Workshop Numerical Analysis for Singularly Perturbed Problems
(dedicated to the 60th birthday of Martin Stynes)

Workshop Singularly Perturbed Problems

Talks & Slides

Some results for singularly perturbed problems with discontinuous data and interior layers: uniform convergence on special meshes

Carmelo Clavero (University of Zaragoza)

In this talk we are interested in the construction and the convergence analysis of a numerical method used to solve a type of singularly perturbed parabolic convection-diffusion initial-boundary value problems. In the problem under consideration, the convective coefficient, which for simplicity depends only on the spatial variable, is such that the convective flux is directed from the lateral boundary inside the domain, and also we assume that this coefficient degenerates at an interior point of the spatial domain. Finally, we will assume that the right-hand side and/or the coefficient of the time derivative term have a first kind discontinuity on the degeneration line. It is wellknown that if the diffusion parameter ε, ε ∈ (0; 1], is sufficiently small, an interior layer appears in a neighborhood of the set where the discontinuity occurs. To solve the problem, a finite difference scheme is constructed by using a standard monotone approximation of the differential equation, which combines the implicit Euler method, on a uniform mesh in time, and the classical central finite difference scheme in space. If the spatial mesh is uniform, then the difference scheme converges only under the rather restrictive condition N-1 = o(ε), N0-1 = o(1), where N + 1 and N0 + 1 are, respectively, the numbers of nodes in the space and time meshes. On the other hand, if a finite difference scheme is defined on a special non-uniform grid of Shishkin or Bakhvalov type condensing in a neighborhood of the interior layer, the solution of the difference scheme converges ε-uniformly at the same rate as a singularly perturbed parabolic problem with the continuous data. We present some numerical results confirming the good behavior of the numerical method and the order of the ε-uniform convergence theoretically proved.

Last change: 21 November 2011