We present a scheme for solving two-dimensional semilinear reaction-diffusion equations using a expanded mixed finite element method. To linearize the mixed-method equations, we use a two grid algorithms based on the Newton iteration method. The solution of nonlinear system on the fine space is reduced to the solution of two small (one linear and one nonlinear) systems on the coarse space and a linear system on the fine space. It is shown that the coarse grid can be much coarser than the fine grid and achieve asymptotically optimal approximation as long as the mesh sizes satisfy H=O (h13). As a result, solving such a large class of nonlinear equation will not be much more difficult than solving one single linearized equation.

A singularly perturbed two-point boundary value problem with an exponential boundary layer is solved numerically by using an adaptive grid method. The mesh is constructed adaptively by equidistributing a monitor function based on the arc-length of the exact solution. The error analysis for this approach was carried out by Qiu et al. (J. Comput. Appl. Math. 101 (1999) 1-25). In this work, their error bound will be improved to the optimal order which is independent of the perturbation parameter. The main ingredient used to obtain the improved result is the theory of the discrete Green's function.

The miscible displacement of one incompressible fluid by another in a porous medium is governed by a system of two equations. One is an elliptic equation for the pressure and the other is parabolic equation for the concentration of one of the fluids. Since the pressure appears in the concentration only through its velocity field, we choose a mixed finite element to approximate the pressure equation and for the concentration we use the standard Galerkin method. We shall obtain an explicit a posteriori error estimator in L2 (L2) for the semi-discrete scheme applied to the nonlinear coupled system.

A singularly perturbed two-point boundary value problem with an exponential boundary layer is solved numerically by using an adaptive grid method. The mesh is constructed adaptively by equidistributing a monitor function based on the arc-length of the approximated solutions. A first-order rate of convergence, independent of the perturbation parameter, is established by using the theory of the discrete Green's function. Unlike some previous analysis for the fully discretized approach, the present problem does not require the conservative form of the underlying boundary value problem.

In this paper, we investigate the full discretization of general convex optimal control problems using mixed finite element methods. The state and co-state are discretized by lowest order Raviart-Thomas element and the control is approximated by piecewise constant functions. We derive error estimates for both the control and the state approximation. Moreover, we present the superconvergence analysis for mixed finite element approximation of the optimal control problems.

We study the numerical approximation of convex optimal control problems governed by elliptic partial differential equations with oscillating coefficients. Since the objective functional contains flux, we approximate the problems using the mixed finite element methods. We first analyze the standard finite element approximation schemes. Then, motivated by the numerical simulation of the primal variable and the flux in highly heterogeneous porous media, we use a mixed multiscale finite element method for solving the state equations. The multiscale finite element bases are constructed by locally solving Dirichlet boundary value problems. The analysis of the approximate control problems is carried out under the assumption that the oscillating coefficients are locally periodic, which allows us to use homogenization theory to obtain the asymptotic structure of the solutions, although the numerical schemes are designed for general cases.

In this paper, we present an a posteriori error analysis for mixed finite element approximation of convex optimal control problems. We derive a posteriori error estimates for the coupled state and control approximations under some assumptions which hold in many applications. Such estimates can be used to construct reliable adaptive mixed finite elements schemes for the control problems.