An adaptive finite element method for a time‐dependent Stokes problem

Abstract

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 ℙ2 − ℙ1 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 L2‐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.