A novel approach of superconvergence analysis of the bilinear-constant scheme for time-dependent Stokes equations
type: Item
instance of: scholarly article
title: A novel approach of superconvergence analysis of the bilinear-constant scheme for time-dependent Stokes equations
zbMATH Open document ID: 1486.76062
author: Huaijun Yang
published in: Applied Numerical Mathematics
publication date: 2022-02-03T00:00:00Z
review text: The paper considers the superconvergence of a low-order fully-discretized time-dependent Stokes discretization in two space dimensions. In the introduction, the time-dependent Stokes problem is discussed with its importance as a stepping stone for the Navier-Stokes equations. The author mentions various publications on different discretization techniques for the non-transient Stokes equations, including superconvergence results for a bilinear-constant finite element pair on two-dimensional rectangular meshes. The main result of this paper is to extend this result to the time-dependent Stokes equations discretized by a backward Euler time discretization. The second chapter states some preliminaries detailing the assumptions on mesh and ansatz functions as well as introducing a crucial interpolation operators for velocity and pressure. Furthermore, well-known interpolation results for a Stokes projection operator are stated. Finally, the results for the interpolation operators originally proven by \textit{J. Pan} [SIAM J. Numer. Anal. 34, No. 6, 2424--2430 (1997; Zbl 0894.76045)] are formulated in Lemma 2.1. With these preliminaries, the author proves in Lemma 2.2 error bounds between the Stokes projection operator and the interpolation operator making use of a discrete inf-sup condition. Lemma 2.3 then introduces a discrete Gronwall inequality, and finally the fully discretized weak form (using a backward Euler scheme) is presented. The third section is concerned with the paper's main result, superconvergence for the proposed scheme. Theorem 3.1. provides error bounds for the fully discretized scheme not taking any postprocessing into account. The proof uses the results from Section 2, together with standard techniques for error estimates of the time discretization, so that the error bounds with respect to time are a direct consequence of the approximation properties of the Stokes projection operator. Using the post-processing approach previously discussed in the pub…
reviewed by: Daniel Arndt
Mathematics Subject Classification ID: 76M10
Mathematics Subject Classification ID: 76M20
Mathematics Subject Classification ID: 76D07
Mathematics Subject Classification ID: 65M12
zbMATH DE Number: 7468863
zbMATH Keywords: time-dependent Stokes equations
zbMATH Keywords: bilinear-constant finite element space
zbMATH Keywords: supercloseness
zbMATH Keywords: energy norm
zbMATH Keywords: backward Euler time discretization
zbMATH Keywords: interpolation operator
MaRDI profile type: Publication
full work available at URL: j.apnum.2021.12.003
OpenAlex ID: W4200516922
cites work: Q5318420
cites work: A finite element pressure gradient stabilization for the Stokes equations based on local projections
cites work: Finite difference streamline diffusion method using nonconforming space for incompressible time-dependent Navier-Stokes equations
cites work: A stabilized finite element method for the Stokes problem based on polynomial pressure projections
cites work: Optimal error analysis of Galerkin FEMs for nonlinear Joule heating equations
cites work: A Note on Parallel Preconditioning for the All-at-Once Solution of Riesz Fractional Diffusion Equations
cites work: Efficient Preconditioned Iterative Linear Solvers for 3-D Magnetostatic Problems Using Edge Elements
cites work: Stabilized finite element method for the non-stationary Navier-Stokes problem
cites work: Finite-Element Approximation of the Nonstationary Navier–Stokes Problem. Part IV: Error Analysis for Second-Order Time Discretization
cites work: Q5716601
cites work: Q5866757
cites work: A stabilized finite element method for the time-dependent Stokes equations based on Crank-Nicolson scheme
cites work: Finite Element Methods for Incompressible Flow Problems
cites work: Unconditional convergence and superconvergence analysis for the transient Stokes equations with damping
cites work: A new local stabilized nonconforming finite element method for the Stokes equations
cites work: A new stabilized finite volume method for the stationary Stokes equations
cites work: A stabilized finite element method based on two local Gauss integrations for the Stokes equations
cites work: A pressure-Poisson stabilized finite element method for the non-stationary Stokes equations to circumvent the inf-sup condition
cites work: A new stabilized finite element method for the transient Navier-Stokes equations
cites work: Superconvergence of a stabilized finite element approximation for the Stokes equations using a local coarse mesh L 2 projection
cites work: Superconvergence analysis of the nonconforming quadrilateral linear-constant scheme for Stokes equations
cites work: Global Superconvergence for the Bilinear-Constant Scheme for the Stokes Problem
cites work: Stability of time-splitting schemes for the Stokes problem with stabilized finite elements
cites work: Simple nonconforming quadrilateral Stokes element
cites work: High accuracy analysis of a new nonconforming mixed finite element scheme for Sobolev equations
cites work: Unconditional superconvergence analysis of a new mixed finite element method for nonlinear Sobolev equation
cites work: Stabilised bilinear-constant velocity-pressure finite elements for the conjugate gradient solution of the Stokes problem
cites work: Two-level stabilized finite element method for the transient Navier–Stokes equations
cites work: Low order nonconforming mixed finite element method for nonstationary incompressible Navier-Stokes equations
cites work: A new streamline diffusion finite element method for the generalized Oseen problem
cites work: A Modified Weak Galerkin Method for Stokes Equations
DOI: 10.1016/J.APNUM.2021.12.003
description: scientific article; zbMATH DE number 7468863