Guangfu Sun’s research while affiliated with University College Cork and other places

This paper considers a simple central dierence scheme for a singu- larly perturbed semilinear reaction{diusion problem, which may have multi- ple solutions. Asymptotic properties of solutions to this problem are discussed and analyzed. To compute accurate approximations to these solutions, we consider a piecewise equidistant mesh of Shishkin type, which contains O(N) points. On such a mesh, we prove existence of a solution to the discretization and show that it is accurate of order N 2 ln2N, in the discrete maximum norm, where the constant factor in this error estimate is independent of the perturbation parameter " and N. Numerical results are presented that verify this rate of convergence.

This paper is concerned with a high order convergent discretization for the semilinear reaction-diffusion problem: , for , subject to , where . We assume that on , which guarantees uniqueness of a solution to the problem. Asymptotic properties of this solution are discussed. We consider a polynomial-based three-point difference scheme on a simple piecewise equidistant mesh of Shishkin type. Existence and local uniqueness of a solution to the scheme are analysed. We prove that the scheme is almost fourth order accurate in the discrete maximum norm, uniformly in the perturbation parameter . We present numerical results in support of this result.

We consider linear second order singularly perturbed two-point boundary value problems with interior turning points. Piecewise linear Galerkin finite element methods are constructed on various piecewise equidistant meshes designed for such problems. These methods are proved to be convergent, uniformly in the singular perturbation parameter, in a weighted energy norm and the usualL 2 norm. Supporting numerical results are presented.