On stabilized finite element methods for linear systems of convection-diffusion-reaction equations

Abstract

A stabilized finite element method for solving systems of convection-diffusion-reaction equations is studied in this paper. The method is based on the subgrid scale approach and an algebraic approximation to the subscales. After presenting the formulation of the method, it is analyzed how it behaves under changes of variables, showing that it relies on the law of change of the matrix of stabilization parameters associated to the method. An expression for this matrix is proposed for the case of general coupled systems of equations that is an extension of the expression proposed for a 1D model problem. Applications of the stabilization technique to the Stokes problem with convection and to the bending of Reissner-Mindlin plates are discussed next. The design of the matrix of stabilization parameters is based on the identification of the stability deficiencies of the standard Galerkin method applied to these two problems.