Workshop Singularly Perturbed Problems |
Talks & SlidesSolution decomposition method as an approach for constructing ε-uniformly convergent difference schemes, which are stable to perturbation in the data, for singularly perturbed parabolic convection-diffusion equationsGrigorii Ivanovich Shishkin (Institute of Mathematics and Mechanics, Russian Academy of Sciences) Standard difference schemes on uniform meshes (as well as the finite element method schemes) used to solve singularly perturbed problems do not converge ε-uniformly in the maximum norm. Moreover, these schemes are not ε-uniformly stable to perturbations in the data of the discrete problem, in particular, to the perturbations arising in process of solving the problem on a computer. For example, in the case of a singularly perturbed ordinary differential convection-diffusion equation, a monotone difference scheme on a uniform grid, under the condition of its convergence (for N >> ε^{-1} where the value N defines the number of nodes in the mesh in x), is not ε-uniformly well conditioned. As a result, when solving such a problem with the required accuracy for ε → 0, it is necessary: (a) unboundedly to increase N (N >> ε^{-3/2} for stability of the scheme) and (b) to use a computer with unboundedly growing number of digits in the computer word. This is not possible in real computations. Note that the regular component of the discrete solution is ε-uniformly well conditioned. But the singular component is ε-uniformly well conditioned only under the condition when the interval, on which the problem is considered, is a value of order O(ε ψ(N)), ψ(N)=o(N). Such a behaviour of the discrete regular and singular components motivates the construction of a numerical method for singularly perturbed problems based on the special decomposition of the discrete solution.
In this talk, for a Dirichlet problem to a singularly perturbed parabolic convection-diffusion equation, a difference scheme of the solution decomposition method is constructed. This method uses decomposition of the discrete solution into the regular and singular components which are solutions of discrete subproblems solving on uniform meshes. The constructed difference scheme is ε-uniformly well conditioned, it converges ε-uniformly in the maximum norm, and also it is ε-uniformly stable to perturbations in the data of the discrete problem, in particular, to the perturbations arising in process of solving the problem on a computer. Slides |