×

On the h-, p- and h-p versions of the boundary element method. - Numerical results. (English) Zbl 0732.65101

The authors present numerical results on two types of boundary integral equations of the first kind. The first one is Symm’s integral equation with logarithmic kernel and the second one is a hypersingular integral equation with the normal derivative of the double layer potential as kernel.
They compare the pure h- and p-versions with the h-p-version on a geometric mesh and the adaptive h-p-version in solving both equations. The superiority of the combined versions (exponential convergence rate) on the pure versions (algebraic convergence rate) is emphasized.
The numerical experiments are performed with the authors’ own code. The theoretical background consists of some important results many of them being the work of the second author.

MSC:

65N38 Boundary element methods for boundary value problems involving PDEs
65N12 Stability and convergence of numerical methods for boundary value problems involving PDEs
65R20 Numerical methods for integral equations
35J05 Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation
35C15 Integral representations of solutions to PDEs
45E10 Integral equations of the convolution type (Abel, Picard, Toeplitz and Wiener-Hopf type)
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Babuška, I.; Dorr, M., Error estimates for the combined \(h\) and \(p\) version of the finite element method, Numer. Math., 37, 257-277 (1981) · Zbl 0487.65058
[2] Babuška, I.; Szabo, B. A.; Katz, I. N., The \(p\)-version of the finite element method, SIAM J. Numer. Anal., 18, 515-545 (1981) · Zbl 0487.65059
[3] Babuška, I., The \(p\) and \(h-p\) versions of the finite method, The state of the art, Institute for Physical Science and Technology, University of Maryland, College Park, MD, Technical Note BN-1156 (1986) · Zbl 0614.65089
[4] I. Babuška, B.Q. Guo and E.P. Stephan, On the exponential convergence of the \(hp\); I. Babuška, B.Q. Guo and E.P. Stephan, On the exponential convergence of the \(hp\)
[5] Stephan, E. P.; Suri, M., On the convergence of the \(p\)-version of the boundary element Galerkin method, Math. Comp., 52, 31-48 (1989) · Zbl 0661.65118
[6] E.P. Stephan and M. Suri, The \(hp\); E.P. Stephan and M. Suri, The \(hp\) · Zbl 0744.65073
[7] Stephan, E. P., The \(h-p\) version of the Galerkin boundary element method for integral equations on polygons and open arcs, (Brebbia, C. A., Proc. Conference BEM-10. Proc. Conference BEM-10, Southampton (1988))
[8] Babuška, I.; Guo, B. Q.; Stephan, E. P., Comput. Methods Appl. Mech. Engrg., 80, 319-325 (1990) · Zbl 0727.65097
[9] B.Q. Guo, T. von Petersdorff and E.P. Stephan, On the exponential convergence of the \(hp\); B.Q. Guo, T. von Petersdorff and E.P. Stephan, On the exponential convergence of the \(hp\)
[10] Stephan, E. P.; Wendland, W. L., Remarks to Galerkin and least squares methods with finite elements for general elliptic problems, Manuscripta Geodaetica, 1, 93-123 (1976) · Zbl 0353.65067
[11] Alarcón, E.; Abia, L.; Reverter, A., On the possibility of adaptive boundary elements, (Accuracy Estimates and Adaptive Refinements in Finite Element Computations (1984), AFREC: AFREC Lisbon) · Zbl 0593.65068
[12] Alarcón, E.; Reverter, A.; Molina, J., Hierarchical boundary elements, Comput. & Structures, 20, 151-156 (1985) · Zbl 0581.73095
[13] Alarcón, E.; Reverter, A., \(p\)-adaptive boundary elements, Internat. J. Numer. Methods Engrg., 23, 801-829 (1986) · Zbl 0593.65068
[14] Lera, S. Gómez; Cerrolaza, M.; Alarcón, E., Adaptive refinements in BEM, (Brebbia, C. A.; Wendland, W. L.; Kuhn, G., Boundary Elements 9, Vol. 1 (1987), Springer: Springer Berlin), 337-349
[15] Parreira, P., Self-adaptive \(p\)-hierarchical boundary elements in elastostatics, (Brebbia, C. A.; Wendland, W. L.; Kuhn, G., Boundary Elements 9, Vol. 1 (1987), Springer: Springer Heidelberg), 351-373
[16] Rank, E., Adaptive boundary element methods, (Brebbia, C. A.; Wendland, W. L.; Kuhn, G., Boundary Elements 9, Vol. 1 (1987), Springer: Springer Heidelberg), 259-273
[17] Stephan, E. P.; Wendland, W. L., An augmented Galerkin procedure for the boundary integral method applied to two-dimensional screen and crack problems, Appl. Analysis, 18, 183-219 (1984) · Zbl 0582.73093
[18] E.P. Stephan and W.L. Wendland, A hypersingular boundary integral method for two-dimensional screen and crack problems, Arch. Rational Mech. Anal., to appear.; E.P. Stephan and W.L. Wendland, A hypersingular boundary integral method for two-dimensional screen and crack problems, Arch. Rational Mech. Anal., to appear. · Zbl 0725.73091
[19] Lions, J. L.; Magenes, E., Non-homogeneous boundary value problems and applications, I (1972), Springer: Springer Berlin · Zbl 0223.35039
[20] von Petersdorff, T., Elasticity problems in polyhedra—Singularities and approximations with boundary element methods, (PhD Thesis (1989), TH Darmstadt) · Zbl 0694.35048
[21] Costabel, M.; Stephan, E. P., Duality estimates for the numerical approximation of boundary integral equations, Numer. Math., 54, 339-353 (1988) · Zbl 0663.65141
[22] Gui, W.; Babuška, I., The \(h-p\) versions of the finite element method in one dimension. Part 3—The adaptive \(h-p\) version, Numer. Math., 49, 658-683 (1986) · Zbl 0614.65090
[23] Wendland, W. L.; Yu, D.-H., Adaptive boundary element methods for strongly elliptic integral equations, Numer. Math., 53, 539-558 (1988) · Zbl 0657.65138
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.