Examples that use MFEM capabilities

Description

As part of the MOOSE test suite, we solve a number of sample problems using MFEM that may serve as a useful starting point for users to adapt:

Steady State Diffusion: Steady state diffusion problem for the temperature profile across a mug with fixed temperatures on two boundaries.
Transient Heat Transfer: Transient heat conduction problem with a boundary parameterized by a heat transfer coefficient that exchanges heat with a thermal reservoir.

Linear Elasticity: Solves a 3D linear elasticity problem for the deformation of a multi-material cantilever beam. This example is based on MFEM Example 2.

Definite Maxwell: Solves a 3D electromagnetic diffusion problem for the electric field on a cube missing an octant, discretized using $var element = document.getElementById("moose-equation-400d4028-c166-471c-8fa1-66351415af37");katex.render("H(\\mathrm{curl})", element, {displayMode:false,throwOnError:false,macros:{"\\eqc":"\\,,","\\eqp":"\\,.","\\pd":"\\frac{\\partial #1}{\\partial #2}","\\pr":"\\left(#1\\right)","\\ddt":"\\frac{d #1}{d t}"}});$ conforming Nédélec elements. This example is based on MFEM Example 3.
Grad-div: Solves a diffusion problem for a vector field on a cuboid domain, discretized using $var element = document.getElementById("moose-equation-5c3954a9-a587-468c-9f87-294b79fbdce0");katex.render("H(\\mathrm{div})", element, {displayMode:false,throwOnError:false,macros:{"\\eqc":"\\,,","\\eqp":"\\,.","\\pd":"\\frac{\\partial #1}{\\partial #2}","\\pr":"\\left(#1\\right)","\\ddt":"\\frac{d #1}{d t}"}});$ conforming Raviart-Thomas elements. This example is based on MFEM Example 4 and is relevant for solving Maxwell's equations using potentials without the Coulomb gauge.

Darcy: Solves a 2D mixed Darcy problem. This is a saddle point problem, discretized using Raviart-Thomas finite elements (velocity $var element = document.getElementById("moose-equation-4968ea78-e383-442e-9b4e-755fc619c26a");katex.render("\\vec u", element, {displayMode:false,throwOnError:false,macros:{"\\eqc":"\\,,","\\eqp":"\\,.","\\pd":"\\frac{\\partial #1}{\\partial #2}","\\pr":"\\left(#1\\right)","\\ddt":"\\frac{d #1}{d t}"}});$ ) and piecewise discontinuous polynomials (pressure $var element = document.getElementById("moose-equation-2b03ee5c-0dd2-474c-810c-2158bfcfa133");katex.render("p", element, {displayMode:false,throwOnError:false,macros:{"\\eqc":"\\,,","\\eqp":"\\,.","\\pd":"\\frac{\\partial #1}{\\partial #2}","\\pr":"\\left(#1\\right)","\\ddt":"\\frac{d #1}{d t}"}});$ ). This example demonstrates the use of transposition in the input file for mixed problems with different trial and test variables. Transposition flips the matrix associated a mixed bilinear form over its diagonal, swapping the roles of the test and trial variables and avoiding the need to define a second kernel. This example is based on MFEM Example 5.