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

TL;DR

The paper tackles the challenge of solving boundary-value problems on curved boundaries using polygonal meshes within a discontinuous Galerkin framework. It introduces the DG-ROD boundary data reconstruction to compensate for geometric mismatch and establishes existence, uniqueness, and optimal error estimates for both convex and non-convex 2D domains, supported by numerical experiments on disk, annulus, and rose-shaped geometries. For convex domains, the method achieves the optimal $O(h^{N+1})$ convergence in the $L^2$-norm and $O(h^N)$ in the DG norm; for non-convex domains, optimality depends on regularity and how well the polygonal mesh approximates the curved boundary, with $N=2$ allowing $O(h^3)$ in $L^2$. The results demonstrate that DG-ROD robustly restores high-order accuracy on polygonal meshes and can be extended to 3D and other boundary conditions.

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.

Figures (6)

• Figure 1: Element $T^k$ with an edge $e^{kB}$ on the computational boundary $\partial \Omega_h$, for the convex case where $T^k \subset \Omega$ (left panel) and for the concave case, where $T^k \not\subset \Omega$ (right panel).
• Figure 2: Discontinuous Galerkin nodal set $\{\boldsymbol{x}_i^k\}_{i=1}^{N_p}$ denoted by the black dots (left panel) and points $P^k_r$, $r=1,\ldots. N-1$, denoted by the dots with diagonal lines pattern (right panel).
• Figure 3: Left panel: Example of non-convex domain $\Omega$ (solid), polygonal mesh $\Omega_h$ (dashed) and extended smooth domain $\tilde{\Omega}$ (dotted). Right panel: Example of $\Omega \cap \Omega_h$.
• Figure 4: Unstructured meshes generated for the disk domain. Mesh with $K = 14$ and $h = 9.34\text{E$-$}01$ (left panel) and mesh with $K = 262$ and $h = 2.34\text{E$-$}01$ (right panel).
• Figure 5: Unstructured mesh generated for the annulus domain. Mesh with $K = 40$ and $h = 5.00\text{E$-$}01$ (left panel) and mesh with $K = 608$ and $h = 1.31\text{E$-$}01$ (right panel).

Theorems & Definitions (14)

• Lemma 2.1
• Theorem 3.1: Inf-sup condition
• proof
• Theorem 4.1: DG-norm estimate
• proof
• Theorem 4.2: $L^2$-norm estimate
• proof
• Theorem 4.3: DG-norm estimate - $L^2$ residual
• proof
• Theorem 4.4: DG-norm estimate - $L^\infty$ residual