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On finite element methods for elliptic equations on domains with corners (1982)

Blum, H. ; Dobrowolski, M.

Springer

Computing 28 (1982), S. 53-63

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ISSN: 1436-5057

Keywords: Polygonal domains ; singular expansion ; 65N30

Source: Springer Online Journal Archives 1860-2000

Topics: Computer Science

Description / Table of Contents: Zusammenfassung Es wird eine Finite Elemente Methode zur Approximation elliptischer Differentialgleichungen auf Eckengebieten vorgeschlagen. Das Verfahren benutzt die Singulärfunktionen des Problems im Raum der Ansatzfunktionen und die Kernfunktionen des adjungierten Operators im Testraum. Dadurch erhält man gute Näherungen der Koeffizienten, der Singulärfunktionen. In einem numerischen Beispiel wird das Verfahren mit der bekannten Methode der Singulärfunktionen verglichen.

Notes: Abstract A finite element method for approximating elliptic equations on domains with corners is proposed. The method makes use of the singular functions of the problem in the trial space and the kernel functions of the adjoint problem in the test space. This leads to good approximates of the coefficients of the singular functions. In the numerical computations, the method is compared with the well known Singular Function Method.

Type of Medium: Electronic Resource

URL: http://dx.doi.org/10.1007/BF02237995

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On the boundary value problem of the biharmonic operator on domains with angular corners (1980)

Blum, H. ; Rannacher, R. ; Leis, R.

Chichester, West Sussex : Wiley-Blackwell

Mathematical Methods in the Applied Sciences 2 (1980), S. 556-581

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ISSN: 0170-4214

Keywords: Mathematics and Statistics ; Applied Mathematics

Source: Wiley InterScience Backfile Collection 1832-2000

Topics: Mathematics

Notes: The paper is concerned with boundary singularities of weak solutions of boundary value problems governed by the biharmonic operator. The presence of angular corner points or points at which the type of boundary condition changes in general causes local singularities in the solution. For that case the general theory of V. A. Kondrat'ev provides a priori estimates in weighted Sobolev norms and asymptotic singular representations for the solution which essentially depend on the zeros of certain transcendental functions. The distribution of these zeros will be analysed in detail for the biharmonic operator under several boundary conditions. This leads to sharp a priori estimates in weighted Sobolev norms where the weight function is characterized by the inner angle of the boundary corner. Such estimates for “negative” Sobolev norms are used to analyse also weakly nonlinear perturbations of the biharmonic operator as, for instance, the von Kármán model in plate bending theory and the stream function formulation of the steady state Navier-Stokes problem. It turns out that here the structure of the corner singularities is essentially the same as in the corresponding linear problem.

Additional Material: 1 Ill.