On Finite Difference Fitted Schemes for Singularly Perturbed Boundary Value Problems with a Parabolic Boundary Layer

AbstractA method to construct grid approximations for singularly perturbed boundary value problems for elliptic and parabolic equations, whose solutions contain a parabolic boundary layer, is considered. The grid approximations are based on the fitted operator method. Finite difference schemes, finite element, or finite volume techniques are included in the term grid approximation methods. It is shown that there exists no grid approximation method on uniform grids from the class of fitted operator methods, whose solutions converge, in the discrete maximum norm, uniformly with respect to the perturbation parameter to the solution of the boundary value problem

A technique to prove parameter-uniform convergence for a singularly perturbed convection–diffusion equation

AbstractA priori parameter explicit bounds on the solution of singularly perturbed elliptic problems of convection–diffusion type are established. Regular exponential boundary layers can appear in the solution. These bounds on the solutions and its derivatives are obtained using a suitable decomposition of the solution into regular and layer components. By introducing extensions of the coefficients to a larger domain, artificial compatibility conditions are not imposed in the derivation of these decompositions

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

A parameter-robust finite difference method for singularly perturbed delay parabolic partial differential equations

AbstractA Dirichlet boundary value problem for a delay parabolic differential equation is studied on a rectangular domain in the x-t plane. The second-order space derivative is multiplied by a small singular perturbation parameter, which gives rise to parabolic boundary layers on the two lateral sides of the rectangle. A numerical method comprising a standard finite difference operator (centred in space, implicit in time) on a rectangular piecewise uniform fitted mesh of Nx×Nt elements condensing in the boundary layers is proved to be robust with respect to the small parameter, or parameter-uniform, in the sense that its numerical solutions converge in the maximum norm to the exact solution uniformly well for all values of the parameter in the half-open interval (0,1]. More specifically, it is shown that the errors are bounded in the maximum norm by C(Nx-2ln2Nx+Nt-1), where C is a constant independent not only of Nx and Nt but also of the small parameter. Numerical results are presented, which validate numerically this theoretical result and show that a numerical method consisting of the standard finite difference operator on a uniform mesh of Nx×Nt elements is not parameter-robust