Solving mixed methods for second order elliptic problems

Abstract

In this paper we deal with a way of solving linear equations arising from the mixed formulation of second order elliptic problems. We follow the method devised by Arnold and Brezzi which leads to more tractable forms of linear equations. Thereby we show that the strongly indefinite linear systems of mixed methods can be reduced to symmetric and positive definite systems which are derived from certain modified conforming or nonconforming finite element methods. We carry out this analysis for the well-known RTN and BDM mixed elements. In particular, for the lowest-order RTN space on triangles we can apply previously known multigrid algorithms to this system.