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

TL;DR

This work develops an $hp$-version interior-penalty discontinuous Galerkin discretization for the steady Stokes equations on meshes with arbitrarily shaped boundary elements, enabling high-order accuracy without mapping to a reference element. By extending trace and inverse estimates to boundary polytopes and employing an $L^2$ projection-based bilinear form, the authors establish discrete inf–sup stability and hp-optimal a priori error bounds. Numerical experiments on rectangular and circular domains confirm hp-convergence for velocity and pressure and demonstrate competitive performance against unfitted ghost-penalty DG methods, while avoiding complex boundary-area stabilization. The framework enhances geometric flexibility for viscous flows on complex boundaries and lays groundwork for extensions to time-dependent and nonlinear fluid problems, with potential preconditioning and reduced-order modeling avenues.

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.

Figures (5)

• Figure 1: Example geometries $\Omega^{\sharp}$ and their boundaries $\Gamma$.
• Figure 2: Meshes$\mathcal{T}^{\sharp}$ based on arbitrarily-shaped boundary elements, the covering meshes$\mathcal{T}^{cov}$ from Definition \ref{['def:covering_domain']}, examples of refined Figure \ref{['geometry']}'s geometries$\Omega^\sharp$ tessellations $\mathcal{T}_a^\sharp$, $\mathcal{T}_b^\sharp$, $\mathcal{T}_c^\sharp$ and the covering domains, $\Omega^{cov}$.
• Figure 3: (i) Curved boundary elements for $d = 2$ with one curved face and vertices $\mathbf{v}_{k,i}$ and the unit outward normal vector to $F_i$ at $x \in F_i$ where $K_{F_i}$ is star-shaped and (ii) the covering mesh $\mathcal{T}^{cov}$, and the mesh $\mathcal{T}^\sharp$ corresponding to the truth geometry with arbitrary shape boundary elements.
• Figure 4: Rectangular geometry: Visualization of the $H^1$-norm velocity errors and $L^2$-norm pressure errors with respect to the discretization size.
• Figure 5: Circular geometry: Visualization of the $H^1$-norm velocity errors and $L^2$-norm pressure errors with respect to the discretization size.

Theorems & Definitions (26)

• Lemma 3.2
• Remark 3.3
• Lemma 3.4
• Definition 3.5
• Lemma 3.6: Georgoulis17
• Definition 3.7
• Theorem 3.9
• Lemma 3.10
• proof
• Lemma 4.1