Analysis of estimators for Stokes problem using a mixed approximation

In this work, we introduce the steady Stokes equations with a new boundary condition, generalizes the Dirichlet and the Neumann conditions. Then we derive an adequate variational formulation of Stokes equations. It includes algorithms for discretization by mixed finite element methods. We use a block diagonal preconditioners for Stokes problem. We obtain a faster convergence when applying the preconditioned MINRES method. Two types of a posteriori error indicator are introduced and are shown to give global error estimates that are equivalent to the true discretization error. In order to evaluate the performance of the method, the numerical results are compared with some previously published works or with others coming from commercial code like Adina system.

Superconvergence of the Space-Time Continuous Finite Elements with Interpolated Coefficients for Semilinear Parabolic Problems

In this article, a space-time continuous finite element method with interpolated coefficients for a class of semilinear parabolic problem is introduced and analyzed. Basic global error estimates are established under the convergence assumption for linear problem. Further application of the orthogonal expansion method which is to construct some superapproximate interpolating functions, the supperconvergence on mesh nodes is proved. Finally the result is tested by a numerical example.

Semiclassical Analysis of Discretizations of Schrödinger-type Equations

We apply Wigner-transform techniques to the analysis of difference methods for Schrödinger-type equations in the case of a small Planck constant. In this way we are able to obtain sharp conditions on the spatial-temporal grid which guarantee convergence for average values of observables as the Planck constant tends to zero. The theory developed in this paper is not based on local and global error estimates and does not depend on whether caustics develop or not.Numerical examples are presented to help interpret the theory.