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Necessary conditions for uniform convergence of finite difference schemes for convection-diffusion problems with exponential and parabolic layers. (English) Zbl 0870.65091
The authors derive necessary conditions for uniform convergence of the finite difference method for solving elliptic boundary value problems with a small diffusion term of the unit square. It is known that these problems have hyperbolic character, which is the main source of numerical difficulties. Exponential and parabolic layers are investigated. The authors prove the non-existence of a five-point scheme, which would be uniformly convergent with respect to the perturbation parameter \(\epsilon \) and discretization parameter \(h.\) They also present necessary conditions for coefficients of a nine-point scheme to ensure uniform convergence.

65N12 Stability and convergence of numerical methods for boundary value problems involving PDEs
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