Uniformly convergent finite element and finite difference methods for singularly perturbed ordinary differential equations

This thesis is concerned with uniformly convergent finite element and finite difference methods for numerically solving singularly perturbed two-point boundary value problems. We examine the following four problems: (i) high order problem of reaction-diffusion type; (ii) high order problem of convection-diffusion type; (iii) second order interior turning point problem; (iv) semilinear reaction-diffusion problem. Firstly, we consider high order problems of reaction-diffusion type and convection-diffusion type. Under suitable hypotheses, the coercivity of the associated bilinear forms is proved and representation results for the solutions of such problems are given. It is shown that, on an equidistant mesh, polynomial schemes cannot achieve a high order of convergence which is uniform in the perturbation parameter. Piecewise polynomial Galerkin finite element methods are then constructed on a Shishkin mesh. High order convergence results, which are uniform in the perturbation parameter, are obtained in various norms. Secondly, we investigate linear second order problems with interior turning points. Piecewise linear Galerkin finite element methods are generated on various piecewise equidistant meshes designed for such problems. These methods are shown to be convergent, uniformly in the singular perturbation parameter, in a weighted energy norm and the usual L2 norm. Finally, we deal with a semilinear reaction-diffusion problem. Asymptotic properties of solutions to this problem are discussed and analysed. Two simple finite difference schemes on Shishkin meshes are applied to the problem. They are proved to be uniformly convergent of second order and fourth order respectively. Existence and uniqueness of a solution to both schemes are investigated. Numerical results for the above methods are presented

A fully discrete ε-uniform method for convection-diffusion problem on equidistant meshes

For a singularly-perturbed two-point boundary value problem, we propose an ε-uniform finite difference method on an equidistant mesh which requires no exact solution of a differential equation. We start with a full-fitted operator method reflecting the singular perturbation nature of the problem through a local boundary value problem. However, to solve the local boundary value problem, we employ an upwind method on a Shishkin mesh in local domain, instead of solving it exactly. We further study the convergence properties of the numerical method proposed and prove it nodally converges to the true solution for any ε

A Mixed Finite Element Method for Singularly Perturbed Fourth Oder Convection-Reaction-Diffusion Problems on Shishkin Mesh

This paper introduces an approach to decoupling singularly perturbed boundary value problems for fourth-order ordinary differential equations that feature a small positive parameter $\epsilon$ multiplying the highest derivative. We specifically examine Lidstone boundary conditions and demonstrate how to break down fourth-order differential equations into a system of second-order problems, with one lacking the parameter and the other featuring $\epsilon$ multiplying the highest derivative. To solve this system, we propose a mixed finite element algorithm and incorporate the Shishkin mesh scheme to capture the solution near boundary layers. Our solver is both direct and of high accuracy, with computation time that scales linearly with the number of grid points. We present numerical results to validate the theoretical results and the accuracy of our method.Comment: 15 pages, 7 figure

An efficient numerical scheme for 1D parabolic singularly perturbed problems with an interior and boundary layers

In this paper we consider a 1D parabolic singularly perturbed reaction-convection-diffusion problem, which has a small parameter in both the diffusion term (multiplied by the parameter e2) and the convection term (multiplied by the parameter µ) in the differential equation (e¿(0, 1], µ¿0, 1], µ=e). Moreover, the convective term degenerates inside the spatial domain, and also the source term has a discontinuity of first kind on the degeneration line. In general, for sufficiently small values of the diffusion and the convection parameters, the exact solution exhibits an interior layer in a neighborhood of the interior degeneration point and also a boundary layer in a neighborhood of both end points of the spatial domain. We study the asymptotic behavior of the exact solution with respect to both parameters and we construct a monotone finite difference scheme, which combines the implicit Euler method, defined on a uniform mesh, to discretize in time, together with the classical upwind finite difference scheme, defined on an appropriate nonuniform mesh of Shishkin type, to discretize in space. The numerical scheme converges in the maximum norm uniformly in e and µ, having first order in time and almost first order in space. Illustrative numerical results corroborating in practice the theoretical results are showed