DGKernels System

Overview

DGKernels are the kernels defined on internal sides. DGKernels are typically for elemental variables, i.e. variables that allow solutions to be discontinuous across element sides. DGKernels along with normal kernels allow the definition of weak forms arising from discontinuous finite element methods (DFEM). DGKernels can be block restricted for calculations with DFEM on subdomains. Internal sides are visited once during residual or Jacobian evaluations by MOOSE. DGKernels handle two pieces of residual, marked as Element and Neighbor, on an internal side and corresponding four pieces of Jacobian, marked as ElementElement, ElementNeighbor, NeighborElement and NeighborNeighbor. The normals on internal sides are pointing towards neighboring element from the current element. Typically DGKernels are irrelevant with the normal direction. When there are mesh refinement, MOOSE visits all the active internal sides, meaning that if there is a hanging node for an internal side, MOOSE visit the child internal sides. DGKernels can make use of the material properties defined on both Element and Neighbor sides. The DGKernel with interior penalty (IP) method for diffusion equations can be found at DGDiffusion. The DGKernel with upwinding scheme for hyperbolic equations can be found at DGConvection.

Extension for Hybrid Finite Element Methods

DGKernels are extended to support hybrid finite element methods (HFEM) (Raviart and Thomas, 1977).

Considering Poisson's equation of the form

$var element = document.getElementById("moose-equation-a828a02a-9d5d-400e-b517-7fcab8b601e9");katex.render("\\begin{aligned} -\\nabla^2 u &= f && \\quad \\in \\Omega \\\\ u &= g && \\quad \\in \\partial \\Omega_D \\\\ \\frac{\\partial u}{\\partial n} &= h && \\quad \\in \\partial \\Omega_N, \\\\ \\alpha u + \\frac{\\partial u}{\\partial n} &= c && \\quad \\in \\partial \\Omega_R, \\end{aligned}", element, {displayMode:true,throwOnError:false});$

where $var element = document.getElementById("moose-equation-f468cf5d-a178-4aa5-92de-120f3d5b5f74");katex.render("\\Omega \\subset \\mathbb{R}^n", element, {displayMode:false,throwOnError:false});$ is the domain, and $var element = document.getElementById("moose-equation-3f5355ac-46a6-414e-838e-b06281dbc59b");katex.render("\\partial \\Omega = \\partial \\Omega_D \\cup \\partial \\Omega_N \\cup \\partial \\Omega_R", element, {displayMode:false,throwOnError:false});$ is its boundary. $var element = document.getElementById("moose-equation-c0ade9d6-2c43-4cb4-9df9-ab5cd2b42a81");katex.render("\\alpha", element, {displayMode:false,throwOnError:false});$ is a given function on $var element = document.getElementById("moose-equation-84bbd4a9-3b5b-495e-a206-dbff4a42a2fc");katex.render("\\Omega_R", element, {displayMode:false,throwOnError:false});$ , which has typical constant value of $var element = document.getElementById("moose-equation-171b6eca-3536-492d-86f0-acb19bc2171e");katex.render("1/2", element, {displayMode:false,throwOnError:false});$ .

The weak form with HFEM for this PDE is to find a triple $var element = document.getElementById("moose-equation-79845ce7-6cb2-476f-b059-8422c10a084e");katex.render("(u, \\lambda, \\lambda_D)", element, {displayMode:false,throwOnError:false});$ , in discontinuous function spaces on $var element = document.getElementById("moose-equation-7fcfccef-1ecf-46c9-94ee-52fa473dd3fd");katex.render("\\Omega", element, {displayMode:false,throwOnError:false});$ , all internal sides $var element = document.getElementById("moose-equation-3d048bda-2309-4bd4-9cc9-7af97ffbc649");katex.render("\\Gamma", element, {displayMode:false,throwOnError:false});$ and $var element = document.getElementById("moose-equation-6f7d2f3b-5fee-4ac9-be32-95e2e4f4e773");katex.render("\\partial \\Omega_D", element, {displayMode:false,throwOnError:false});$ respectively, such that, $var element = document.getElementById("moose-equation-76ce8a1e-0234-4aa7-b661-6829515fa25f");katex.render("\\forall (u^\\ast, \\lambda^\\ast, \\lambda^\\ast_D)", element, {displayMode:false,throwOnError:false});$ in the same function spaces,

