The p-version of the finite element method for domains with corners and for infinite domains. (English) Zbl 0717.65084

Babuška, Ivo; Oh, Hae-Soo

Numer. Methods Partial Differ. Equations 6, No. 4, 371-392 (1990).

A special approach to deal with elliptic boundary value problems with singularities is introduced which is called an auxiliary mapping technique. It is based on the p-version of the finite element method which, in contrast to the standard h-version, uses a fix mesh and an increasing degree p of the elements.
As model problems two-dimensional Laplace equations are studied in domains with corners (mixed Dirichlet and Neumann boundary conditions). The essence of the auxiliary mapping technique involves locally transforming a region around the corner to a new domain by use of a mapping such as \(\xi =z^{1/\alpha}\) which locally transforms the exact singular solution to a smoother or even locally analytic function.
It is shown that one can obtain an exponential rate of convergence at no extra cost. Finally, the technique is applied to infinite domains (mapping \(\xi =1/z)\). Numerical examples for L-shaped domains demonstrate application and convergence properties of the method.

Reviewer: J.Weisel

MSC:

65N30	Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
65N15	Error bounds for boundary value problems involving PDEs
35J25	Boundary value problems for second-order elliptic equations

Keywords:

singularities; auxiliary mapping technique; p-version of the finite element method; Laplace equations; domains with corners; Dirichlet and Neumann boundary conditions; exact singular solution; exponential rate of convergence; Numerical examples; L-shaped domains

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