Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Penalty-free discontinuous Galerkin method

Jan Jaśkowiec, N. Sukumar

Abstract

In this paper, we present a new high-order discontinuous Galerkin (DG) method, in which neither a penalty parameter nor a stabilization parameter is needed. We refer to this method as penalty-free DG (\PFDG). In this method, the trial and test functions belong to the broken Sobolev space, in which the functions are in general discontinuous on the mesh skeleton and do not meet the Dirichlet boundary conditions. However, a subset can be distinguished in this space, where the functions are continuous and satisfy the Dirichlet boundary conditions, and this subset is called admissible. The trial solution is chosen to lie in an \emph{augmented} admissible subset, in which a small violation of the continuity condition is permitted. This subset is constructed by applying special augmented constraints to the linear combination of finite element basis functions. In this approach, all the advantages of the DG method are retained without the necessity of using stability parameters or numerical fluxes. Several benchmark problems in two dimensions (Poisson equation, linear elasticity, hyperelasticity, and biharmonic equation) on polygonal (triangles, quadrilateral and weakly convex polygons) meshes as well as a three-dimensional Poisson problem on hexahedral meshes are considered. Numerical results are presented that affirm the sound accuracy and optimal convergence of the method in the $L^2$ norm and the energy seminorm.

Penalty-free discontinuous Galerkin method

Abstract

In this paper, we present a new high-order discontinuous Galerkin (DG) method, in which neither a penalty parameter nor a stabilization parameter is needed. We refer to this method as penalty-free DG (\PFDG). In this method, the trial and test functions belong to the broken Sobolev space, in which the functions are in general discontinuous on the mesh skeleton and do not meet the Dirichlet boundary conditions. However, a subset can be distinguished in this space, where the functions are continuous and satisfy the Dirichlet boundary conditions, and this subset is called admissible. The trial solution is chosen to lie in an \emph{augmented} admissible subset, in which a small violation of the continuity condition is permitted. This subset is constructed by applying special augmented constraints to the linear combination of finite element basis functions. In this approach, all the advantages of the DG method are retained without the necessity of using stability parameters or numerical fluxes. Several benchmark problems in two dimensions (Poisson equation, linear elasticity, hyperelasticity, and biharmonic equation) on polygonal (triangles, quadrilateral and weakly convex polygons) meshes as well as a three-dimensional Poisson problem on hexahedral meshes are considered. Numerical results are presented that affirm the sound accuracy and optimal convergence of the method in the norm and the energy seminorm.
Paper Structure (25 sections, 8 theorems, 147 equations, 21 figures)

This paper contains 25 sections, 8 theorems, 147 equations, 21 figures.

Table of Contents

  1. Introduction
  2. Discontinuous Galerkin method without stability parameter
  3. Mathematical formulation
  4. Approximate solution in the PF-DG method
  5. Construction of the constraint equations
  6. PF-DG for linear elasticity
  7. PF-DG for biharmonic problem
  8. Numerical examples
  9. Example 1: Poisson problem with trigonometric solution
  10. Example 2: Poisson problem with exponential solution
  11. Example 3: Poisson problem on L-shaped domain
  12. Example 4: Linear elasticity problem on L-shaped domain
  13. Example 5: Cook’s membrane problem for hyperelasticity
  14. Example 6: Fourth-order biharmonic problem
  15. Example 7: 3D benchmark problem
  16. ...and 10 more sections

Key Result

Theorem 1

The null space of the $D$ is the same as the space ${V^{c}_{0}}$, i.e. $\ker(D)={V^{c}_{0}}$.

Figures (21)

  • Figure 1: Surface and contour plots of the exact solution given in \ref{['eq:bench_sin_exact']} for Example 1.
  • Figure 2: Representative meshes in Example 1. (a) Triangular mesh, (a) quadtree mesh and (b) polygonal mesh.
  • Figure 3: Convergence of the PF-DG method in the $L^2$ norm (top) and the energy seminorm (bottom) for Example 1. Computations are shown for (a) polygonal, (b) quadrilateral and (c) triangular meshes. Scale is $\frac{1}{2}\log$--$\log$.
  • Figure 4: Convergence study for Example 1 on the polygonal meshes using the three DG methods (DG,ODEN1998491 DGFD and PF-DG). The plots are presented for (a) $p=3$, (b) $p=5$ and (c) $p=10$. Scale is $\frac{1}{2}\log$--$\log$.
  • Figure 5: Condition numbers for the matrices used in Example 1, for the three DG methods (DG,ODEN1998491 DGFD and PF-DG). The plots are presented for (a) $p=3$, (b) $p=5$ and (c) $p=10$.
  • ...and 16 more figures

Theorems & Definitions (16)

  • Theorem 1
  • proof
  • Theorem 2
  • Theorem 3
  • proof
  • Conjecture 1
  • proof
  • Theorem 4
  • proof
  • Theorem 5
  • ...and 6 more