$var element = document.getElementById("moose-equation-8aa8c7ed-c769-4abb-bd79-34c3af2279ec");katex.render("\\begin{aligned} \\left( \\nabla u^\\ast, \\nabla u \\right)_\\Omega + \\left( \\lambda^\\ast, [ u ] \\right)_\\Gamma + \\left( [ u^\\ast ], \\lambda \\right)_\\Gamma +\\left( u^\\ast, \\lambda_D \\right)_{\\partial \\Omega_D} + \\left( \\lambda_D^\\ast, u - g \\right)_{\\partial \\Omega_D} -\\left( u^\\ast, h \\right)_{\\partial \\Omega_N} +\\left( u^\\ast, \\alpha u - c \\right)_{\\partial \\Omega_R} -\\left( u^\\ast, f \\right)_\\Omega = 0. \\end{aligned}", element, {displayMode:true,throwOnError:false});$

$var element = document.getElementById("moose-equation-bf3b599b-2c93-4fb8-b723-9b4c9ff7c70d");katex.render("[u]", element, {displayMode:false,throwOnError:false});$ represents the jump of $var element = document.getElementById("moose-equation-7fbd9f24-b4c0-4a6e-9621-2aa95991288e");katex.render("u", element, {displayMode:false,throwOnError:false});$ on an internal side. It is noted that the orientation of normals on internal sides does not affect the solution of $var element = document.getElementById("moose-equation-a4f230f2-5dd2-4973-8794-28627982f14e");katex.render("u", element, {displayMode:false,throwOnError:false});$ but flips the sign of $var element = document.getElementById("moose-equation-90d7a632-236b-48e6-a252-270c51bfa4fe");katex.render("\\lambda", element, {displayMode:false,throwOnError:false});$ . $var element = document.getElementById("moose-equation-ed67e1d3-f4ef-4f1c-aca9-96ed8f1b76fe");katex.render("\\lambda", element, {displayMode:false,throwOnError:false});$ and $var element = document.getElementById("moose-equation-84ae307f-21eb-4f34-a115-eedbf4a7fbbb");katex.render("\\lambda_D", element, {displayMode:false,throwOnError:false});$ are also known as Lagrange multipliers for weakly imposing the continuity of $var element = document.getElementById("moose-equation-94244103-0325-492e-938f-99a3e87dec82");katex.render("u", element, {displayMode:false,throwOnError:false});$ across internal sides on $var element = document.getElementById("moose-equation-50e9b021-aa16-4820-86ce-94b7dd2917eb");katex.render("\\Gamma", element, {displayMode:false,throwOnError:false});$ and imposing the Dirichlet boundary condition at $var element = document.getElementById("moose-equation-64f45500-dc8b-4533-b4d1-dfa24bd29486");katex.render("\\Omega_D", element, {displayMode:false,throwOnError:false});$ . They resemble the current ( $var element = document.getElementById("moose-equation-5d515b8e-af58-4be6-9299-69d421f391d2");katex.render("-\\frac{\\partial u}{\\partial n}", element, {displayMode:false,throwOnError:false});$ ) and converge to the current when discretization error gets smaller and smaller with mesh refinement. HFEM has explicit local conservation, which can be seen if we substitute a test function of $var element = document.getElementById("moose-equation-38e97966-206b-49e8-8c83-2bc669674a10");katex.render("u^\\ast", element, {displayMode:false,throwOnError:false});$ with constant value of one element of interest and zero elsewhere. The local conservation is evaluated with Lagrange multiplier $var element = document.getElementById("moose-equation-6368da3b-2635-441a-9aa4-41888e79a59e");katex.render("\\lambda", element, {displayMode:false,throwOnError:false});$ and the source function $var element = document.getElementById("moose-equation-eb0a17fa-1f4e-499d-ae5d-4723af6e9e0d");katex.render("f", element, {displayMode:false,throwOnError:false});$ for an element inside of the domain $var element = document.getElementById("moose-equation-1e9b9cce-54f4-411a-a463-1190daa683bf");katex.render("\\Omega", element, {displayMode:false,throwOnError:false});$ .

