Mixed Finite Element Methods for the Unsteady Partial Differential Equations

In real-world scientific engineering, a great number of problems are dynamic, namely the corresponding partial differential equations (PDEs) are unsteady. To this end, in this chapter, we will focus on the nonstandard finite element (FE) methods, namely the mixed FE (MFE) methods for the unsteady PDEs such as the unsteady Burgers equation, the regularized long wave (RLW) equation, the unsaturated flow equation, the unsteady Boussinesq equation, and the improved system of time-domain Maxwell’s equations, and use some numerical examples to verify the correction of the MFE methods. There are two approaches to create the fully discretized MFE (FDMFE) format. One is first to discretize spatial variable by the FE method, and then, to discretize time derivative by the finite difference (FD) method. Another is first to discretize time derivative by the FD method, and then, to discretize spatial variable by the FE method. By comparing with two approaches we find that the latter seems more convenient than the former.