesh]
  type = GeneratedMesh
  dim = 2
  xmin = 0
  xmax = 10
  ymin = 0
  ymax = 10
  elem_type = QUAD4
  nx = 8
  ny = 8
[]

# the minimum eigenvalue of this problem is 2*(PI/a)^2;
# Its inverse is 0.5*(a/PI)^2 = 5.0660591821169. Here a is equal to 10.

[Variables]
  [u]
    order = FIRST
    family = LAGRANGE
    components = 2
  []
  [v]
    order = FIRST
    family = LAGRANGE
    components = 2
  []
[]

[Kernels]
  [diffu]
    type = ArrayDiffusion
    variable = u
    diffusion_coefficient = dc
  []

  [reactionu]
    type = ArrayReaction
    variable = u
    reaction_coefficient = rc
    extra_vector_tags = 'eigen'
  []

  [diffv]
    type = ArrayDiffusion
    variable = v
    diffusion_coefficient = dc
  []

  [reactionv]
    type = ArrayReaction
    variable = v
    reaction_coefficient = rc
    extra_vector_tags = 'eigen'
  []
[]

[Materials]
  [dc]
    type = GenericConstantArray
    prop_name = dc
    prop_value = '1. 1.'
  []
  [rc]
    type = GenericConstantArray
    prop_name = rc
    prop_value = '-1 -1'
  []
[]

[BCs]
  [hom_u]
    type = ArrayDirichletBC
    variable = u
    values = '0 0'
    boundary = '0 1 2 3'
  []

  [eigenhom_u]
    type = EigenArrayDirichletBC
    variable = u
    boundary = '0 1 2 3'
  []

  [hom_v]
    type = ArrayDirichletBC
    variable = v
    values = '0 0'
    boundary = '0 1 2 3'
  []

  [eigenhom_v]
    type = EigenArrayDirichletBC
    variable = v
    boundary = '0 1 2 3'
  []
[]

[Executioner]
  type = Eigenvalue
[]

[VectorPostprocessors]
  [./eigenvalues]
    type = Eigenvalues
    execute_on = 'timestep_end'
  [../]
[]

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

(test/tests/problems/eigen_problem/eigensolvers/gipm_ibc.i)

[Mesh]
  type = GeneratedMesh
  dim = 2
  xmin = 0
  xmax = 100
  ymin = 0
  ymax = 100
  elem_type = QUAD4
  nx = 64
  ny = 64

  displacements = 'x_disp y_disp'
[]

[Variables]
  [./u]
    order = first
    family = LAGRANGE
  [../]
[]

[AuxVariables]
  [./x_disp]
  [../]
  [./y_disp]
  [../]
[]

[AuxKernels]
  [./x_disp]
    type = FunctionAux
    variable = x_disp
    function = x_disp_func
  [../]
  [./y_disp]
    type = FunctionAux
    variable = y_disp
    function = y_disp_func
  [../]
[]

[Functions]
  [./x_disp_func]
    type = ParsedFunction
    value = 0
  [../]
  [./y_disp_func]
    type = ParsedFunction
    value = 0
  [../]
[]

[Kernels]
  [./diff]
    type = Diffusion
    variable = u
    use_displaced_mesh = true
  [../]

  [./rea]
    type = CoefReaction
    variable = u
    coefficient = 2.0
    use_displaced_mesh = true
  [../]

  [./rhs]
    type = CoefReaction
    variable = u
    use_displaced_mesh = true
    coefficient = -1.0
    extra_vector_tags = 'eigen'
  [../]
[]

[BCs]
  [./nbc_homogeneous]
    type = DirichletBC
    variable = u
    boundary = '0'
    value = 0
    use_displaced_mesh = true
  [../]

  [./nbc_eigen]
    type = EigenDirichletBC
    variable = u
    boundary = '0'
    use_displaced_mesh = true
  [../]

  [./ibc]
    type = VacuumBC
    variable = u
    boundary = '1 2 3'
    extra_vector_tags = 'eigen'
    use_displaced_mesh = true
  []
[]

[Executioner]
  type = Eigenvalue
  eigen_problem_type = gen_non_hermitian
  which_eigen_pairs = SMALLEST_MAGNITUDE
  n_eigen_pairs = 1
  n_basis_vectors = 18
  solve_type = jacobi_davidson
  petsc_options = '-eps_view'
[]

