$hp$-Version Analysis for Arbitrarily Shaped Elements on the Boundary Discontinuous Galerkin Method for Stokes Systems

Abstract

In the present work, we examine and analyze an $hp$-version interior penalty discontinuous Galerkin finite element method for the numerical approximation of a steady fluid system on computational meshes consisting of polytopic elements on the boundary. This approach is based on the discontinuous Galerkin method, enriched by arbitrarily shaped elements techniques as has been introduced in [13]. In this framework, and employing extensions of trace, Markov-type, and $H^1/L^2$-type inverse estimates to arbitrary element shapes, we examine a stationary Stokes fluid system enabling the proof of the inf/sup condition and the $hp$- a priori error estimates, while we investigate the optimal convergence rates numerically. This approach recovers and integrates the flexibility and superiority of the discontinuous Galerkin methods for fluids whenever geometrical deformations are taking place by degenerating the edges, facets, of the polytopic elements only on the boundary, combined with the efficiency of the $hp$-version techniques based on arbitrarily shaped elements without requiring any mapping from a given reference frame.