The $h-p$ version of the boundary element method on polygonal domains with quasiuniform meshes

— We investigate the h-p version of the boundary element method for intégral équation formulations for PDEs over polygonal domains where both the mesh size h and the polynomial degree p are changed to improve accuracy. Under the assumption of quasiuniform meshes, we obtain estimâtes for the rate of convergence which show the effects on the error of changing h andp either separately or together. Using précise resul ts for the singular behaviour of the solution near corners of the domain, it is shown that the rate of convergence for the p-version (h fïxed) is twice that of the h version (p fixed) for most problems, which agrées with computational results reported in the literature. Interior estimâtes are also derived. Résumé. — Nous étudions la version h-p des éléments finis pour la méthode intégrale appliquée à la résolution d'équations aux dérivées partielles sur des domaines polygonaux. Le pas de maillage h ainsi que le degré des polynômes p sont modifiés afin d'améliorer la précision. Pour des maillages quasi uniformes, nous obtenons des estimations de convergence qui montrent l'incidence sur l'erreur si on change h et p en même temps ou séparément. En utilisant des résultats précis sur la singularité de la solution près des coins du domaine, on montre que la vitesse de convergence de la version p (pour h fixé) est deux fois supérieure à celle de la version (pour p fixé) pour la plupart des problèmes, ce qui confirme les résultats numériques rencontrés dans la littérature. On en déduit des estimations pour l'intérieur du domaine. (*) Received February 1990. O School of Mathematics, Georgia Instituts of Technology, Atlanta, GA 30332, U.S.A. () Department of Mathematics and Statistics, University of Maryland Baltimore County, Baltimore, MD 21228, U.S.A. (**) The first author was supported by the National Science Foundation under Grants DMS8603954 and DMS-8704463 and the second author was supported by the Air Force Office of Scientific Research, Air Force Systems Command, USAF, under Grant AFOSR-85-0322. This paper was partly written while the first author was visiting at IPST, University of Maryland, College Park, Maryland. MAN Modélisation mathématique et Analyse numérique 0764-583X/91/06/783/25/$ 4.50 Mathematical Modelling and Numerical Analysis Œ) AFCET Gauthier-Villars 784 E. P. STEPHAN, M. SURI

[1]  Ernst P. Stephan,et al.  A hypersingular boundary integral method for two-dimensional screen and crack problems , 1990 .

[2]  J. Lions,et al.  Non-homogeneous boundary value problems and applications , 1972 .

[3]  Ernst P. Stephan,et al.  An augmented galerkin procedure for the boundary integral method applied to two-dimensional screen and crack problems , 1984 .

[4]  Ivo Babuška,et al.  The h-p version of the finite element method , 1986 .

[5]  Ernst P. Stephan,et al.  Remarks to Galerkin and least squares methods with finite elements for general elliptic problems , 1976 .

[6]  Martin Costabel,et al.  Boundary integral equations for mixed boundary value problems in polygonal domains and Galerkin approximation , 1985 .

[7]  I. Babuska,et al.  Theh,p andh-p versions of the finite element method in 1 dimension , 1986 .

[8]  Ernst P. Stephan,et al.  On the Convergence of the p-Version of the Boundary Element Galerkin Method. , 1989 .

[9]  M. Costabel,et al.  The normal dervative of the double layer potential on polygons and galerkin approximation , 1983 .

[10]  Ivo Babuška,et al.  The p - and h-p version of the finite element method, an overview , 1990 .

[11]  Milo R. Dorr,et al.  The Approximation Theory for the p-Version of the Finite Element Method , 1984 .

[12]  Ivo Babuška,et al.  Error estimates for the combinedh andp versions of the finite element method , 1981 .

[13]  Stefan Hildebrandt,et al.  Constructive proofs of representation theorems in separable Hilbert space , 1964 .

[14]  E. Alarćon,et al.  Hierarchical boundary elements , 1985 .

[15]  Wolfgang L. Wendland,et al.  ON SOME MATHEMATICAL ASPECTS OF BOUNDARY ELEMENT METHODS FOR ELLIPTIC PROBLEMS , 1985 .

[16]  Manil Suri,et al.  The treatment of nonhomogeneous Dirichlet boundary conditions by thep-version of the finite element method , 1989 .

[17]  E. Rank,et al.  Adaptive boundary element methods , 1987 .