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Fully discrete error analysis of finite element discretizations of time-dependent Stokes equations in a stream-function formulation

Dmitriy Leykekhman, Boris Vexler, Jakob Wagner

TL;DR

The paper develops a rigorous fully discrete Galerkin framework for the time-dependent Stokes problem in a stream-function formulation, employing discontinuous Galerkin in time and a general spatial discretization that satisfies Galerkin orthogonality. It proves a best-approximation type error bound for the gradient of the stream-function error, $\ abla(\psi-\psi_{kh})$, in terms of spatial and temporal projection errors, and demonstrates optimal convergence rates for conforming $C^1$ and $C^0$ interior-penalty spaces under minimal regularity assumptions. The analysis leverages a robust stability theory and a duality-based error decomposition, and confirms that the stream-function approach yields pressure-robust velocity errors consistent with the best-approximation framework. The numerical results corroborate the theoretical rates and illustrate the method's effectiveness for optimal control contexts and general space discretizations.

Abstract

In this paper we establish best approximation type error estimates for the fully discrete Galerkin solutions of the time-dependent Stokes problem using the stream-function formulation. For the time discretization we use the discontinuous Galerkin method of arbitrary degree, whereas we present the space discretization in a general framework. This makes our result applicable for a wide variety of space discretization methods, provided some Galerkin orthogonality conditions are satisfied. As an example, conformal $C^1$ and $C^0$ interior penalty methods are covered by our analysis. The results do not require any additional regularity assumptions beyond the natural regularity given by the domain and data and can be used for optimal control problems.

Fully discrete error analysis of finite element discretizations of time-dependent Stokes equations in a stream-function formulation

TL;DR

The paper develops a rigorous fully discrete Galerkin framework for the time-dependent Stokes problem in a stream-function formulation, employing discontinuous Galerkin in time and a general spatial discretization that satisfies Galerkin orthogonality. It proves a best-approximation type error bound for the gradient of the stream-function error, , in terms of spatial and temporal projection errors, and demonstrates optimal convergence rates for conforming and interior-penalty spaces under minimal regularity assumptions. The analysis leverages a robust stability theory and a duality-based error decomposition, and confirms that the stream-function approach yields pressure-robust velocity errors consistent with the best-approximation framework. The numerical results corroborate the theoretical rates and illustrate the method's effectiveness for optimal control contexts and general space discretizations.

Abstract

In this paper we establish best approximation type error estimates for the fully discrete Galerkin solutions of the time-dependent Stokes problem using the stream-function formulation. For the time discretization we use the discontinuous Galerkin method of arbitrary degree, whereas we present the space discretization in a general framework. This makes our result applicable for a wide variety of space discretization methods, provided some Galerkin orthogonality conditions are satisfied. As an example, conformal and interior penalty methods are covered by our analysis. The results do not require any additional regularity assumptions beyond the natural regularity given by the domain and data and can be used for optimal control problems.

Paper Structure

This paper contains 17 sections, 20 theorems, 221 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Continuous problem
  3. Notation
  4. Function spaces
  5. Results on the time-dependent Stokes problem
  6. Results for stationary biharmonic equation
  7. Results for time-dependent problems
  8. Interpolated Regularity
  9. Fully discrete approximation
  10. Spatial discretization
  11. Full space-time discretization
  12. Fully discrete error analysis
  13. Specific spatial finite element spaces
  14. Conformal $C^1$ elements
  15. The $C^0$ interior penalty method
  16. ...and 2 more sections

Key Result

Lemma 2.1

On the function space $H^2_0(\Omega)$, the norm $\lVert v\rVert_{H^2_0(\Omega)} := \lVert\Delta v\rVert_{L^2(\Omega)}$ is equivalent to $\lVert v\rVert_{H^2(\Omega)}$.

Figures (1)

  • Figure 1: Convergence of discretization with MINI element for different right hand sides (black/red) and interior penalty method for stream-function formulation (blue).

Theorems & Definitions (43)

  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof
  • Proposition 2.3
  • Lemma 2.4
  • proof
  • Proposition 2.5
  • Corollary 2.6
  • Lemma 2.7
  • ...and 33 more