Abstract:

In this paper, two-dimensional parabolic equations are studied with mixed method. A type of  new mixed variational formulations, time-semi-discrete mixed equations, and fully discrete mixed finite element formulations are derived, where Brezzi--Babu\v{s}ka's condition is automatically satisfied. And some rigorous error analyses are provided. The degrees of freedom are not only minimum for these mixed finite element formulations, but their error estimates also optimal order. Moreover, it is shown that the methods here are the improvements and renovations for existing all methods.

Key words: Parabolic equations, New mixed variational formulation, New mixed finite element formulation, Error estimate