hp-DGFEM on Shape-Irregular Meshes: Reaction-Diffusion Problems

We consider the hp-version of the discontinuous Galerkin finite element method (DGFEM) for second-order elliptic reaction-diffusion equations with mixed Dirichlet and Neumann boundary conditions. For simplicity of the presentation, we only consider boundary-value problems defined on an axiparallel polygonal domain whose solutions are approximated on subdivisions consisting of axiparallel elements. Our main concern is the generalisation of the error analysis of the hp-DGFEM for the case when shape-irregular (anisotropic) meshes and anisotropic polynomial degrees for the element basis functions are used. We shall present a general framework for deriving error bounds for the approximation error and we shall consider two important special cases. In the first of these we derive an error bound that is simultaneously optimal in h and p, for shape-regular elements and isotropic polynomial degrees, provided that the solution belongs to a certain anisotropic Sobolev space. The second result deals with the case where we have a uniform polynomial degree in every space direction and a shape-irregular mesh. Again we derive an error bound that is optimal both in h and in p. For element-wise analytic solutions the method exhibits exponential rates of convergence under p-refinement, in both cases considered. Finally, numerical experiments using shape-regular and shape-irregular elements are presented