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).

var element = document.getElementById("moose-equation-3f682dc8-c64e-44a1-9a80-981c2152b611");katex.render("{\\left(\\mathcal{L}(u), v\\right)_{\\Omega}=\\left(f,v\\right)_{\\Omega}\\,\\,\\,\\forall v \\in V}", element, {displayMode:true,throwOnError:false});

Discretizing the trial variable $var element = document.getElementById("moose-equation-c49feba0-70d5-4bc1-8d5d-97fa92bdd7dd");katex.render("u", element, {displayMode:false,throwOnError:false});$ by $var element = document.getElementById("moose-equation-d7e99b33-c0e3-482f-b5f3-55ecfc5f95ca");katex.render("u\\approx u_h = \\sum_i u_i \\phi_i(\\vec x)", element, {displayMode:false,throwOnError:false});$ , and approximating the test space $var element = document.getElementById("moose-equation-9e3eb09e-5326-46e6-a8f6-9d6144476c11");katex.render("V", element, {displayMode:false,throwOnError:false});$ by a finite dimensional subspace spanned by the basis $var element = document.getElementById("moose-equation-ee6d5a86-28e9-4af6-bc9e-2d482215e4b2");katex.render("\\{\\varphi_i\\}", element, {displayMode:false,throwOnError:false});$ , the weak form becomes

var element = document.getElementById("moose-equation-78765338-8db8-4119-be88-051aba5c068b");katex.render("{\\sum_{j}u_j\\left(\\mathcal{L}(\\phi_j),\\varphi_i\\right)_{\\Omega}=\\left(f,\\varphi_i\\right)_{\\Omega}}", element, {displayMode:true,throwOnError:false});

after variation over the test variable $var element = document.getElementById("moose-equation-51a60eda-64ae-4441-b362-5faddb011b9f");katex.render("v", element, {displayMode:false,throwOnError:false});$ . This can be expressed in matrix form by defining $var element = document.getElementById("moose-equation-335870de-ee54-40a9-9b67-b6cbb361e8f9");katex.render("A_{ij} = \\left(\\mathcal{L}(\\phi_j), \\varphi_i\\right)_{\\Omega}", element, {displayMode:false,throwOnError:false});$ and $var element = document.getElementById("moose-equation-a7142a7f-a29e-4965-b251-bdd43f13254b");katex.render("{b_i=\\left(f,\\varphi_i\\right)_{\\Omega}}", element, {displayMode:false,throwOnError:false});$ .

var element = document.getElementById("moose-equation-61c3ca5d-23e7-4190-a6f2-6de3c95d2615");katex.render("{\\sum_j A_{ij} u_j = b_i}", element, {displayMode:true,throwOnError:false});

MFEMKernels contribute domain integrators to the weak form, and MFEMIntegratedBCs contribute boundary integrators to the weak form. mfem::BilinearFormIntegrators add contributions to $var element = document.getElementById("moose-equation-1bd643fe-ace3-4268-a109-2341ed65df13");katex.render("A_{ij}(\\varphi_i, \\phi_j)", element, {displayMode:false,throwOnError:false});$ and mfem::LinearFormIntegrators add contributions to $var element = document.getElementById("moose-equation-9ce62449-3872-4f07-b569-3275d6535635");katex.render("b_i(\\varphi_i)", element, {displayMode:false,throwOnError:false});$ when assembled.

TimeDependentEquationSystem

For transient problems, time derivatives $var element = document.getElementById("moose-equation-b87d8d56-b658-4e0f-8649-2293a356b4ef");katex.render("\\dot{u}", element, {displayMode:false,throwOnError:false});$ of the trial variables $var element = document.getElementById("moose-equation-1999f9fc-4959-443f-af22-a51495b44df5");katex.render("u", element, {displayMode:false,throwOnError:false});$ will also be present in the weak form.

var element = document.getElementById("moose-equation-cf5ffa88-2f53-416a-8320-b454ce4d910a");katex.render("\\left(\\mathcal{T}(\\dot{u}), v\\right)_{\\Omega} + {\\left(\\mathcal{L}(u), v\\right)_{\\Omega} =\\left(f,v\\right)_{\\Omega}\\,\\,\\,\\forall v \\in V}", element, {displayMode:true,throwOnError:false});

Contributions to $var element = document.getElementById("moose-equation-5e0f26de-d284-4ab9-b315-12b4d397bff5");katex.render("\\left(\\mathcal{T}(\\phi_j), \\varphi_i\\right)_{\\Omega}", element, {displayMode:false,throwOnError:false});$ are given by time derivative kernels such as MFEMTimeDerivativeMassKernel

Currently, transient problems in Platypus are solved using an implicit backwards Euler method. Denoting $var element = document.getElementById("moose-equation-1f103dbe-ad32-4382-ad1f-11305a9eff0e");katex.render("u_n=u(t+\\delta t)", element, {displayMode:false,throwOnError:false});$ , $var element = document.getElementById("moose-equation-cebd065f-f70e-46d6-a16c-340cc5dc5ec4");katex.render("u_{n-1}=u(t)", element, {displayMode:false,throwOnError:false});$ and approximating $var element = document.getElementById("moose-equation-fd42c1d7-8b7a-4ec1-8a7d-b786e82fe7e6");katex.render("u(t+\\delta t) \\approx u(t) + \\delta t \\dot{u}(t+\\delta t)", element, {displayMode:false,throwOnError:false});$ , we have

var element = document.getElementById("moose-equation-767b6206-edca-4649-b420-e90210a7e7f2");katex.render("\\left([\\mathcal{T}+\\delta t\\mathcal{L}](\\dot{u}_n), v\\right)_{\\Omega} =\\left([f-\\mathcal{L}(u_{n-1})],v\\right)_{\\Omega}\\,\\,\\,\\forall v \\in V", element, {displayMode:true,throwOnError:false});

TimeDependentEquationSystem