A difference scheme of improved accuracy for a quasilinear singularly perturbed elliptic convection-diffusion equation.

A Dirichlet boundary value problem for a quasilinear singularly perturbed elliptic convection-diffusion equation on a strip is considered. For such a problem, a difference scheme based on classical approximations of the problem on piecewise uniform meshes condensing in the layer converges epsilon-uniformly with an order of accuracy not more than 1. We construct a linearized iterative scheme based on the nonlinear Richardson scheme, where the nonlinear term is computed using the sought function taken at the previous iteration; the solution of the iterative scheme converges to the solution of the nonlinear Richardson scheme at the rate of a geometry progression epsilon-uniformly with respect to the number of iterations. The use of lower and upper solutions of the linearized iterative Richardson scheme as a stopping criterion allows us during the computational process to define a current iteration under which the same epsilon-uniform accuracy of the solution is achieved as for the nonlinear Richardson schem

High-order time-accurate schemes for parabolic singular perturbation problems with convection

We consider the first boundary value problem for a singularly perturbed para-bo-lic PDE with convection on an interval. For the case of sufficiently smooth data, it is easy to construct a standard finite difference operator and a piecewise uniform mesh, condensing in the boundary layer, which gives an $epsilon$-uniformly convergent difference scheme. The order of convergence for such a scheme is exactly one and up to a small logarithmic factor one with respect to the time and space variables, respectively. In this paper we construct high-order time-accurate $epsilon$-uniformly convergent schemes by a defect correction technique. The efficiency of the new defect-correction scheme is confirmed by numerical experiments

An a-posteriori adaptive mesh technique for singularly perturbed convection-diffusion problems with a moving interior layer

We study numerical approximations for a class of singularly perturbed problems of convection-diffusion type with a moving interior layer. In a domain (a segment) with a moving interface between two subdomains, we consider an initial boundary value problem for a singularly perturbed parabolic convection-diffusion equation. Convection fluxes on the subdomains are directed towards the interface. The solution of this problem has a moving transition layer in the neighbourhood of the interface. Unlike problems with a stationary layer, the solution exhibits singular behaviour also with respect to the time variable. Well-known upwind finite difference schemes for such problems do not~converge $eps$-uniformly in the uniform norm, even under the condition $N^{-1}+N_0^{-1}approx eps$, where $eps$ is the perturbation parameter and $N$ and $N_0$ denote the number of mesh points with respect to $x$ and $t$. In the case of rectangular meshes which are ({it a~priori,} or {it a~posteriori,}) locally refined in the transition layer, there are no schemes that convergence uniformly in $eps$ even under the {it very restrictive,} condition $N^{-2}+N_0^{-2} approx eps$. However, the condition for convergence can be {it essentially weakened} if we take the geometry of the layer into account, i.e., if we introduce a new coordinate system which captures the interface. For the problem in such a coordinate system, one can use either an {it a~priori,}, or an {it a~posteriori,} adaptive mesh technique. Here we construct a scheme on {it a~posteriori,} adaptive meshes (based on the gradient of the solution), whose solution converges `almost $eps$-uniformly', viz., under the condition $N^{-1}=o(eps^{ u})$, where $u>0$ is an arbitrary number from the half-open interval $(0,1]$