A Generalization of the Local Projection Stabilization for Convection-Diffusion-Reaction Equations

Abstract

We introduce a generalization of the local projection stabilization for steady scalar convection-diffusion-reaction equations which allows us to use local projection spaces defined on overlapping sets. This enables us to define the local projection method without the need of a mesh refinement or an enrichment of the finite element space and increases the robustness of the local projection method with respect to the choice of the stabilization parameter. The stabilization term is slightly modified, which leads to an optimal estimate of the consistency error even if the stabilization parameters scale correctly with respect to convection, diffusion, and mesh width. We prove that the bilinear form corresponding to the method satisfies an inf-sup condition with respect to the SUPG norm and establish an optimal error estimate in this norm. The theoretical considerations are illustrated by numerical results.