This weak form requires a compatibility condition to have a unique solution (Raviart and Thomas, 1977). Typically we satisfy this condition by letting the order of the shape function for $var element = document.getElementById("moose-equation-b92f2dc8-032e-4ecd-b64e-a9aad06cae59");katex.render("u", element, {displayMode:false,throwOnError:false});$ two order higher (including two) than the order for $var element = document.getElementById("moose-equation-becbd5de-85da-44c9-93e3-89e35a21c536");katex.render("\\lambda", element, {displayMode:false,throwOnError:false});$ .

An alternative way of imposing the Robin boundary condition ( $var element = document.getElementById("moose-equation-904cca55-cf5b-42da-82c4-b60534a52980");katex.render("\\alpha u + \\frac{\\partial u}{\\partial n} = c", element, {displayMode:false,throwOnError:false});$ ) is to replace $var element = document.getElementById("moose-equation-0c3942d8-ad5b-4b3d-94ed-41f659d0dd49");katex.render("\\left( u^\\ast, \\alpha u - c \\right)_{\\partial \\Omega_R}", element, {displayMode:false,throwOnError:false});$ with

$var element = document.getElementById("moose-equation-aee65469-f5b2-42a4-94a2-25b41f48394d");katex.render("\\begin{aligned} \\left( u^\\ast, \\lambda_R \\right)_{\\partial \\Omega_R} + \\left( \\lambda_R^\\ast, u - u_R \\right)_{\\partial \\Omega_R} + \\left( u_R^\\ast, \\alpha u_R - c - \\lambda_R \\right)_{\\partial \\Omega_R}, \\end{aligned}", element, {displayMode:true,throwOnError:false});$

with Lagrange multiplier $var element = document.getElementById("moose-equation-a622ee73-611f-41e4-8a97-976621321a21");katex.render("\\lambda_R", element, {displayMode:false,throwOnError:false});$ and the projected solution $var element = document.getElementById("moose-equation-56ceabc1-3e3f-4b1a-a9ec-6b1d410f23b3");katex.render("u_R", element, {displayMode:false,throwOnError:false});$ on $var element = document.getElementById("moose-equation-8f31fa29-e5e2-4d70-bb3d-09aed1ec83cc");katex.render("\\partial \\Omega_R", element, {displayMode:false,throwOnError:false});$ and their corresponding test functions $var element = document.getElementById("moose-equation-a8c2f2a6-7b73-43c4-a2f9-d9ad9cb276df");katex.render("\\lambda_R^\\ast", element, {displayMode:false,throwOnError:false});$ and $var element = document.getElementById("moose-equation-6b7136b3-70dc-4a58-9902-cae12c637116");katex.render("u_R^\\ast", element, {displayMode:false,throwOnError:false});$ .

A variable for the Lagrange multiplier defined on all interior sides $var element = document.getElementById("moose-equation-6e8a3f82-32a8-48fd-be12-91ae0d32122f");katex.render("\\Gamma", element, {displayMode:false,throwOnError:false});$ can be coupled in DGKernels with a lower-dimensional mesh derived from the main mesh for $var element = document.getElementById("moose-equation-ea52dae3-d4d9-458d-95af-859c8b9d4ae2");katex.render("\\Gamma", element, {displayMode:false,throwOnError:false});$ . With this extension, DGKernels can handle three pieces of residual, marked as Element, Neighbor and Lower, on an internal side and corresponding nine pieces of Jacobian, marked as ElementElement, ElementNeighbor, NeighborElement, NeighborNeighbor, PrimaryLower, SecondaryLower, LowerPrimary, LowerSecondary and LowerLower. Similarly, a variable for the Lagrange multiplier on boundary $var element = document.getElementById("moose-equation-b8111b87-9c87-421f-9334-403fa2b4a684");katex.render("\\Omega_D", element, {displayMode:false,throwOnError:false});$ can be coupled in integrated boundary conditions.