[VectorPostprocessors]
  [./eigenvalues]
    type = Eigenvalues
    execute_on = 'timestep_end'
  [../]
[]

[Outputs]
  csv = true
  execute_on = 'timestep_end'
  [./console]
    type = Console
    outlier_variable_norms = false
  [../]
[]

(test/tests/problems/eigen_problem/eigensolvers/gipm.i)

[Mesh]
  type = GeneratedMesh
  dim = 2
  xmin = 0
  xmax = 100
  ymin = 0
  ymax = 100
  elem_type = QUAD4
  nx = 64
  ny = 64

  displacements = 'x_disp y_disp'
[]

#The minimum eigenvalue for this problem is 2*(pi/a)^2 + 2 with a = 100.
#Its inverse will be 0.49950700634518.

[Variables]
  [./u]
    order = first
    family = LAGRANGE
  [../]
[]

[AuxVariables]
  [./x_disp]
  [../]
  [./y_disp]
  [../]
[]

[AuxKernels]
  [./x_disp]
    type = FunctionAux
    variable = x_disp
    function = x_disp_func
  [../]
  [./y_disp]
    type = FunctionAux
    variable = y_disp
    function = y_disp_func
  [../]
[]

[Functions]
  [./x_disp_func]
    type = ParsedFunction
    value = 0
  [../]
  [./y_disp_func]
    type = ParsedFunction
    value = 0
  [../]
[]

[Kernels]
  [./diff]
    type = Diffusion
    variable = u
    use_displaced_mesh = true
  [../]

  [./rea]
    type = CoefReaction
    variable = u
    coefficient = 2.0
    use_displaced_mesh = true
  [../]

  [./rhs]
    type = CoefReaction
    variable = u
    coefficient = -1.0
    use_displaced_mesh = true
    extra_vector_tags = 'eigen'
  [../]
[]

[BCs]
  [./homogeneous]
    type = DirichletBC
    variable = u
    boundary = '0 1 2 3'
    value = 0
    use_displaced_mesh = true
  [../]
  [./eigen_bc]
    type = EigenDirichletBC
    variable = u
    boundary = '0 1 2 3'
    use_displaced_mesh = true
  [../]
[]

[Executioner]
  type = Eigenvalue
  eigen_problem_type = gen_non_hermitian
  which_eigen_pairs = SMALLEST_MAGNITUDE
  n_eigen_pairs = 1
  n_basis_vectors = 18
  solve_type = jacobi_davidson
  petsc_options = '-eps_view'
[]

[VectorPostprocessors]
  [./eigenvalues]
    type = Eigenvalues
    execute_on = 'timestep_end'
  [../]
[]

[Outputs]
  csv = true
  execute_on = 'timestep_end'
  [./console]
    type = Console
    outlier_variable_norms = false
  [../]
[]

(test/tests/problems/eigen_problem/initial_condition/ne_ic.i)

[Mesh]
  type = GeneratedMesh
  dim = 2
  xmin = 0
  xmax = 10
  ymin = 0
  ymax = 10
  elem_type = QUAD4
  nx = 8
  ny = 8
[]

# the minimum eigenvalue of this problem is 2*(PI/a)^2;
# Its inverse is 0.5*(a/PI)^2 = 5.0660591821169. Here a is equal to 10.

[Variables]
  [./u]
    order = FIRST
    family = LAGRANGE
  [../]
[]

[Kernels]
  [./diff]
    type = Diffusion
    variable = u
  [../]

  [./rhs]
    type = CoefReaction
    variable = u
    coefficient = -1.0
    extra_vector_tags = 'eigen'
  [../]
[]

[BCs]
  [./homogeneous]
    type = DirichletBC
    variable = u
    boundary = '0 1 2 3'
    value = 0
  [../]
  [./eigen]
    type = EigenDirichletBC
    variable = u
    boundary = '0 1 2 3'
  [../]
[]

[Executioner]
  type = Eigenvalue
  solve_type = PJFNK
  nl_abs_tol = 1e-8
  nl_rel_tol = 1e-6
[]

[VectorPostprocessors]
  [./eigenvalues]
    type = Eigenvalues
    execute_on = 'timestep_end'
  [../]
[]

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

(test/tests/problems/eigen_problem/eigensolvers/ane.i)

[Mesh]
  [gmg]
    type = GeneratedMeshGenerator
    dim = 2
    xmin = 0
    xmax = 10
    ymin = 0
    ymax = 10
    nx = 8
    ny = 8
    elem_type = QUAD4
  []
[]

# the minimum eigenvalue is (2*PI*(p-1)^(1/p)/a/p/sin(PI/p))^p;
# Its inverse is 35.349726539758187. Here a is equal to 10.

[Variables]
  [./u]
  []
[]

# Set an random initial condition is necessary for this problem
# A constant initial condition will not work for this problem since
# the problem is ill-conditioned at a constant vector.
# We observed bad convergence when using a constant initial condition
[ICs]
  [./uic]
    type = RandomIC
    variable = u
  [../]
[]

[Kernels]
  [./diff]
    type = PHarmonic
    variable = u
    p = 3
  [../]

  [./rhs]
    type = PMassKernel
    extra_vector_tags = 'eigen'
    variable = u
    coefficient = -1.0
    p = 3
  [../]
[]

[BCs]
  [./homogeneous]
    type = DirichletBC
    variable = u
    boundary = '0 2'
    value = 0
  [../]
  [./eigen]
    type = EigenDirichletBC
    variable = u
    boundary = '0 2'
  [../]
[]

[Executioner]
  type = Eigenvalue
  free_power_iterations = 10
  solve_type = PJFNK
[]

[VectorPostprocessors]
  [./eigenvalues]
    type = Eigenvalues
    execute_on = 'timestep_end'
  [../]
[]

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

(test/tests/problems/eigen_problem/eigensolvers/scalar.i)

[Mesh]
  type = GeneratedMesh
  dim = 1
  xmin = 0
  xmax = 1
  nx = 1
[]

[Variables]
  [./f1]
    family = SCALAR
    order = FIRST
    initial_condition = 1
  [../]
  [./f2]
    family = SCALAR
    order = FIRST
    initial_condition = 1
  [../]
[]

[ScalarKernels]
  [./row1]
    type = ParsedODEKernel
    variable = f1
    function = '5*f1 + 2*f2'
    args = 'f2'
  [../]

  [./row2]
    type = ParsedODEKernel
    variable = f2
    function = '2*f1 + 5*f2'
    args = 'f1'
  [../]
[]

[VectorPostprocessors]
  [./eigenvalues]
    type = Eigenvalues
    execute_on = 'timestep_end'
  [../]
[]

[Preconditioning]
  [./smp]
    type = SMP
    full = true
  [../]
[]

[Executioner]
  type = Eigenvalue
  which_eigen_pairs = LARGEST_MAGNITUDE
  eigen_problem_type = HERMITIAN
  n_eigen_pairs = 2
  n_basis_vectors = 4
  eigen_max_its = 10
  solve_type = KRYLOVSCHUR
  petsc_options = '-eps_view'
[]

[Outputs]
  csv = true
[]

(test/tests/problems/eigen_problem/eigensolvers/ne_coupled_picard.i)

[Mesh]
  type = GeneratedMesh
  dim = 2
  xmin = 0
  xmax = 10
  ymin = 0
  ymax = 10
  elem_type = QUAD4
  nx = 8
  ny = 8
[]

[Variables]
  [./u]
    order = FIRST
    family = LAGRANGE
    # set this so that the Picard initial norm is not zero
    initial_condition = 1
  [../]
[]

[AuxVariables]
  [./T]
    order = FIRST
    family = LAGRANGE
    # set this so that the Picard initial norm is not zero
    initial_condition = 1
  [../]
  [./power]
    order = FIRST
    family = LAGRANGE
  [../]
[]

[Kernels]
  [./diff]
    type = DiffMKernel
    variable = u
    mat_prop = diffusion
    offset = 0.0
  [../]

  [./rhs]
    type = CoefReaction
    variable = u
    coefficient = -1.0
    extra_vector_tags = 'eigen'
  [../]
[]

[AuxKernels]
  [./power_ak]
    type = NormalizationAux
    variable = power
    source_variable = u
    normalization = unorm
    # this coefficient will affect the eigenvalue.
    normal_factor = 10
    execute_on = timestep_end
  [../]
[]

