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Methods of constructing grid approximations for singularly perturbed boundary-value problems. Condensing-grid methods. (English) Zbl 0816.65072
Summary: The Dirichlet boundary-value problem for an elliptic equation is treated on a rectangle. The highest derivatives of the equation contain a parameter taking on the values on a half-open interval (0,1]. For the zero value of the parameter, the elliptic equation degenerates into a first-order one. The method of special condensing grids is used to solve the boundary-value problem.
The conditions providing uniform convergence of the difference schemes in the parameter are specified for the node distribution of the grids being constructed and for the grid approximations of differential equations. Various methods of constructing such grid approximations of the boundary- value problem that meet the conditions sufficient for the schemes being constructed to converge uniformly in the parameter, are given. Several schemes are constructed for the boundary-value problem involved.

MSC:
65N06 Finite difference methods for boundary value problems involving PDEs
65N12 Stability and convergence of numerical methods for boundary value problems involving PDEs
35J25 Boundary value problems for second-order elliptic equations
35B25 Singular perturbations in context of PDEs
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