The Mixed Finite Element Method And Numerical Analysis For Two Kinds Of Quasilinear Evolution Equations

In this paper , we consider the Expanded Mixed Finite Element Method and mixed covolume method for the quasilinear parabolic integro-differential equation and quasilinear parabolic problem. The Expanded Mixed Finite Element Method expands the standard mixed formulation in the sense that three variable are explicitly treated:the scalar unknwon,its gradient and its flux. Optimal order error estimates for the scalar unknown,its gradient and its flux in L~2-norms are obtained. The mixed covolume method was introduced by Russel[ll] and has been extensively tested by Jones et al[12,13]. Optimal order error estimates for the scalar unknown in L~2-norms are obtained.In chapter one,we consider the mixed covolume method for the following quasilinear parabolic problemIn this chapter,we propose the mixed covolume method for the parabolic problem ,Optimal order error estimates for the scaler unknown,its gradient and its flux in L~2-norms are obtained.In chapter two ,we consider the Expanded Mixed Finite Method for the following quasilinear parabolic integro-differential equation.(a) ut - V.{a(u)Vw + /04 b(x, t, r, u(x, r))Vu(x, r)dr}+c(u) = f(x,t), (x,t) efix J, (b)u(x,O) =uq(x), xefi, (c) u(x,t) = —5, x e dQ. x J.In this chapter,we propose the expanded mixed formulation for the parabolic problem ,and prove that the discrete formulation has a unique solution. Optimal order error datimates for the scalar unknown, ita gradient and its flux in .L2-norms are obtained.