[BCs]
  [./homogeneous]
    type = DirichletBC
    variable = u
    boundary = '0 1 2 3'
    value = 0
  [../]

  [./eigenU]
    type = EigenDirichletBC
    variable = u
    boundary = '0 1 2 3'
  [../]
[]

[Materials]
  [./dc]
    type = VarCouplingMaterial
    var = T
    block = 0
    base = 1.0
    coef = 1.0
  [../]
[]

[Executioner]
  type = Eigenvalue
  solve_type = PJFNK
  nl_abs_tol = 1e-8
  nl_rel_tol = 1e-6
  fixed_point_max_its = 10
  fixed_point_rel_tol = 1e-6
[]

[Postprocessors]
  [./unorm]
    type = ElementIntegralVariablePostprocessor
    variable = u
    execute_on = linear
  [../]
[]

[VectorPostprocessors]
  [./eigenvalues]
    type = Eigenvalues
    execute_on = 'timestep_end'
  [../]
[]

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

[MultiApps]
  [./sub]
    type = FullSolveMultiApp
    input_files = ne_coupled_picard_sub.i
    execute_on = timestep_end
  [../]
[]

[Transfers]
  [./T_from_sub]
    type = MultiAppMeshFunctionTransfer
    direction = from_multiapp
    multi_app = sub
    source_variable = T
    variable = T
  [../]
  [./power_to_sub]
    type = MultiAppMeshFunctionTransfer
    direction = to_multiapp
    multi_app = sub
    source_variable = power
    variable = power
  [../]
[]

(test/tests/problems/eigen_problem/eigensolvers/ne_hmg.i)

[Mesh]
  type = GeneratedMesh
  dim = 2
  xmin = 0
  xmax = 10
  ymin = 0
  ymax = 10
  elem_type = QUAD4
  nx = 8
  ny = 8
[]

# the minimum eigenvalue of this problem is 2*(PI/a)^2;
# Its inverse is 0.5*(a/PI)^2 = 5.0660591821169. Here a is equal to 10.

[Variables]
  [./u]
    order = FIRST
    family = LAGRANGE
  [../]
  [./v]
    order = FIRST
    family = LAGRANGE
  [../]
[]

[Kernels]
  [./diff]
    type = Diffusion
    variable = u
  [../]

  [./rhs]
    type = CoefReaction
    variable = u
    coefficient = -1.0
    extra_vector_tags = 'eigen'
  [../]

  [./diffv]
    type = Diffusion
    variable = v
  [../]

  [./rhsv]
    type = CoefReaction
    variable = v
    coefficient = -1.0
    extra_vector_tags = 'eigen'
  [../]
[]

[BCs]
  [./homogeneous]
    type = DirichletBC
    variable = u
    boundary = '0 1 2 3'
    value = 0
  [../]
  [./eigen]
    type = EigenDirichletBC
    variable = u
    boundary = '0 1 2 3'
  [../]

  [./homogeneousv]
    type = DirichletBC
    variable = v
    boundary = '0 1 2 3'
    value = 0
  [../]
  [./eigenv]
    type = EigenDirichletBC
    variable = v
    boundary = '0 1 2 3'
  [../]
[]

[Executioner]
  type = Eigenvalue
  solve_type = PJFNK
  petsc_options_iname = '-pc_type
                         -pc_hmg_use_subspace_coarsening'
  petsc_options_value = 'hmg true'
  petsc_options = '-eps_view'
[]

[VectorPostprocessors]
  [./eigenvalues]
    type = Eigenvalues
    execute_on = 'timestep_end'
  [../]
[]

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

(test/tests/problems/eigen_problem/eigensolvers/ipm.i)

[Mesh]
  type = GeneratedMesh
  dim = 2
  xmin = 0
  xmax = 100
  ymin = 0
  ymax = 100
  elem_type = QUAD4
  nx = 8
  ny = 8

  uniform_refine = 0

  displacements = 'x_disp y_disp'
[]

[Variables]
  [./u]
    order = FIRST
    family = LAGRANGE
  [../]
[]

[AuxVariables]
  [./x_disp]
  [../]
  [./y_disp]
  [../]
[]

[AuxKernels]
  [./x_disp]
    type = FunctionAux
    variable = x_disp
    function = x_disp_func
  [../]
  [./y_disp]
    type = FunctionAux
    variable = y_disp
    function = y_disp_func
  [../]
[]

