EquationSystem

The EquationSystem is responsible for defining and assembling the weak form of the PDE into an mfem::Operator used to solve an iteration of the FE problem. This operator is passed to an mfem::NewtonSolver in a ProblemOperator, which handles the update of the state of all variables (including any required nonlinear iterations).

Discretizing the trial variable by , and approximating the test space by a finite dimensional subspace spanned by the basis , the weak form becomes

after variation over the test variable . This can be expressed in matrix form by defining and .

MFEMKernels contribute domain integrators to the weak form, and MFEMIntegratedBCs contribute boundary integrators to the weak form. mfem::BilinearFormIntegrators add contributions to and mfem::LinearFormIntegrators add contributions to when assembled.

MFEMKernels can also contribute to domain integrators for non-linear actions. This allows to form the residual for non-linear Newton's method as shown below

where is the Jacobian, and is the incremental solution.