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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SpringerOpen
2017-09-01
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Series: | Boundary Value Problems |
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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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issn | 1687-2770 |
language | English |
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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 |