[Functions]
  [./x_disp_func]
    type = ParsedFunction
    value = 0
  [../]
  [./y_disp_func]
    type = ParsedFunction
    value = 0
  [../]
[]

[Kernels]
  [./diff]
    type = Diffusion
    variable = u
    use_displaced_mesh = true
  [../]

  [./rea]
    type = CoefReaction
    variable = u
    coefficient = 2.0
    use_displaced_mesh = true
  [../]
[]

[BCs]
  [./homogeneous]
    type = DirichletBC
    variable = u
    boundary = '0 1 2 3'
    value = 0
    use_displaced_mesh = true
  [../]
[]

[Executioner]
  type = Eigenvalue
  which_eigen_pairs = largest_magnitude
  eigen_problem_type = NON_HERMITIAN
  n_eigen_pairs = 5
  n_basis_vectors = 15
  solve_type = krylovschur
  petsc_options = '-eps_view'
[]

[VectorPostprocessors]
  [./eigenvalues]
    type = Eigenvalues
    execute_on = 'timestep_end'
  [../]
[]

[Outputs]
  csv = true
  execute_on = 'timestep_end'
  [./console]
    type = Console
    outlier_variable_norms = false
  [../]
[]

(test/tests/problems/eigen_problem/eigensolvers/ne_deficient_b.i)

[Mesh]
  type = GeneratedMesh
  dim = 2
  xmin = 0
  xmax = 10
  ymin = 0
  ymax = 10
  elem_type = QUAD4
  nx = 8
  ny = 8
[]

[Variables]
  [./u]
    order = FIRST
    family = LAGRANGE
  [../]
  [./v]
    order = FIRST
    family = LAGRANGE
    eigen = true
  [../]
[]

[Kernels]
  [./diff_u]
    type = Diffusion
    variable = u
  [../]
  [./diff_v]
    type = Diffusion
    variable = v
  [../]

  [./rhs]
    type = CoupledForce
    variable = u
    v = v
    extra_vector_tags = 'eigen'
  [../]
  [./src_v]
    type = CoupledForce
    variable = v
    v = u
  [../]
[]

[BCs]
  [./homogeneous_u]
    type = DirichletBC
    variable = u
    boundary = '0 1 2 3'
    value = 0
  [../]
  [./homogeneous_v]
    type = DirichletBC
    variable = v
    boundary = '0 1 2 3'
    value = 0
  [../]
  [./eigenBC_u]
    type = EigenDirichletBC
    variable = u
    boundary = '0 1 2 3'
  [../]
  [./eigenBC_v]
    type = EigenDirichletBC
    variable = v
    boundary = '0 1 2 3'
  [../]

[]

[Preconditioning]
  [./smp]
    type = SMP
    full = true
  [../]
[]

[Executioner]
  type = Eigenvalue
  solve_type = PJFNK
[]

[VectorPostprocessors]
  [./eigenvalues]
    type = Eigenvalues
    execute_on = 'timestep_end'
  [../]
[]

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

(test/tests/problems/eigen_problem/eigensolvers/dg_krylovschur.i)

[Mesh]
  type = GeneratedMesh
  dim = 2
  xmin = 0
  xmax = 1
  ymin = 0
  ymax = 1
  nx = 10
  ny = 10
  elem_type = QUAD4
[]

[Variables]
  [./u]
    order = FIRST
    family = MONOMIAL
  [../]
[]

[DGKernels]
  [./dg_diff]
    type = DGDiffusion
    variable = u
    sigma = 6
    epsilon = -1
  []
[]

[Kernels]
  [./rhs]
    type = Reaction
    variable = u
    extra_vector_tags = 'eigen'
  [../]
[]

[Executioner]
  type = Eigenvalue
  solve_type = KRYLOVSCHUR
  eigen_problem_type = HERMITIAN
  which_eigen_pairs = LARGEST_MAGNITUDE
[]

[VectorPostprocessors]
  [./eigenvalues]
    type = Eigenvalues
    execute_on = 'timestep_end'
  [../]
[]

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

(test/tests/problems/eigen_problem/eigensolvers/ne-coupled-resid-scaling.i)

[Mesh]
  type = GeneratedMesh
  dim = 2
  xmin = 0
  xmax = 10
  ymin = 0
  ymax = 10
  elem_type = QUAD4
  nx = 8
  ny = 8
[]

