Summary
It is well known that Galerkin discretizations based on hp-Finite Element Spaces are converging exponentially with respect to the degrees of freedom for elliptic problems with piecewise analytic data. However, the question whether these methods can be realized for general situations such that the exponential convergence is preserved also with respect to the computing time is very essential.
In this paper, we will show how the numerical quadrature can be realized in order that the resulting fully discrete hp-Boundary Element Method converges exponentially with algebraically growing work. The key point is to approximate the integrals constituting the stiffness matrix by exponentially converging cubature methods.
partially supported by the NSF under grant DMS 91-20877
partially supported by the AFOSR under grant F49620-J-0100
This is a preview of subscription content, log in via an institution.
Buying options
Tax calculation will be finalised at checkout
Purchases are for personal use only
Learn about institutional subscriptionsPreview
Unable to display preview. Download preview PDF.
References
P. Davis and P. Rabinowitz. Methods of Numerical Integration. Academic Press, 1984.
J. Elschner. The Double Layer Potential Operator over Polyhedral Domains II: Spline Galerkin Methods. Math. Meth. Appl. Sci., 15:23–37, 1992.
J. Elschner. On the Exponential Convergence of some Boundary Element Methods for Laplace’s Equation in Nonsmooth Domains. In M. Costabel, M. Dauge, and S. Nicaise, editors, Boundary Value Problems and Integral Equations in Nonsmooth Domains, New York, 1995. Marcel Dekker.
B. Guo and I. Babuska. The hp-Version of the Finite Element Method. Part i: The Basic Approximation Results. Computational Mechanics, 1:21–41, 1986.
B. Guo and I. Babuska. The hp-Version of the Finite Element Method. Part II: General Results and Applications. Computational Mechanics, 1:203–220, 1986.
W. Hackbusch. Integral Equations. ISNM 120, Birkhäuser Publ. Basel, 1995.
W. Hackbusch and S. Sauter. On the Efficient Use of the Galerkin Method to Solve Fredholm Integral Equations. Applications of Mathematics, 38(4–5):301–322, 1993.
H. Han. The Boundary Integro-Differential Equations of Three-Dimensional Neuman Problem in Linear Elasticity. Numer. Math., 68(2):269–281, 1994.
C. Lage. Software Development for Boundary Element Mehtods: Analysis and Design of Efficient Techniques (in German). PhD thesis, Lehrstuhl Prakt. Math., Universität Kiel, 1995.
M. Maischak and E. Stephan. The hp-Version of the Boundary Element Method in R3 — the Basic Approximation Results. Math. Meth. Appl. Sci, to appear, 1996.
J. Nédélec. Integral Equations with Non Integrable Kernels. Integral Equations Oper. Theory, 5:562–572, 1982.
S. Sauter. Cubature Techniques for 3-d Galerkin BEM. In W. Hackbusch and G. Wittum, editors, BEM: Implementation and Analysis of Advanced Algorithms. Vieweg Verlag, 1996.
S. A. Sauter. Über die effiziente Verwendung des Galerkinverfahrens zur Lösung Fredholmscher Integralgleichungen. PhD thesis, Inst. f. Prakt. Math., Universität Kiel, 1992.
S. A. Sauter and A. Krapp. On the Effect of Numerical Integration in the Galerkin Boundary Element Method. Technical Report 95–4, Lehrstuhl Praktische Mathematik, Universität Kiel, Germany, 1995, to appear in Numer. Math.
S. A. Sauter and C. Schwab. Quadrature for hp-Galerkin BEM in 3-d. Technical Report 96–02, Seminar for Appl. Math., ETH Zürich, March 1996.
C. Schwab. Variable order composite quadrature of singular and nearly singular integrals. Computing, 53:173–194, 1994.
C. Schwab and W. Wendland. Kernel Properties and Representations of Boundary Integral Operators. Math. Nachr., 156:187–218, 1992.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 1996 Friedr. Vieweg & Sohn Verlagsgesellschaft mbH, Braunschweig/Wiesbaden
About this chapter
Cite this chapter
Sauter, S.A., Schwab, C. (1996). Realization of hp-Galerkin BEM in ℝ3 . In: Hackbusch, W., Wittum, G. (eds) Boundary Elements: Implementation and Analysis of Advanced Algorithms. Notes on Numerical Fluid Mechanics (NNFM), vol 50. Vieweg+Teubner Verlag. https://doi.org/10.1007/978-3-322-89941-5_16
Download citation
DOI: https://doi.org/10.1007/978-3-322-89941-5_16
Publisher Name: Vieweg+Teubner Verlag
Print ISBN: 978-3-322-89943-9
Online ISBN: 978-3-322-89941-5
eBook Packages: Springer Book Archive