We consider the mixed covolume method combining with the expanded mixed element for a system of first-order partial differential equations resulting from the mixed formulation of a general self-adjoint elliptic problem with a full diffusion tensor. The system can be used to model the transport of a contaminant carried by a flow in porous media. We use the lowest order Raviart-Thomas mixed element space. We show the first-order error estimate for the approximate solution in L2 norm. We show the superconvergence both for pressure and velocity in certain discrete norms. We also get a finite difference scheme by using proper approximate integration formulas. Finally we give some numerical examples.

Two least-squares Galerkin finite element schemes are formulated to solve parabolic integro-differential equations. The advantage of this method is that it is not subject to the LBB condition. The convergence analysis shows that the least-squares mixed element schemes yield the approximate solution with optimal accuracy in H(div; Ω)×H1(Ω)and (L2(Ω))2×L2(Ω), respectively.

Consider the following convection di.usion equation,{∂u/∂t+ b1(x; y) ∂u/∂x+ b2(y) ∂u/∂y-(a1∂2u/∂x2 + a2∂2u/∂y2) = f in Ω×J, u(x; y; t) = φ(x; t) onΩ×J,u(x; y; 0) = u0(x; y) inΩ(1)} whereΩ= (0, 1) × (0, 1), J = (0, T), b1(x; y), b2(y) are smooth functions and a1; a2 are positive constants. When convection dominates di.usion, i.e. 0 < a1; a2 << |b|, the general finite di.erence or finite element methods often result numerical oscillation[1]. The upwind method is an e¡Àcient method but is only first order accurate. For one dimensional stable problem with constant coe¡Àcient, [3] presented a high order upwind scheme, but it is di¡Àcult to extend the method to variable coe¡Àcient problem and two dimensional problem. In this paper we give an alternative direction iterative method combining with one dimensional second order upwind scheme for two dimensional problem. It can be as a high speed algorithm on parallel computer. The maximum principle and the second order uniform norm error estimate are obtained. Finally we give some numerical examples.

Based on domain decomposition, we give a multiplicative Schwarz domain decomposition method for semi-linear parabolic problems. We consider the relation between the convergence rate and discretization parameters, including the diameter of the subdomain. We give the error estimate, which tells us that the convergence of the approximate solution is independent of the iteration number at each time level. Finally we give a numerical example.

We present a mixed covolume method for a system of first order partial differential equations resulting from the mixed formulation of the general self-adjoint parabolic problem with a variable nondiagonal diffusion tensor. The lowest order Raviart-Thomas mixed element space on rectangles is used. We prove the first order optimal rate of convergence for approximate pressure as well as for approximate velocity. We also prove the second order superconvergence both for approximate velocity and pressure in certain discrete norms.