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A low-cost, parameter-free, and pressure-robust enriched Galerkin method for the Stokes equations

Seulip Lee, Lin Mu

TL;DR

The paper addresses efficiently solving the Stokes equations with a parameter-free, pressure-robust scheme. It develops a modified enriched Galerkin (mEG) method that replaces velocity derivatives with weak derivatives, removing the need for penalty parameters while preserving stability and optimal convergence. A velocity reconstruction-based pressure-robust variant (PR-mEG) is then introduced, delivering velocity errors that are independent of the viscosity $\nu$ and reduced pressure error as $\nu$ decreases. Numerical experiments in two and three dimensions confirm the theoretical results, demonstrating stable parameter-free performance and clear advantages in velocity accuracy for small viscosities.

Abstract

In this paper, we propose a low-cost, parameter-free, and pressure-robust Stokes solver based on the enriched Galerkin (EG) method with a discontinuous velocity enrichment function. The EG method employs the interior penalty discontinuous Galerkin (IPDG) formulation to weakly impose the continuity of the velocity function. However, the symmetric IPDG formulation, despite of its advantage of symmetry, requires a lot of computational effort to choose an optimal penalty parameter and to compute different trace terms. In order to reduce such effort, we replace the derivatives of the velocity function with its weak derivatives computed by the geometric data of elements. Therefore, our modified EG (mEG) method is a parameter-free numerical scheme which has reduced computational complexity as well as optimal rates of convergence. Moreover, we achieve pressure-robustness for the mEG method by employing a velocity reconstruction operator on the load vector on the right-hand side of the discrete system. The theoretical results are confirmed through numerical experiments with two- and three-dimensional examples.

A low-cost, parameter-free, and pressure-robust enriched Galerkin method for the Stokes equations

TL;DR

The paper addresses efficiently solving the Stokes equations with a parameter-free, pressure-robust scheme. It develops a modified enriched Galerkin (mEG) method that replaces velocity derivatives with weak derivatives, removing the need for penalty parameters while preserving stability and optimal convergence. A velocity reconstruction-based pressure-robust variant (PR-mEG) is then introduced, delivering velocity errors that are independent of the viscosity and reduced pressure error as decreases. Numerical experiments in two and three dimensions confirm the theoretical results, demonstrating stable parameter-free performance and clear advantages in velocity accuracy for small viscosities.

Abstract

In this paper, we propose a low-cost, parameter-free, and pressure-robust Stokes solver based on the enriched Galerkin (EG) method with a discontinuous velocity enrichment function. The EG method employs the interior penalty discontinuous Galerkin (IPDG) formulation to weakly impose the continuity of the velocity function. However, the symmetric IPDG formulation, despite of its advantage of symmetry, requires a lot of computational effort to choose an optimal penalty parameter and to compute different trace terms. In order to reduce such effort, we replace the derivatives of the velocity function with its weak derivatives computed by the geometric data of elements. Therefore, our modified EG (mEG) method is a parameter-free numerical scheme which has reduced computational complexity as well as optimal rates of convergence. Moreover, we achieve pressure-robustness for the mEG method by employing a velocity reconstruction operator on the load vector on the right-hand side of the discrete system. The theoretical results are confirmed through numerical experiments with two- and three-dimensional examples.
Paper Structure (17 sections, 11 theorems, 99 equations, 8 figures, 3 tables, 3 algorithms)

This paper contains 17 sections, 11 theorems, 99 equations, 8 figures, 3 tables, 3 algorithms.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. A Modified Enriched Galerkin Method
  4. Standard enriched Galerkin method with interior penalty
  5. Modified enriched Galerkin method with weak derivatives
  6. Well-Posedness and Error Analysis
  7. Well-posedness
  8. Error estimates
  9. A Pressure-Robust Modified Enriched Galerkin Method
  10. Numerical Experiments
  11. Two dimensional examples
  12. Parameter-free test
  13. Pressure-robustness test
  14. Three dimensional examples
  15. Parameter-free test
  16. ...and 2 more sections

Key Result

Lemma 4.1

For any ${\bf v}\in{\bf V}_h$, there are positive constants $\gamma_*$ and $\gamma^*$ independent of $h:=\max_{T\in\mathcal{T}_h}h_T$ such that

Figures (8)

  • Figure 1: Errors and condition numbers of EG and mEG for $0.1\leq\rho,\rho_\texttt{m}\leq 5$ ($\nu=1$, $h=1/16$).
  • Figure 2: Comparison of the numerical solutions with $h=1/16$ and $\nu=1$.
  • Figure 3: Comparison of EG and mEG for $0.1\leq\rho,\rho_\texttt{m}\leq 5$ on various meshes ($\nu=1$).
  • Figure 4: Error profiles of the mEG and PR-mEG methods with varying $\nu$ values and a fixed mesh size $h=1/32$.
  • Figure 5: Errors and condition numbers of EG and mEG for $0.1\leq\rho\leq 5$ ($\nu=1$, $h=1/4$) in the 3D case.
  • ...and 3 more figures

Theorems & Definitions (26)

  • Definition 3.1
  • Remark 3.2
  • Remark 3.3
  • Remark 3.4
  • Lemma 4.1
  • proof
  • Lemma 4.2
  • proof
  • Lemma 4.3
  • proof
  • ...and 16 more