Multilevel diagonal scaling preconditioners for boundary element equations on locally refined meshes
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We study a multilevel preconditioner for the Galerkin boundary element matrix arising from a symmetric positive-definite bilinear form. The associated energy norm is assumed to be equivalent to a Sobolev norm of positive, possibly fractional, order m on a bounded (open or closed) surface of dimension d, with \(0<2m\le d\). We consider piecewise linear approximation on triangular elements. Successive levels of the mesh are created by selectively subdividing elements within local refinement zones. Hanging nodes may be created and the global mesh ratio can grow exponentially with the number of levels. The coarse-grid correction consists of an exact solve, and the correction on each finer grid amounts to a simple diagonal scaling involving only those degrees of freedom whose associated nodal basis functions overlap the refinement zone. Under appropriate assumptions on the choice of refinement zones, the condition number of the preconditioned system is shown to be bounded by a constant independent of the number of degrees of freedom, the number of levels and the global mesh ratio. In addition to applying to Galerkin discretisation of hypersingular boundary integral equations, the theory covers finite element methods for positive-definite, self-adjoint elliptic problems with Dirichlet boundary conditions.
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