Analysis of Finite Element Approximation of Evolution Problems in Mixed Form

Abstract

This paper deals with the finite element approximation of evolution problems in mixed form. Following [D. Boffi, F. Brezzi, and L. Gastaldi, Math. Comp., 69 (2000), pp. 121--140], we handle separately two types of problems. A model for the first case is the heat equation in mixed form, while the time dependent Stokes problem fits within the second one. For either case, we give sufficient conditions for a good approximation in the natural functional spaces. The results are not obvious in the first situation. In this case, the well-known conditions for the well posedness and convergence of the corresponding steady problem are not sufficient for the good approximation of the time dependent problem. This issue is demonstrated with a numerical (counter-) example and justified analytically.