Two Mixed Finite Element Methods And Its Numerical Analysis

In the first part of this paper, with an H~1-Galerkin mixed finite element method combined with expanded method, we consider the second-order linear parabolic partial differential equationand the second-order semilinear parabolic partial differential equationBy introducing two medial variables, this method first split the initial problem into a first order system and then approximates the scalar unknown, its gradient and its flux (the coefficient times the gradient) optimally and stimultaneously. The approximating finite element spaces V_h and W_h are allowed to be of differing polynomial degrees for the proposed method. Moreover, complicated boundary problems and small-parameter problems can be solved by this method. We obtain the optimal order of convergence theoretically. A numerical example conforms the efficiency of our method.Then we consider the hyperbolic problem(a) pa - div(K(x)Vp + b(x)p) + c(x) = /, (a:, t) € ft x (0, T],(b) {K(x)Vp + b(x)p) ? n = 0, (x, t) € dft x [0, T],(c) p(x,0) =po(a), a;6fi,(d) pt(x,0) =Pi(x), xeQ, which is stimulated by the mixed covolume method. The lowest order R-T mixed element space on rectangles is used. We prove the optimal L2-norm error estimates for approximating pressure and approximating velocity.