Stabilized element residual method (SERM): A posteriori error estimation for the advection-diffusion equation

Residual-based a posteriori error estimation techniques have been developed for linear elliptic symmetric positive-definite problems. One asymptotically-exact error estimator for the elliptic Laplacian operator relies on solving local Neumann problems in each element. This technique is extended to the unsymmetric and positive semi-definite advection-diffusion (AD) operator. Here, a novel approach, the Stabilized Element Residual Method (SERM) is presented. In this method, the unsymmetric advection terms are retained in the formulation of the local error problem through the use of stabilized methods. The selection of the optimum stabilization parameter is discussed.