Optimal error estimates for a discontinuous Galerkin method on curved boundaries with polygonal meshes

TL;DR

This paper provides a rigorous theoretical analysis of a discontinuous Galerkin method with boundary data reconstruction, achieving optimal error estimates for curved boundary problems on polygonal meshes in two dimensions.

Contribution

It extends the DG-ROD method's theoretical foundation, proving optimal convergence rates for curved boundary problems with polygonal approximations.

Findings

The DG-ROD method achieves optimal convergence rates on polygonal meshes.

Existence and uniqueness of the discrete solution are established.

Numerical benchmarks confirm the theoretical error estimates.

Abstract

We consider a discontinuous Galerkin method for the numerical solution of boundary value problems in two-dimensional domains with curved boundaries. A key challenge in this setting is the potential loss of convergence order due to approximating the physical domain by a polygonal mesh. Unless boundary conditions can be accurately transferred from the true boundary to the computational one, such geometric approximation errors generally lead to suboptimal convergence. To overcome this limitation, a higher-order strategy based on polynomial reconstruction of boundary data was introduced for classical finite element methods in [28, 29] and in the finite volume context in [7, 11]. More recently, this approach was extended to discontinuous Galerkin methods in [32], leading to the DG-ROD method, which restores optimal convergence rates on polygonal approximations of domains with curved boundaries. In this work, we provide a rigorous theoretical analysis of the DG-ROD method, establishing existence and uniqueness of the discrete solution and deriving error estimates for a two-dimensional linear advection-diffusion-reaction problem with homogeneous Dirichlet boundary conditions on both convex and non-convex domains. Following and extending techniques from classical finite element methods [29], we prove that, under suitable regularity assumptions on the exact solution, the DG-ROD method achieves optimal convergence despite polygonal approximations. Finally, we illustrate and confirm the theoretical results with a numerical benchmark.