Anisotropic mesh refinement: The conditioning of Galerkin boundary element matrices and simple preconditioners

In this paper we obtain upper and lower bounds on the spectrum of the stiffness matrix arising from a finite element Galerkin approximation (using nodal basis functions) of a bounded, symmetric bilinear form which is elliptic on a Sobolev space of real index m is an element of [- 1, 1]. The key point is that the finite element mesh is required to be neither quasi-uniform nor shape-regular, so that our theory allows anisotropic meshes often used in practice. (However, we assume that the polynomial degree of the elements is fixed.) Our bounds indicate the ill-conditioning which can arise from anisotropic mesh refinement. In addition we obtain spectral bounds for the diagonally scaled stiffness matrix, which indicate the improvement provided by this simple preconditioning. For the special case of boundary integral operators on a two-dimensional screen in R-3, numerical experiments show that our bounds are sharp. We find that diagonal scaling essentially removes the ill-conditioning due to mesh degeneracy, leading to the same asymptotic growth in the condition number as arises for a quasi-uniform mesh refinement. Our results thus generalize earlier work by Bank and Scott [SIAM J. Numer. Anal., 26 (1989), pp. 1383 - 1394] and Ainsworth, McLean, and Tran [SIAM J. Numer. Anal., 36 ( 1999), pp. 1901 - 1932] for the shape-regular case.