Affiliation:
1. Faculty of Civil Engineering Cracow University of Technology Cracow Poland
2. Department of Civil and Environmental Engineering University of California Davis California USA
Abstract
AbstractIn this article, 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. 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 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.
Reference64 articles.
1. ReedW HillT.Triangular mesh methods for the neutron transport equation. Report LA‐UR‐73‐479. Los Alamos Scientific Laboratory; 1973.
2. Discontinuous Galerkin Methods
3. Mathématiques et Applications;Di Pietro D,2011
4. Discontinuous Galerkin Methods for Solving Elliptic and Parabolic Equations