TY - JOUR

T1 - The conditioning of boundary element equations on locally refined meshes and preconditioning by diagonal scaling

AU - Ainsworth, Mark

AU - Mclean, William

AU - Tran, Thanh

PY - 1999

Y1 - 1999

N2 - Consider a boundary integral operator on a bounded, d-dimensional surface in ℝd+1. Suppose that the operator is a pseudodifferential operator of order 2ℳ, ℳ ∈ ℝ, and that the associated bilinear form is symmetric and positive-definite. (The surface may be open or closed, and ℳ may be positive or negative.) Let B denote the stiffness matrix arising from a Galerkin boundary element method with standard nodal basis functions. If local mesh refinement is used, then the partition may contain elements of very widely differing sizes, and consequently B may be very badly conditioned. In fact, if the elements are nondegenerate and 2|ℳ| < d, then the ℓ2 condition number of B satisfies cond(B) < CN2|ℳ|/d(hmax/hmin)d-2ℳ, where hmax and hmin are the sizes of the largest and smallest elements in the partition, and N is the number of degrees of freedom. However, if B is preconditioned using a simple diagonal scaling, then the condition number is reduced to script O sign(N2lℳl/d). That is, diagonal scaling restores the condition number of the linear system to the same order of growth as that for a uniform partition. The growth in the critical case 2|ℳ| = d is worse by only a logarithmic factor.

AB - Consider a boundary integral operator on a bounded, d-dimensional surface in ℝd+1. Suppose that the operator is a pseudodifferential operator of order 2ℳ, ℳ ∈ ℝ, and that the associated bilinear form is symmetric and positive-definite. (The surface may be open or closed, and ℳ may be positive or negative.) Let B denote the stiffness matrix arising from a Galerkin boundary element method with standard nodal basis functions. If local mesh refinement is used, then the partition may contain elements of very widely differing sizes, and consequently B may be very badly conditioned. In fact, if the elements are nondegenerate and 2|ℳ| < d, then the ℓ2 condition number of B satisfies cond(B) < CN2|ℳ|/d(hmax/hmin)d-2ℳ, where hmax and hmin are the sizes of the largest and smallest elements in the partition, and N is the number of degrees of freedom. However, if B is preconditioned using a simple diagonal scaling, then the condition number is reduced to script O sign(N2lℳl/d). That is, diagonal scaling restores the condition number of the linear system to the same order of growth as that for a uniform partition. The growth in the critical case 2|ℳ| = d is worse by only a logarithmic factor.

KW - Boundary element method

KW - Condition numbers

KW - Diagonal scaling

KW - Preconditioning

UR - http://www.scopus.com/inward/record.url?scp=0001451993&partnerID=8YFLogxK

U2 - 10.1137/S0036142997330809

DO - 10.1137/S0036142997330809

M3 - Article

SN - 0036-1429

VL - 36

SP - 1901

EP - 1932

JO - SIAM Journal on Numerical Analysis

JF - SIAM Journal on Numerical Analysis

IS - 6

ER -