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Sufficient conditions for uniform convergence on layer-adapted grids. (English) Zbl 0931.65085
This paper deals with convergence properties of the simple upwind difference scheme and a Galerkin finite element method (FEM) on generalized Shishkin grids. Conditions on the mesh-characterizing function that are sufficient for the convergence of the method, uniformly with respect to the perturbation parameter are presented. The authors verify experimentally the theoretical results for the Galerkin FEM and also test the performance of the simple upwind scheme on various meshes when applied to the test second-order layer problem for ordinary differential equations with homogeneous Dirichlet boundary conditions.

MSC:
65L10 Numerical solution of boundary value problems involving ordinary differential equations
34E15 Singular perturbations, general theory for ordinary differential equations
65L20 Stability and convergence of numerical methods for ordinary differential equations
65Y20 Complexity and performance of numerical algorithms
34B15 Nonlinear boundary value problems for ordinary differential equations
65L12 Finite difference and finite volume methods for ordinary differential equations
65L60 Finite element, Rayleigh-Ritz, Galerkin and collocation methods for ordinary differential equations
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