TY - JOUR
T1 - Spectral equivalence and proper clusters for matrices from the boundary element method
AU - TYRTYSHNIKOV, E. E.
AU - CHAN, R. H.
PY - 2000/11/30
Y1 - 2000/11/30
N2 - The Galerkin matrices An from applications of the boundary element method to integral equations of the first kind usually need to be preconditioned. In the Laplace equation context, we highlight a family of preconditioners Cn that simultaneously enjoy two important properties: (a) An and Cn are spectrally equivalent, and (b) the eigenvalues of Cn-1An have a proper cluster at unity. In the Helmholtz equation context, we prove the spectral equivalence for the so-called second Galerkin matrices and that the eigenvalues of Cn-1An still have a proper cluster at unity. We then show that some circulant integral approximate operator (CIAO) preconditioners belong to this family, including the well-known optimal CIAO. Consequently, if we use the preconditioned conjugate gradients to solve the problems, the number of iterations for a prescribed accuracy does not depend on n, and, what is more, the convergence rate is superlinear.
AB - The Galerkin matrices An from applications of the boundary element method to integral equations of the first kind usually need to be preconditioned. In the Laplace equation context, we highlight a family of preconditioners Cn that simultaneously enjoy two important properties: (a) An and Cn are spectrally equivalent, and (b) the eigenvalues of Cn-1An have a proper cluster at unity. In the Helmholtz equation context, we prove the spectral equivalence for the so-called second Galerkin matrices and that the eigenvalues of Cn-1An still have a proper cluster at unity. We then show that some circulant integral approximate operator (CIAO) preconditioners belong to this family, including the well-known optimal CIAO. Consequently, if we use the preconditioned conjugate gradients to solve the problems, the number of iterations for a prescribed accuracy does not depend on n, and, what is more, the convergence rate is superlinear.
UR - http://www.scopus.com/inward/record.url?scp=0034320844&partnerID=8YFLogxK
U2 - 10.1002/1097-0207(20001130)49:9<1211::AID-NME998>3.0.CO;2-X
DO - 10.1002/1097-0207(20001130)49:9<1211::AID-NME998>3.0.CO;2-X
M3 - Journal Article (refereed)
AN - SCOPUS:0034320844
SN - 0029-5981
VL - 49
SP - 1211
EP - 1224
JO - International Journal for Numerical Methods in Engineering
JF - International Journal for Numerical Methods in Engineering
IS - 9
ER -