Eugene O'Riordan

A numerical method for a system of singularly perturbed reaction-diffusion equations

A Dirichlet problem for a system of two coupled singularly perturbed reaction-diffusion ordinary differential equations is examined. A numerical method whose solutions converge pointwise at all points of the domain independently of the singular perturbation parameters is constructed and analysed. Numerical results are presented, which illustrate the theoretical results.

A globally uniformly convergent finite element method for a singularly perturbed elliptic problem in two dimensions

We analyze a new Galerkin finite element method for numerically solving a linear convection-dominated convection-diffusion problem in two dimensions. The method is shown to be convergent, uniformly in the perturbation parameter, of order h1/2 in a global energy norm which is stronger than the L2 norm. This order is optimal in this norm for our choice of trial functions.

Global maximum norm parameter-uniform numerical method for a singularly perturbed convection-diffusion problem with discontinuous convection coefficient

A singularly perturbed convection-diffusion problem, with a discontinuous convection coefficient and a singular perturbation parameter @e, is examined. Due to the discontinuity an interior layer appears in the solution. A finite difference method is constructed for solving this problem, which generates @e-uniformly convergent numerical approximations to the solution. The method uses a piecewise uniform mesh, which is fitted to the interior layer, and the standard upwind finite difference operator on this mesh. The main theoretical result is the @e-uniform convergence in the global maximum norm of the approximations generated by this finite difference method. Numerical results are presented, which are in agreement with the theoretical results.

A parameter robust numerical method for a two dimensional reaction-diffusion problem

In this paper a singularly perturbed reaction-diffusion partial differential equation in two space dimensions is examined. By means of an appropriate decomposition, we describe the asymptotic behaviour of the solution of problems of this kind. A central finite difference scheme is constructed for this problem which involves an appropriate Shishkin mesh. We prove that the numerical approximations are almost second order uniformly convergent (in the maximum norm) with respect to the singular perturbation parameter. Some numerical experiments are given that illustrate in practice the theoretical order of convergence established for the numerical method.

A finite element method for a singularly perturbed boundary value problem

SummaryWe examine the problem:εu″+a(x)u′−b(x)u=f(x) for 00,b(x)>β,α2 = 4εβ>0,a, b andf inC2 [0, 1], ε in (0, 1],u(0) andu(1) given. Using finite elements and a discretized Greens function, we show that the El-Mistikawy and Werle difference scheme on an equidistant mesh of widthh is uniformly second order accurate for this problem (i.e., the nodal errors are bounded byCh2, whereC is independent ofh and ε). With a natural choice of trial functions, uniform first order accuracy is obtained in theL∞ (0, 1) norm. On choosing piecewise linear trial functions (“hat” functions), uniform first order accuracy is obtained in theL1 (0, 1) norm.

Error Analysis of a Finite Difference Method on Graded Meshes for a Time-Fractional Diffusion Equation

A reaction-diffusion problem with a Caputo time derivative of order

Uniformly convergent difference schemes for singularly perturbed parabolic diffusion-convection problems without turning points

\alpha\in (0,1)

Numerical methods for time-dependent convection-diffusion equations

is considered. The solution of such a problem is shown in general to have a weak singularity near the initial time

An analysis of a singularly perturbed two-point boundary value problem using only finite element techniques

t=0

Parameter-uniform finite difference schemes for singularly perturbed parabolic diffusion-convection-reaction problems

, and sharp pointwise bounds on certain derivatives of this solution are derived. A new analysis of a standard finite difference method for the problem is given, taking into account this initial singularity. This analysis encompasses both uniform meshes and meshes that are graded in time, and includes new stability and consistency bounds. The final convergence result shows clearly how the regularity of the solution and the grading of the mesh affect the order of convergence of the difference scheme, so one can choose an optimal mesh grading. Numerical results are presented that confirm the sharpness of the error analysis.