Convergence of a linearly extrapolated BDF2 finite element scheme for viscoelastic fluid flow

Abstract The stability and convergence of a linearly extrapolated second order backward difference (BDF2-LE) time-stepping scheme for solving viscoelastic fluid flow in R d $\mathbb{R}^{d}$ , d = 2 , 3 $d=2,3$ , are presented in this paper. The time discretization is based on the implicit scheme for...

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Main Authors: Yunzhang Zhang, Chao Xu, Jiaquan Zhou
Format: Article
Language:English
Published: SpringerOpen 2017-09-01
Series:Boundary Value Problems
Subjects:
Online Access:http://link.springer.com/article/10.1186/s13661-017-0872-z
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author Yunzhang Zhang
Chao Xu
Jiaquan Zhou
author_facet Yunzhang Zhang
Chao Xu
Jiaquan Zhou
author_sort Yunzhang Zhang
collection DOAJ
description Abstract The stability and convergence of a linearly extrapolated second order backward difference (BDF2-LE) time-stepping scheme for solving viscoelastic fluid flow in R d $\mathbb{R}^{d}$ , d = 2 , 3 $d=2,3$ , are presented in this paper. The time discretization is based on the implicit scheme for the linear term and the two-step linearly extrapolated scheme for the nonlinear term. Mixed finite element (MFE) method is applied for the spatial discretization. The approximations of stress tensor σ, velocity vector u and pressure p are P m $P_{m}$ -discontinuous, P k $P_{k}$ -continuous and P q $P_{q}$ -continuous elements, respectively. Upwinding needed for convection of σ is made by a discontinuous Galerkin (DG) FE method. For the time step △t small enough, the existence of an approximate solution is proven. If m , k ⩾ d 2 $m, k \geqslant \frac{d}{2}$ , q + 1 ⩾ d 2 $q+1\geqslant\frac{d}{2}$ , and △ t ⩽ C 0 h d 4 $\triangle t \leqslant C_{0} h^{\frac{d}{4}}$ , then the discrete H 1 $H^{1}$ and L 2 $L^{2}$ errors for the velocity and stress, and L 2 $L^{2}$ error for the pressure, are bounded by C ( △ t 2 + h min { m , k , q + 1 } ) $C(\triangle t^{2}+h^{\min\{m,k,q+1\}})$ , where h denotes the mesh size. The derived theoretical results are supported by numerical tests.
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spelling doaj.art-0a20e251f2404fa689b0e9acca78be412022-12-22T03:55:13ZengSpringerOpenBoundary Value Problems1687-27702017-09-012017113510.1186/s13661-017-0872-zConvergence of a linearly extrapolated BDF2 finite element scheme for viscoelastic fluid flowYunzhang Zhang0Chao Xu1Jiaquan Zhou2School of Mathematics and Statistics, Henan University of Science and TechnologyFaculty of Mathematics and Physics Education, Luoyang Institute of Science and TechnologyFaculty of Mathematics and Physics Education, Luoyang Institute of Science and TechnologyAbstract The stability and convergence of a linearly extrapolated second order backward difference (BDF2-LE) time-stepping scheme for solving viscoelastic fluid flow in R d $\mathbb{R}^{d}$ , d = 2 , 3 $d=2,3$ , are presented in this paper. The time discretization is based on the implicit scheme for the linear term and the two-step linearly extrapolated scheme for the nonlinear term. Mixed finite element (MFE) method is applied for the spatial discretization. The approximations of stress tensor σ, velocity vector u and pressure p are P m $P_{m}$ -discontinuous, P k $P_{k}$ -continuous and P q $P_{q}$ -continuous elements, respectively. Upwinding needed for convection of σ is made by a discontinuous Galerkin (DG) FE method. For the time step △t small enough, the existence of an approximate solution is proven. If m , k ⩾ d 2 $m, k \geqslant \frac{d}{2}$ , q + 1 ⩾ d 2 $q+1\geqslant\frac{d}{2}$ , and △ t ⩽ C 0 h d 4 $\triangle t \leqslant C_{0} h^{\frac{d}{4}}$ , then the discrete H 1 $H^{1}$ and L 2 $L^{2}$ errors for the velocity and stress, and L 2 $L^{2}$ error for the pressure, are bounded by C ( △ t 2 + h min { m , k , q + 1 } ) $C(\triangle t^{2}+h^{\min\{m,k,q+1\}})$ , where h denotes the mesh size. The derived theoretical results are supported by numerical tests.http://link.springer.com/article/10.1186/s13661-017-0872-zviscoelastic fluid flowlinearly extrapolated BDF2mixed finite elementdiscontinuous Galerkinstability analysiserror estimate
spellingShingle Yunzhang Zhang
Chao Xu
Jiaquan Zhou
Convergence of a linearly extrapolated BDF2 finite element scheme for viscoelastic fluid flow
Boundary Value Problems
viscoelastic fluid flow
linearly extrapolated BDF2
mixed finite element
discontinuous Galerkin
stability analysis
error estimate
title Convergence of a linearly extrapolated BDF2 finite element scheme for viscoelastic fluid flow
title_full Convergence of a linearly extrapolated BDF2 finite element scheme for viscoelastic fluid flow
title_fullStr Convergence of a linearly extrapolated BDF2 finite element scheme for viscoelastic fluid flow
title_full_unstemmed Convergence of a linearly extrapolated BDF2 finite element scheme for viscoelastic fluid flow
title_short Convergence of a linearly extrapolated BDF2 finite element scheme for viscoelastic fluid flow
title_sort convergence of a linearly extrapolated bdf2 finite element scheme for viscoelastic fluid flow
topic viscoelastic fluid flow
linearly extrapolated BDF2
mixed finite element
discontinuous Galerkin
stability analysis
error estimate
url http://link.springer.com/article/10.1186/s13661-017-0872-z
work_keys_str_mv AT yunzhangzhang convergenceofalinearlyextrapolatedbdf2finiteelementschemeforviscoelasticfluidflow
AT chaoxu convergenceofalinearlyextrapolatedbdf2finiteelementschemeforviscoelasticfluidflow
AT jiaquanzhou convergenceofalinearlyextrapolatedbdf2finiteelementschemeforviscoelasticfluidflow