Mixed Element Methods For Two Classes Of Partial Differential Equations

In this paper, first, a standard mixed finite element method is proposed to investi-gate the convergence of the initial-boundary value problem of pseudo-hyperbolic integro-differential equation based on the Raviart-Thomas space Vh×Wh.Compared with the usual finite element method, the unknown scalar and the adjoint vector function are approximated optimally and simultaneously. By introducing the projection of generalized mixed element, optimal order L2 estimates are obtained for the approximation of the unknown functions u,ut,uu, the associated velocityσand divσ. L∞estimates are also obtained for the ap-proximations of u andσ; And second, a new H1-Galerkin mixed finite element scheme is selected for Sobolev equation. Optimal error estimates are derived in both semidiscrete and fully discrete case in one space dimension, an extension to problems are also discussed in two-and three space variables. The argumentation shows that the method dose not require the LBB consistency condition.