[Variables]
  [u]
    initial_condition = 1
  []
  [T]
    initial_condition = 1
  []
[]

[AuxVariables]
  [power][]
[]

[Kernels]
  [./diff]
    type = DiffMKernel
    variable = u
    mat_prop = diffusion
    offset = 0.0
  [../]

  [./rhs]
    type = CoefReaction
    variable = u
    coefficient = -1.0
    extra_vector_tags = 'eigen'
  [../]

  [./diff_T]
    type = CoefDiffusion
    variable = T
    coef = 1e30
  [../]
  [./src_T]
    type = CoupledForce
    variable = T
    v = power
    coef = 1e30
  [../]
[]

[AuxKernels]
  [./power_ak]
    type = NormalizationAux
    variable = power
    source_variable = u
    normalization = unorm
    # this coefficient will affect the eigenvalue.
    normal_factor = 10
    execute_on = linear
  [../]
[]

[BCs]
  [./homogeneous]
    type = DirichletBC
    variable = u
    boundary = '0 1 2 3'
    value = 0
  [../]

  [./eigenU]
    type = EigenDirichletBC
    variable = u
    boundary = '0 1 2 3'
  [../]

  [./homogeneousT]
    type = DirichletBC
    variable = T
    boundary = '0 1 2 3'
    value = 0
  [../]

  [./eigenT]
    type = EigenDirichletBC
    variable = T
    boundary = '0 1 2 3'
  [../]
[]

[Materials]
  [./dc]
    type = VarCouplingMaterial
    var = T
    block = 0
    base = 1.0
    coef = 1.0
  [../]
[]

[Executioner]
  type = Eigenvalue
  solve_type = PJFNK
  automatic_scaling = true
  petsc_options = '-pc_svd_monitor'
  petsc_options_iname = '-pc_type'
  petsc_options_value = 'svd'
  verbose = true
  resid_vs_jac_scaling_param = 1
[]

[Postprocessors]
  [./unorm]
    type = ElementIntegralVariablePostprocessor
    variable = u
    execute_on = linear
  [../]
[]

[VectorPostprocessors]
  [./eigenvalues]
    type = Eigenvalues
    execute_on = 'timestep_end'
  [../]
[]

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

(test/tests/problems/eigen_problem/eigensolvers/ne_coupled_picard_subT_sub.i)

[Mesh]
  type = GeneratedMesh
  dim = 2
  xmin = 0
  xmax = 10
  ymin = 0
  ymax = 10
  elem_type = QUAD4
  nx = 10
  ny = 10
[]

[Variables]
  [./u]
    order = FIRST
    family = LAGRANGE
  [../]
[]

[AuxVariables]
  [./T]
    order = FIRST
    family = LAGRANGE
  [../]
  [./power]
    order = FIRST
    family = LAGRANGE
  [../]
[]

[Kernels]
  [./diff]
    type = DiffMKernel
    variable = u
    mat_prop = diffusion
    offset = 0.0
  [../]

  [./rhs]
    type = CoefReaction
    variable = u
    coefficient = -1.0
    extra_vector_tags = 'eigen'
  [../]
[]

[AuxKernels]
  [./power_ak]
    type = NormalizationAux
    variable = power
    source_variable = u
    normalization = unorm
    normal_factor = 10
    execute_on = timestep_end
  [../]
[]

[BCs]
  [./homogeneous]
    type = DirichletBC
    variable = u
    boundary = '0 1 2 3'
    value = 0
  [../]

  [./eigenU]
    type = EigenDirichletBC
    variable = u
    boundary = '0 1 2 3'
  [../]
[]

[Materials]
  [./dc]
    type = VarCouplingMaterial
    var = T
    block = 0
    base = 1.0
    coef = 1.0
  [../]
[]

[Executioner]
  type = Eigenvalue
  solve_type = PJFNK
  nl_abs_tol = 1e-8
  nl_rel_tol = 1e-6
[]

[Postprocessors]
  [./power]
    type = ElementIntegralVariablePostprocessor
    variable = power
    execute_on = timestep_end
  [../]
  [./unorm]
    type = ElementIntegralVariablePostprocessor
    variable = u
    execute_on = linear
  [../]
[]

