Optimal Operator Preconditioning for Galerkin Boundary Element Methods on 3-Dimensional Screens

Author: 

Hiptmair, R
Jerez-Hanckes, C
Urzúa-Torres, C

Publication Date: 

27 January 2020

Journal: 

SIAM Journal on Numerical Analysis

Last Updated: 

2020-08-10T13:04:29.37+01:00

Issue: 

1

Volume: 

58

DOI: 

10.1137/18m1196029

page: 

834-857

abstract: 

We consider first-kind weakly singular and hypersingular boundary integral operators for the Laplacian on screens in R3 and their Galerkin discretization by means of low-order piecewise polynomial boundary elements. For the resulting linear systems of equations we propose novel Calder´on-type preconditioners based on (i) new boundary integral operators, which provide the exact inverses of the weakly singular and hypersingular operators on flat disks, and (ii) stable duality pairings relying on dual meshes. On screens obtained as images of the unit disk under bi-Lipschitz transformations, this approach achieves condition numbers uniformly bounded in the meshwidth even on locally refined meshes. Comprehensive numerical tests also confirm its excellent pre-asymptotic performance.

Symplectic id: 

1058980

Submitted to ORA: 

Submitted

Publication Type: 

Journal Article