Definite Maxwell Problem

Summary

Solves a 3D electromagnetic diffusion problem for the electric field on a cube missing an octant, discretized using conforming Nédélec elements. This example is based on MFEM Example 3.

Description

This problem solves a definite Maxwell equation with strong form:

where

In this example, we solve this using the weak form

where

Example File

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

[Mesh<<<{"href": "../Mesh/index.html"}>>>]
  type = MFEMFileMesh
  file = ../mesh/small_fichera.mesh
[]

[Problem<<<{"href": "../Problem/index.html"}>>>]
  type = MFEMProblem
[]

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

[Variables<<<{"href": "../Variables/index.html"}>>>]
  [e_field]
    type = MFEMVariable
    fespace = HCurlFESpace
  []
[]

[AuxVariables<<<{"href": "../AuxVariables/index.html"}>>>]
  [db_dt_field]
    type = MFEMVariable
    fespace = HDivFESpace
  []
[]

[AuxKernels<<<{"href": "../AuxKernels/index.html"}>>>]
  [curl]
    type = MFEMCurlAux
    variable = db_dt_field
    source = e_field
    scale_factor = -1.0
    execute_on = TIMESTEP_END
  []
[]

[Functions<<<{"href": "../Functions/index.html"}>>>]
  [exact_e_field]
    type = ParsedVectorFunction<<<{"description": "Returns a vector function based on string descriptions for each component.", "href": "../../source/functions/MooseParsedVectorFunction.html"}>>>
    expression_x<<<{"description": "x-component of function."}>>> = 'sin(kappa * y)'
    expression_y<<<{"description": "y-component of function."}>>> = 'sin(kappa * z)'
    expression_z<<<{"description": "z-component of function."}>>> = 'sin(kappa * x)'

    symbol_names<<<{"description": "Symbols (excluding t,x,y,z) that are bound to the values provided by the corresponding items in the symbol_values vector."}>>> = kappa
    symbol_values<<<{"description": "Constant numeric values, postprocessor names, function names, and scalar variables corresponding to the symbols in symbol_names."}>>> = 3.1415926535
  []

  [forcing_field]
    type = ParsedVectorFunction<<<{"description": "Returns a vector function based on string descriptions for each component.", "href": "../../source/functions/MooseParsedVectorFunction.html"}>>>
    expression_x<<<{"description": "x-component of function."}>>> = '(1. + kappa * kappa) * sin(kappa * y)'
    expression_y<<<{"description": "y-component of function."}>>> = '(1. + kappa * kappa) * sin(kappa * z)'
    expression_z<<<{"description": "z-component of function."}>>> = '(1. + kappa * kappa) * sin(kappa * x)'

    symbol_names<<<{"description": "Symbols (excluding t,x,y,z) that are bound to the values provided by the corresponding items in the symbol_values vector."}>>> = kappa
    symbol_values<<<{"description": "Constant numeric values, postprocessor names, function names, and scalar variables corresponding to the symbols in symbol_names."}>>> = 3.1415926535
  []
[]

[BCs<<<{"href": "../BCs/index.html"}>>>]
  [tangential_E_bdr]
    type = MFEMVectorTangentialDirichletBC
    variable = e_field
    vector_coefficient = exact_e_field
  []
[]

[Kernels<<<{"href": "../Kernels/index.html"}>>>]
  [curlcurl]
    type = MFEMCurlCurlKernel
    variable = e_field
  []
  [mass]
    type = MFEMVectorFEMassKernel
    variable = e_field
  []
  [source]
    type = MFEMVectorFEDomainLFKernel
    variable = e_field
    vector_coefficient = forcing_field
  []
[]

[Solvers]
  active = 'gmres ams'
  [ams]
    type = MFEMHypreAMS
    fespace = HCurlFESpace
  []
  [matrix_free_ams]
    type = MFEMMatrixFreeAMS
  []
  [gmres]
    type = MFEMHypreGMRES
    preconditioner = ams
    l_tol = 1e-12
  []
  [cg]
    type = MFEMCGSolver
    preconditioner = matrix_free_ams
    l_tol = 1e-12
  []
[]

[Executioner<<<{"href": "../Executioner/index.html"}>>>]
  type = MFEMSteady
  device = cpu
[]

[VectorPostprocessors<<<{"href": "../VectorPostprocessors/index.html"}>>>]
  [line_sample_e_field]
    type = MFEMVariableLineValueSampler
    variable = 'e_field'
    start_point = '-0.99 -0.99 0.99'
    end_point = '0.99 0.99 -0.99'
    num_points = 114
  []
  [line_sample_db_dt_field]
    type = MFEMVariableLineValueSampler
    variable = 'db_dt_field'
    start_point = '-0.99 -0.99 0.99'
    end_point = '0.99 0.99 -0.99'
    num_points = 114
  []
[]

[Outputs<<<{"href": "../Outputs/index.html"}>>>]
  [CSV]
    type = CSV<<<{"description": "Output for postprocessors, vector postprocessors, and scalar variables using comma seperated values (CSV).", "href": "../../source/outputs/CSV.html"}>>>
    execute_on<<<{"description": "The list of flag(s) indicating when this object should be executed. For a description of each flag, see https://mooseframework.inl.gov/source/interfaces/SetupInterface.html."}>>> = 'timestep_end'
    file_base<<<{"description": "The desired solution output name without an extension. If not provided, MOOSE sets it with Outputs/file_base when available. Otherwise, MOOSE uses input file name and this object name for a master input or uses master file_base, the subapp name and this object name for a subapp input to set it."}>>> = OutputData/CurlCurl/curlcurl
  []
[]
(moose/test/tests/mfem/kernels/curlcurl.i)