Thomas Führer

Pontificia Universidad Católica de Chile

In this talk I present results from one of our recent works where we consider pseudodifferential operators of order minus two in any space dimension and their discretization by piecewise polynomials.

We define and analyze quasi-diagonal preconditioners for the discretized operators where quasi-diagonal means diagonal up to sparse transformations. Considering shape regular simplicial meshes and arbitrary fixed polynomial degrees, we prove, for dimensions larger than one, that our preconditioners are asymptotically optimal.

Our analysis is based on the additive Schwarz theory and key ingredient is the decomposition of piecewise constants into the divergence of lowest-order Raviart-Thomas functions.

Numerical experiments in two, three and four dimensions confirm our theoretical results. We present examples for uniform and adaptive mesh-refinement.