Extension of the Local Problem Error Estimate to the Finite Volume Discretisation

Out of the wide range of a-posteriori error estimates for the Finite Element Method (FEM) of discretisation, 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 Local Problem Error Estimate (LPEE) 1], which includes the elements of the duality theory and consistently produces good results with the ef-fectivity indices consistently close to unity. In this paper, LPEE 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 deenition of a residual in the FV framework, a procedure for the calculation of self-equilibrating uxes based on the conservative properties of the FV solution and a simpliied solution method for the Local Problem. The paper also covers the extensions of the Local Problem Error Estimate to the convection-diiusion and the Navier-Stokes problem, following 2] and 3], respectively. The Local Problem Error Estimate is tested on three test cases with analytical solutions, where its performance is shown to be similar as in the case of the Finite Element discretisation.