[VectorPostprocessors]
  [./eigenvalues]
    type = Eigenvalues
    execute_on = 'timestep_end'
  [../]
[]

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

(test/tests/problems/eigen_problem/eigensolvers/ne_coupled.i)

[Mesh]
  type = GeneratedMesh
  dim = 2
  xmin = 0
  xmax = 10
  ymin = 0
  ymax = 10
  elem_type = QUAD4
  nx = 8
  ny = 8
[]

[Variables]
  [./u]
    order = FIRST
    family = LAGRANGE
  [../]

  [./T]
    order = FIRST
    family = LAGRANGE
  [../]
[]

[AuxVariables]
  [./power]
    order = FIRST
    family = LAGRANGE
  [../]
[]

[Kernels]
  [./diff]
    type = DiffMKernel
    variable = u
    mat_prop = diffusion
    offset = 0.0
  [../]

  [./rhs]
    type = CoefReaction
    variable = u
    coefficient = -1.0
    extra_vector_tags = 'eigen'
  [../]

  [./diff_T]
    type = Diffusion
    variable = T
  [../]
  [./src_T]
    type = CoupledForce
    variable = T
    v = power
  [../]
[]

[AuxKernels]
  [./power_ak]
    type = NormalizationAux
    variable = power
    source_variable = u
    normalization = unorm
    # this coefficient will affect the eigenvalue.
    normal_factor = 10
    execute_on = linear
  [../]
[]

[BCs]
  [./homogeneous]
    type = DirichletBC
    variable = u
    boundary = '0 1 2 3'
    value = 0
  [../]

  [./eigenU]
    type = EigenDirichletBC
    variable = u
    boundary = '0 1 2 3'
  [../]

  [./homogeneousT]
    type = DirichletBC
    variable = T
    boundary = '0 1 2 3'
    value = 0
  [../]
[]

[Materials]
  [./dc]
    type = VarCouplingMaterial
    var = T
    block = 0
    base = 1.0
    coef = 1.0
  [../]
[]

[Executioner]
  type = Eigenvalue
  solve_type = PJFNK
[]

[Postprocessors]
  [./unorm]
    type = ElementIntegralVariablePostprocessor
    variable = u
    execute_on = linear
  [../]
[]

[VectorPostprocessors]
  [./eigenvalues]
    type = Eigenvalues
    execute_on = 'timestep_end'
  [../]
[]

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

(test/tests/problems/eigen_problem/preconditioners/ne_pbp.i)

[Mesh]
  type = GeneratedMesh
  dim = 2
  xmin = 0
  xmax = 10
  ymin = 0
  ymax = 10
  elem_type = QUAD4
  nx = 8
  ny = 8
[]

# the minimum eigenvalue of this problem is 2*(PI/a)^2;
# Its inverse is 0.5*(a/PI)^2 = 5.0660591821169. Here a is equal to 10.

[Variables]
  [./u]
    order = FIRST
    family = LAGRANGE
  [../]

  [./v]
    order = FIRST
    family = LAGRANGE
  [../]
[]

[Kernels]
  [./diffu]
    type = Diffusion
    variable = u
  [../]

  [./difv]
    type = Diffusion
    variable = v
  [../]

  [./rhsu]
    type = CoefReaction
    variable = u
    coefficient = -1.0
    extra_vector_tags = 'eigen'
  [../]

  [./rhsv]
    type = CoefReaction
    variable = v
    coefficient = -1.0
    extra_vector_tags = 'eigen'
  [../]
[]

[BCs]
  [./homogeneousu]
    type = DirichletBC
    variable = u
    boundary = '0 1 2 3'
    value = 0
  [../]

  [./homogeneousv]
    type = DirichletBC
    variable = v
    boundary = '0 1 2 3'
    value = 0
  [../]

  [./eigenu]
    type = EigenDirichletBC
    variable = u
    boundary = '0 1 2 3'
  [../]

  [./eigenv]
    type = EigenDirichletBC
    variable = v
    boundary = '0 1 2 3'
  [../]
[]

[Executioner]
  type = Eigenvalue
  solve_type = JFNK
[]

[VectorPostprocessors]
  [./eigenvalues]
    type = Eigenvalues
    execute_on = 'timestep_end'
  [../]
[]

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

[Preconditioning]
  [./PBP]
    type = PBP
    solve_order = 'u v'
    preconditioner  = 'LU LU'
  [../]
[]

(test/tests/problems/eigen_problem/arraykernels/ne_two_variables.i)

[Mesh]
  type = GeneratedMesh
  dim = 2
  xmin = 0
  xmax = 10
  ymin = 0
  ymax = 10
  elem_type = QUAD4
  nx = 8
  ny = 8
[]

# the minimum eigenvalue of this problem is 2*(PI/a)^2;
# Its inverse is 0.5*(a/PI)^2 = 5.0660591821169. Here a is equal to 10.

