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A local refinement finite-element method for two-dimensional parabolic systems. (English) Zbl 0659.65105
An adaptive finite element method is presented for the numerical solution of initial boundary value problems for parabolic systems in two spatial dimensions. Discretization in space is obtained using a finite element Galerkin procedure with piecewise bilinear finite elements on a rectangular mesh. Spatial error estimates of the solution are calculated using piecewise cubics that utilize nodal superconvergence to improve the computational efficiency. The resulting systems of ordinary differential equations for the solution and error estimate are integrated in time by the existing codes for stiff differential systems. The spatial error estimate is used for the local refinement of the finite element grid. Information employed in the algorithms presented is managed using a dynamic tree data structure that provides ample efficiency. Several numerical examples illustrating the work of the procedure are presented. Some special cases and extensions of the method are discussed.
Reviewer: V.V.Kobkov

65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
65N15 Error bounds for boundary value problems involving PDEs
65N50 Mesh generation, refinement, and adaptive methods for boundary value problems involving PDEs
35K20 Initial-boundary value problems for second-order parabolic equations
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