hp-version analysis for arbitrarily shaped elements on the boundary discontinuous Galerkin method for Stokes systems

TL;DR

This paper develops and analyzes an hp-version interior penalty discontinuous Galerkin method for steady Stokes systems on meshes with arbitrarily shaped boundary elements, providing theoretical error estimates and demonstrating optimal convergence.

Contribution

It introduces a novel hp-DG method for boundary polytopic elements with arbitrary shapes, extending inverse estimates and proving stability and error bounds.

Findings

Proves inf/sup stability for the method.

Establishes hp-a priori error estimates.

Numerical results confirm optimal convergence rates.

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 H1/L2-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.