Mixed and nonconforming finite element methods : implementation, postprocessing and error estimates

Abstract

We discuss a technique of implementing certain mixed finite elements based on the use of Lagrange multipliers to impose interelement continuity. The matrices arising from this implementation are positive definite. Considering some well-known mixed methods, namely the Raviart-Thomas methods for second order elliptic problems and the Hellan-Hermann-Johnson method for biharmonic problems, we show that the computed Lagrange multipliers may be exploited in a simple postprocess to produce better approximation of the original variables. We further extablish an equivalence between the mixed methods and certain modified versions of well-known nonconforming methods, notably the Morley method in the case of the biharmonic problem. The equivalence is exploited to provide error estimates for both the mixed and nonconforming methods.