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\(hp\)-finite element methods for singular perturbations. (English) Zbl 1021.65055
Lecture Notes in Mathematics. 1796. Berlin: Springer. xiv, 318 p. EUR 43.95/net; sFr. 75.50; £31.00; $ 59.80 (2002).
This book presents an \(hp\)-finite element method for singularly perturbed reaction-diffusion equations on curvilinear polygons in the plane, and it gives a thorough analysis of the accuracy of the method. Anisotropic grids are used in the boundary layers, and geometrically graded grids in the corner layers. The theory is complicated by the fact that the error estimates are uniform with respect to the perturbation parameter \(\varepsilon\), the mesh size \(h\), and the order \(p\) of the finite elements.

MSC:
65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
65-02 Research exposition (monographs, survey articles) pertaining to numerical analysis
65N15 Error bounds for boundary value problems involving PDEs
35B25 Singular perturbations in context of PDEs
35J65 Nonlinear boundary value problems for linear elliptic equations
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