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Supercloseness and asymptotic analysis of the Crouzeix-Raviart and enriched Crouzeix-Raviart elements for the Stokes problem

Wei Chen, Hao Han, Limin Ma

TL;DR

The paper develops two pseudostress interpolations that yield a full one-order supercloseness for the nonconforming CR and ECR elements in the Stokes problem, enabling a concise intrinsic asymptotic eigenvalue analysis and proving optimal eigenvalue superconvergence by extrapolation. It also proves optimal postprocessing superconvergence for the Stokes equation on uniform triangular grids and validates the results with numerical experiments. The analysis leverages the RT equivalence to avoid heavy discrete-pressure expansions and yields explicit eigenvalue error expansions and a practical extrapolation formula. Overall, the work improves accuracy for nonconforming Stokes discretizations and provides an effective extrapolation-based route to high-precision eigenvalues and postprocessed solutions.

Abstract

For the Crouzeix-Raviart and enriched Crouzeix-Raviart elements, asymptotic expansions of eigenvalues of the Stokes operator are derived by establishing two pseudostress interpolations, which admit a full one-order supercloseness with respect to the numerical velocity and the pressure, respectively. The design of these interpolations overcomes the difficulty caused by the lack of supercloseness of the canonical interpolations for the two nonconforming elements, and leads to an intrinsic and concise asymptotic analysis of numerical eigenvalues, which proves an optimal superconvergence of eigenvalues by the extrapolation algorithm. Meanwhile, an optimal superconvergence of postprocessed approximations for the Stokes equation is proved by use of this supercloseness. Finally, numerical experiments are tested to verify the theoretical results.

Supercloseness and asymptotic analysis of the Crouzeix-Raviart and enriched Crouzeix-Raviart elements for the Stokes problem

TL;DR

The paper develops two pseudostress interpolations that yield a full one-order supercloseness for the nonconforming CR and ECR elements in the Stokes problem, enabling a concise intrinsic asymptotic eigenvalue analysis and proving optimal eigenvalue superconvergence by extrapolation. It also proves optimal postprocessing superconvergence for the Stokes equation on uniform triangular grids and validates the results with numerical experiments. The analysis leverages the RT equivalence to avoid heavy discrete-pressure expansions and yields explicit eigenvalue error expansions and a practical extrapolation formula. Overall, the work improves accuracy for nonconforming Stokes discretizations and provides an effective extrapolation-based route to high-precision eigenvalues and postprocessed solutions.

Abstract

For the Crouzeix-Raviart and enriched Crouzeix-Raviart elements, asymptotic expansions of eigenvalues of the Stokes operator are derived by establishing two pseudostress interpolations, which admit a full one-order supercloseness with respect to the numerical velocity and the pressure, respectively. The design of these interpolations overcomes the difficulty caused by the lack of supercloseness of the canonical interpolations for the two nonconforming elements, and leads to an intrinsic and concise asymptotic analysis of numerical eigenvalues, which proves an optimal superconvergence of eigenvalues by the extrapolation algorithm. Meanwhile, an optimal superconvergence of postprocessed approximations for the Stokes equation is proved by use of this supercloseness. Finally, numerical experiments are tested to verify the theoretical results.
Paper Structure (13 sections, 15 theorems, 51 equations, 2 figures, 1 table)

This paper contains 13 sections, 15 theorems, 51 equations, 2 figures, 1 table.

Table of Contents

  1. Introduction
  2. Notations and preliminaries
  3. Supercloseness and superconvergence for the nonconforming elements
  4. Nonconforming and mixed element methods for the Stokes equation
  5. Supercloseness for the nonconforming elements
  6. Superconvergence for the nonconforming finite elements
  7. Asymptotic analysis for the Stokes eigenvalue problem
  8. Error expansions for eigenvalues
  9. Asymptotic expansion of interpolation error terms
  10. Asymptotic expansions of eigenvalues and extrapolation algorithm
  11. Numerical examples
  12. Example 1
  13. Example 2

Key Result

Lemma 2.1

Let $(\sigma^f, u^f)$ and $(\sigma_{\rm RT}^f, u_{\rm RT}^f)$ be the solutions of Problems mixCon and RTPi0u, respectively, with $(\sigma^f,u^f)\in W^{2,\infty}(\Omega,\mathbb{R}^{2\times 2})\times H^2(\Omega,\mathbb{R}^2)$. It holds on uniform triangulations with sufficiently small $h$

Figures (2)

  • Figure 1: Errors of CR element and ECR element for the Stokes source problem.
  • Figure 2: The relative errors of the multiple eigenvalues from CR element and ECR element.

Theorems & Definitions (18)

  • Lemma 2.1
  • Lemma 2.2
  • Theorem 2.3
  • Remark 2.4
  • Definition 2.5
  • Lemma 2.6
  • Theorem 2.7
  • Lemma 3.1
  • Lemma 3.2
  • Lemma 3.3
  • ...and 8 more