[Variables]
  [./u]
    order = FIRST
    family = LAGRANGE
  [../]
  [./v]
    order = FIRST
    family = LAGRANGE
  [../]
[]

[Kernels]
  [./diffu]
    type = Diffusion
    variable = u
  [../]

  [./diffv]
    type = Diffusion
    variable = v
  [../]

  [./rhsu]
    type = CoefReaction
    variable = u
    coefficient = -1.0
    extra_vector_tags = 'eigen'
  [../]

  [./rhsv]
    type = CoefReaction
    variable = v
    coefficient = -1.0
    extra_vector_tags = 'eigen'
  [../]
[]

[BCs]
  [./homogeneousu]
    type = DirichletBC
    variable = u
    boundary = '0 1 2 3'
    value = 0
  [../]
  [./homogeneousv]
    type = DirichletBC
    variable = v
    boundary = '0 1 2 3'
    value = 0
  [../]

  [./eigenu]
    type = EigenDirichletBC
    variable = u
    boundary = '0 1 2 3'
  [../]
  [./eigenv]
    type = EigenDirichletBC
    variable = v
    boundary = '0 1 2 3'
  [../]
[]

[Executioner]
  type = Eigenvalue
  solve_type = PJFNK
[]

[VectorPostprocessors]
  [./eigenvalues]
    type = Eigenvalues
    execute_on = 'timestep_end'
  [../]
[]

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

(test/tests/problems/eigen_problem/eigensolvers/ne-coupled-scaling.i)

[Mesh]
  type = GeneratedMesh
  dim = 2
  xmin = 0
  xmax = 10
  ymin = 0
  ymax = 10
  elem_type = QUAD4
  nx = 8
  ny = 8
[]

[Variables]
  [u][]
  [T][]
[]

[AuxVariables]
  [power][]
[]

[Kernels]
  [./diff]
    type = DiffMKernel
    variable = u
    mat_prop = diffusion
    offset = 0.0
  [../]

  [./rhs]
    type = CoefReaction
    variable = u
    coefficient = -1.0
    extra_vector_tags = 'eigen'
  [../]

  [./diff_T]
    type = CoefDiffusion
    variable = T
    coef = 1e30
  [../]
  [./src_T]
    type = CoupledForce
    variable = T
    v = power
    coef = 1e30
  [../]
[]

[AuxKernels]
  [./power_ak]
    type = NormalizationAux
    variable = power
    source_variable = u
    normalization = unorm
    # this coefficient will affect the eigenvalue.
    normal_factor = 10
    execute_on = linear
  [../]
[]

[BCs]
  [./homogeneous]
    type = DirichletBC
    variable = u
    boundary = '0 1 2 3'
    value = 0
  [../]

  [./eigenU]
    type = EigenDirichletBC
    variable = u
    boundary = '0 1 2 3'
  [../]

  [./homogeneousT]
    type = DirichletBC
    variable = T
    boundary = '0 1 2 3'
    value = 0
  [../]
[]

[Materials]
  [./dc]
    type = VarCouplingMaterial
    var = T
    block = 0
    base = 1.0
    coef = 1.0
  [../]
[]

[Executioner]
  type = Eigenvalue
  solve_type = PJFNK
  automatic_scaling = true
  petsc_options = '-pc_svd_monitor'
  petsc_options_iname = '-pc_type'
  petsc_options_value = 'svd'
  verbose = true
[]

[Postprocessors]
  [./unorm]
    type = ElementIntegralVariablePostprocessor
    variable = u
    execute_on = linear
  [../]
[]

[VectorPostprocessors]
  [./eigenvalues]
    type = Eigenvalues
    execute_on = 'timestep_end'
  [../]
[]

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

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