Boundary element preconditioners for a hypersingular integral equation on an interval

We propose an almost optimal preconditioner for the iterative solution of the Galerkin equations arising from a hypersingular integral equation on an interval. This preconditioning technique, which is based on the single layer potential, was already studied for closed curves [11,14]. For a boundary element trial space, we show that the condition number is of order (1 + | log hmin|)2, where hmin is the length of the smallest element. The proof requires only a mild assumption on the mesh, easily satisfied by adaptive refinement algorithms.

[1]  Martin Costabel,et al.  Boundary Integral Operators on Lipschitz Domains: Elementary Results , 1988 .

[2]  P. Grisvard.  Elliptic Problems in Nonsmooth Domains , 1985 .

[3]  E. Hille.  Analytic Function Theory , 1961 .

[4]  B. Silbermann,et al.  Numerical Analysis for Integral and Related Operator Equations , 1991 .

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

[6]  Thanh Tran,et al.  Additive schwarz methods for the H-version boundary element method , 1996 .

[7]  Jacques Chazarain,et al.  Introduction to the theory of linear partial differential equations , 1982 .

[8]  J. Nédélec,et al.  Numerical solution of an exterior Neumann problem using a double layer potential , 1978 .

[9]  Olaf Steinbach,et al.  The construction of some efficient preconditioners in the boundary element method , 1998, Adv. Comput. Math..

[10]  Olaf Steinbach,et al.  On the stability of the $L_2$ projection in fractional Sobolev spaces , 2001, Numerische Mathematik.

[11]  Mark Ainsworth,et al.  The Conditioning of Boundary Element Equations on Locally Refined Meshes and Preconditioning by Diagonal Scaling , 1999 .

[12]  L. Hörmander.  The analysis of linear partial differential operators , 1990 .

[13]  V. Thomée,et al.  The stability in _{} and ¹_{} of the ₂-projection onto finite element function spaces , 1987 .

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

[15]  Gene H. Golub,et al.  Matrix computations , 1983 .

[16]  Thanh Tran,et al.  A preconditioning strategy for boundary element Galerkin methods , 1997 .