Div—Curl—Grad System

Abstract

In this chapter we study and compare different finite element methods for the solution of two- and three-dimensional diffusion-type problems usually governed by second-order elliptic partial differential equations. In the mixed Galerkin method, the second-order scalar equation is decomposed into a first-order grad—div system by introducing additional variables (the fluxes). The conventional least-squares finite element method is also based on the same grad—div system. We shall present theoretical analysis and numerical results to show that this simple procedure of reduction destroys ellipticity and thus the conventional LSFEM is not optimal, that is, the rate of convergence for the fluxes is one order lower than optimal. In order to have an optimal LSFEM, the div—curl—grad system should be employed.