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Sur l’approximation des solutions du problème de Dirichlet dans un ouvert avec coins. (French) Zbl 0575.65102
Singularities and constructive methods for their treatment, Proc. Conf., Oberwolfach 1983, Lect. Notes Math. 1121, 199-206 (1985).
[For the entire collection see Zbl 0547.00025.]
Solutions of Dirichlet problems \((\Delta u=f\) in D, \(u=0\) on \(\partial D)\) in polygonal domains \(D\subset {\mathbb{R}}^ 2\) have singularities at non-convex corners which reduce the convergence rate of finite element approximations. A possibility to avoid this difficulty is demonstrated for an L-shaped domain D (0 is the non-convex corner). The finite element approximation of \(\Delta S=0\) in D, \(S=Im(z^{-2/3})\) on \(\partial D\) allows to calculate an approximation \(\lambda_ h\) of the coefficient of the singular term of u(O(h)). If \(w_ h\) is the finite element approximation of \(\Delta w=f\) in D, \(w=-\lambda_ hIm(z^{2/3})\) on \(\partial D\) then \(u_ h:=w_ h+\lambda_ hIm(z^{2/3})\) is an O(h) approximation of u in \(H_ 1\). Numerical examples are not given.
Reviewer: J.Weisel

MSC:
65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
35J05 Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation