Adaptive grid methods are established as valuable computational technique in approximating effectively the solutions of problems with boundary or interior layers. In this paper, we present the analysis of an upwind scheme for singularly perturbed differential-difference equation on a grid which is formed by equidistributing arc-length monitor function. It is shown that the discrete solution obtained converges uniformly with respect to the perturbation parameter. Numerical experiments illustrate in practice the result of convergence proved theoretically.

In this paper, a class of new immersed interface finite element methods (IIFEM) is developed to solve elasticity interface problems with homogeneous and non-homogeneous jump conditions in two dimensions. Simple non-body-fitted meshes are used. For homogeneous jump conditions, both non-conforming and conforming basis functions are constructed in such a way that they satisfy the natural jump conditions. For non-homogeneous jump conditions, a pair of functions that satisfy the same non-homogeneous jump conditions are constructed using a level-set representation of the interface. With such a pair of functions, the discontinuities across the interface in the solution and flux are removed; and an equivalent elasticity interface problem with homogeneous jump conditions is formulated. Numerical examples are presented to demonstrate that such methods have second order convergence.

In this paper, a-posteriori error estimators are proposed for the Legendre spectral Galerkin method for  two-point boundary value problems. The key idea is to postprocess the Galerkin approximation, and the analysis shows that the postprocess improves the order of convergence. Consequently, we obtain asymptotically exact a-posteriori error estimators based on the postprocessing results. Numerical examples are included to illustrate the theoretical analysis.

A modified polynomial preserving gradient recovery technique is proposed. Unlike the polynomial preserving gradient recovery technique, the gradient recovered with the modified polynomial preserving recovery (MPPR) is constructed element-wise, and it is discontinuous across the interior edges. One advantage of the MPPR technique is that the implementation is easier when adaptive meshes are involved. Superconvergence results of the gradient recovered with MPPR are proved for finite element methods for elliptic boundary problems and eigenvalue problems under adaptive meshes. The MPPR is applied to adaptive finite element methods to construct asymptotic exact a posteriori error estimates. Numerical tests are provided to examine the theoretical results and the effectiveness of the adaptive finite element algorithms.

The state equations of stochastic control problems, which are controlled stochastic differential equations, are proposed to be discretized by the weak midpoint rule and predictor-corrector methods for the Markov chain approximation approach. Local consistency of the methods are proved. Numerical tests on a simplified Merton's portfolio model show better simulation to feedback control rules by these two methods, as compared with the weak Euler-Maruyama discretisation used by Krawczyk. This suggests a new approach of improving accuracy of approximating Markov chains for stochastic control problems.

In this paper we use the simplex B-spline representation of polynomials or piecewise polynomials in terms of their polar forms to construct several differential or discrete bivariate quasi interpolants which have an optimal approximation order. This method provides an efficient tool for describing many approximation schemes involving values and (or) derivatives of a given function.