Element residual error estimate for the finite volume method

Out of the wide range of a-posteriori error estimates for the finite element method (FEM), the group of estimates based on the element residual seems to be the most popular. One recent extension of the element residual method is the element residual error estimate (EREE) [Numerische Mathematik 65 (1993) 23], which includes the elements of the duality theory and consistently produces good results. In this paper, the EREE will be extended to allow its use in conjunction with the finite volume (FV) type of discretisation. The extension consists of three parts: an appropriate definition of the residual in the FV framework, a procedure for calculation of self-equilibrating fluxes based on the conservative properties of the FV solution and a simplified solution method for the Local Problem. The paper covers the extensions of the EREE to the convection-diffusion and the Navier-Stokes problem, following [Comp Meth Appl Mech Engng 101 (1992) 73] and [Comp Meth Appl Mech Engng 111 (1994) 185], respectively. The error estimate is tested on three test cases with analytical solutions, where its performance is shown to be similar to its FEM counterpart. Finally, the estimate is applied to a realistic laminar fluid flow problem. (C) 2002 Elsevier Science Ltd. All rights reserved.