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An adaptive finite element method for a time‐dependent Stokes problem
Author(s) -
Prato Torres Ricardo,
Domínguez Catalina,
Díaz Stiven
Publication year - 2019
Publication title -
numerical methods for partial differential equations
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.901
H-Index - 61
eISSN - 1098-2426
pISSN - 0749-159X
DOI - 10.1002/num.22302
Subject(s) - mathematics , estimator , finite element method , discontinuous galerkin method , a priori and a posteriori , discretization , dirichlet boundary condition , galerkin method , superconvergence , residual , adaptive mesh refinement , boundary value problem , space time , boundary (topology) , mathematical optimization , mathematical analysis , algorithm , physics , thermodynamics , philosophy , statistics , computational science , epistemology , chemical engineering , engineering
In this article, we conduct an a posteriori error analysis of the two‐dimensional time‐dependent Stokes problem with homogeneous Dirichlet boundary conditions, which can be extended to mixed boundary conditions. We present a full time–space discretization using the discontinuous Galerkin method with polynomials of any degree in time and the ℙ 2 − ℙ 1 Taylor–Hood finite elements in space, and propose an a posteriori residual‐type error estimator. The upper bounds involve residuals, which are global in space and local in time, and an L 2 ‐error term evaluated on the left‐end point of time step. From the error estimate, we compute local error indicators to develop an adaptive space/time mesh refinement strategy. Numerical experiments verify our theoretical results and the proposed adaptive strategy.