The DGKernal for $var element = document.getElementById("moose-equation-a8079656-bb70-48ec-800d-39f58e1718aa");katex.render("\\left( \\lambda^\\ast, [ u ] \\right)_\\Gamma + \\left( [ u^\\ast ], \\lambda \\right)_\\Gamma", element, {displayMode:false,throwOnError:false});$ can be found at HFEMDiffusion. The boundary condition for $var element = document.getElementById("moose-equation-974b8837-7c0c-48a4-9d11-86ddb2a3dff0");katex.render("\\left( u^\\ast, \\lambda_D \\right)_{\\partial \\Omega_D} + \\left( \\lambda_D^\\ast, u - g \\right)_{\\partial \\Omega_D}", element, {displayMode:false,throwOnError:false});$ can be found at HFEMDirichletBC with the generalization of $var element = document.getElementById("moose-equation-f87c6770-e80f-4ac0-aeaa-aef5245da4c4");katex.render("g", element, {displayMode:false,throwOnError:false});$ being either a fixed value or a variable defined on boundary. With this generalization, HFEMDirichletBC can be used along with kernels defined on the Robin boundary for $var element = document.getElementById("moose-equation-407aea2c-3094-418e-bf9a-d6be3e10ab85");katex.render("\\left( u_R^\\ast, \\alpha u_R - c - \\lambda_R \\right)_{\\partial \\Omega_R}", element, {displayMode:false,throwOnError:false});$ for the alternative way of imposing the Robin boundary condition above.

warning

MOOSE does not support mesh adaptation with HFEM currently.

Example Input File Syntax

DGKernels are added through DGKernels input syntax.

Available Objects

Moose App
ADDGAdvectionAdds internal face advection flux contributions for discontinuous Galerkin discretizations
ADDGDiffusionDG kernel for diffusion operator
ArrayDGDiffusionImplements interior penalty method for array diffusion equations.
ArrayHFEMDiffusionImposes the constraints on internal sides with HFEM.
DGConvectionDG upwinding for the convection
DGDiffusionComputes residual contribution for the diffusion operator using discontinous Galerkin method.
HFEMDiffusionImposes the constraints on internal sides with HFEM.
HFEMTestJumpImposes constraints for HFEM with side-discontinuous variables.
HFEMTrialJumpImposes constraints for HFEM with side-discontinuous variables.
Zapdos App
DGCoeffDiffusionThe discontinuous Galerkin form of the generic diffusion term(Densities must be in log form)
DGEFieldAdvectionThe discontinuous Galerkin form of the generic electric field driven advection term(Densities must be in log form)
Squirrel App
DGCoupledAdvectionDG upwinding for convection on Navier-Stokes velocity variables
DGFunctionConvection
DGFunctionTemperatureAdvectionDG upwinding for func temp convection
DGTemperatureAdvectionDG upwinding for the convection

Available Actions

Moose App
AddDGKernelActionAdd a DGKernel object to the simulation.

References

P. A. Raviart and J. M. Thomas. Primal hybrid finite element methods for 2nd order elliptic equations. MATHEMATICS OF COMPUTATION, 31(138):391–413, 1977.

@article{RT-HFEM,
    author = "Raviart, P. A. and Thomas, J. M.",
    title = "Primal Hybrid Finite Element Methods for 2nd Order Elliptic Equations",
    journal = "MATHEMATICS OF COMPUTATION",
    volume = "31",
    number = "138",
    pages = "391--413",
    year = "1977"
}

Overview
Extension for Hybrid Finite Element Methods
Example Input File Syntax
Available Objects
Available Actions
References