Parameter uniform numerical solution of elliptic singular perturbation problems with discontinuous data

In the first chapter of this thesis a singularly perturbed convection-diffusion equation with a source term which is continuous but whose first derivative contains a point of discontinuity is considered. The solution features a boundary layer and a weak interior layer. A numerical method is constructed which involves a Shishkin mesh fitted at the boundary layer, but not at the interior layer. The method is shown, both theoretically and by numerical experiments, to be uniformly convergent with respect to the singular perturbation parameter.

In the second chapter we consider the following boundary value problem in a domain Ω, the unit square.

Luε ≡ ε∆uε + p1 ∂uε ∂x + p2 ∂uε ∂y − quε = f in Ω,

uε(x, 0) = gs(x), uε(x, 1) = gn(x), x ∈ (0, 1)

uε(0, y) = gw(y), uε(1, y) = ge(y) y ∈ (0, 1)

gs(0) ̸= gw(0),

where p1, p2 and q are positive constants and 0 < ε ≤ 1, and the boundary data are smooth except at the point (0, 0), Analogously to the methods introduced in [45] we derive a pointwise bound on the solution which highlights the influence of the boundary conditions at the corners of the domain, as well as that of the singular perturbation parameter ε.

In the final chapter we examine experimentally the performance of numerical methods comprising domain decomposition and Schwarz iterative techniques ( [67], [72], [73] and [91]) extended to the class of singularly perturbed convection-diffusion problems with more general boundary conditions as described above.