Galerkin projected residual method applied to diffusion–reaction problems

Abstract

A stabilized finite element method is presented for scalar and linear second-order boundary value problems. The method is obtained by adding to the Galerkin formulation multiple projections of the residual of the differential equation at element level. These multiple projections allow the generation of appropriate number of free stabilization parameters in the element matrix depending on the local space of approximation and on the differential operator. The free parameters can be determined imposing some convergence and/or stability criteria or by postulating the element matrix with the desired stability properties. The element matrix of most stabilized methods (such as, GLS and GGLS methods) can be obtained using this new method with appropriate choices of the stabilization parameters. We applied this formulation to diffusion–reaction problems. Optimal rates of convergency are numerically observed for regular solutions.