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Optimal pressure approximation for the nonstationary Stokes problem by a variational method in time with post-processing

Mathias Anselmann, Markus Bause, Gunar Matthies, Friedhelm Schieweck

TL;DR

The paper develops a space-time variational framework for the nonstationary Stokes equations, employing higher-order finite elements in space and a first-order Galerkin-Petrov discretization in time that enforces global continuity of the velocity while allowing discontinuities in the pressure at interval endpoints. A key contribution is the introduction of a pressure post-processing strategy that yields optimal second-order convergence in time and optimal spatial accuracy, implemented via two variants: collocation-based post-processing (leading to a globally smooth velocity and pressure trajectory) and interpolation-based post-processing (which preserves pressure accuracy through values at midpoints). The velocity discretization achieves optimal-order error in time and space, and the pressure is shown to have a unique mid-interval value while the endpoint pressure degree of freedom remains free, enabling targeted post-processing to recover optimal L2-pressure error. The theoretical developments are supported by numerical experiments demonstrating the predicted convergence rates and the effectiveness of the post-processing schemes in delivering high-accuracy pressure reconstructions.

Abstract

We provide an error analysis for the solution of the nonstationary Stokes problem by a variational method in space and time. We use finite elements of higher order for the approximation in space and a Galerkin-Petrov method with first order polynomials for the approximation in time. We require global continuity of the discrete velocity trajectory in time, while allowing the discrete pressure trajectory to be discontinuous at the endpoints of the time intervals. We show existence and uniqueness of the discrete velocity solution, characterize the set of all discrete pressure solutions and prove an optimal second order estimate in time for the pressure error in the midpoints of the time intervals. The key result and innovation is the construction of approximations to the pressure trajectory by means of post-processing together with the proof of optimal order error estimates. We propose two variants for a post-processed pressure within the set of pressure solutions based on collocation techniques or interpolation. Both variants guarantee that the pressure error measured in the L2-norm converges with optimal second order in time and optimal order in space. For the discrete velocity solution, we prove error estimates of optimal order in time and space. We present some numerical tests to support our theoretical results.

Optimal pressure approximation for the nonstationary Stokes problem by a variational method in time with post-processing

TL;DR

The paper develops a space-time variational framework for the nonstationary Stokes equations, employing higher-order finite elements in space and a first-order Galerkin-Petrov discretization in time that enforces global continuity of the velocity while allowing discontinuities in the pressure at interval endpoints. A key contribution is the introduction of a pressure post-processing strategy that yields optimal second-order convergence in time and optimal spatial accuracy, implemented via two variants: collocation-based post-processing (leading to a globally smooth velocity and pressure trajectory) and interpolation-based post-processing (which preserves pressure accuracy through values at midpoints). The velocity discretization achieves optimal-order error in time and space, and the pressure is shown to have a unique mid-interval value while the endpoint pressure degree of freedom remains free, enabling targeted post-processing to recover optimal L2-pressure error. The theoretical developments are supported by numerical experiments demonstrating the predicted convergence rates and the effectiveness of the post-processing schemes in delivering high-accuracy pressure reconstructions.

Abstract

We provide an error analysis for the solution of the nonstationary Stokes problem by a variational method in space and time. We use finite elements of higher order for the approximation in space and a Galerkin-Petrov method with first order polynomials for the approximation in time. We require global continuity of the discrete velocity trajectory in time, while allowing the discrete pressure trajectory to be discontinuous at the endpoints of the time intervals. We show existence and uniqueness of the discrete velocity solution, characterize the set of all discrete pressure solutions and prove an optimal second order estimate in time for the pressure error in the midpoints of the time intervals. The key result and innovation is the construction of approximations to the pressure trajectory by means of post-processing together with the proof of optimal order error estimates. We propose two variants for a post-processed pressure within the set of pressure solutions based on collocation techniques or interpolation. Both variants guarantee that the pressure error measured in the L2-norm converges with optimal second order in time and optimal order in space. For the discrete velocity solution, we prove error estimates of optimal order in time and space. We present some numerical tests to support our theoretical results.
Paper Structure (10 sections, 3 theorems, 35 equations)

This paper contains 10 sections, 3 theorems, 35 equations.

Table of Contents

  1. Introduction
  2. Notation and preliminaries
  3. Continuous function spaces
  4. Discrete function spaces and operators
  5. Quadrature formulas, interpolation operators and Stokes projection
  6. Proof.
  7. Proof.
  8. Proof.
  9. Approximation scheme and first error estimates
  10. A linear in time Galerkin method with finite elements in space

Key Result

Lemma 2.2

Let $H\subset L^2(\Omega)$ be a Hilbert space with inner product denoted by $\langle \cdot, \cdot \rangle_H$. For $v\in C(\overline{I};H)$, the function $R_\tau^1 v \in C(\overline I;H)$ is continuous in time on $\overline I$ with $R_\tau^1 v(t_n) = v(t_n)$ for all $n=0,\ldots, N$.

Theorems & Definitions (5)

  • Lemma 2.2
  • Lemma 2.3
  • Definition 2.4: Stokes projection
  • Remark 2.5
  • Lemma 2.6: Error estimates for the Stokes projection