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Optimal quasi-diagonal preconditioners for pseudodifferential operators of order minus two. (English) Zbl 1417.65213

Summary: We present quasi-diagonal preconditioners for piecewise polynomial discretizations of pseudodifferential operators of order minus two in any space dimension. Here, quasi-diagonal means diagonal up to a sparse transformation. Considering shape regular simplicial meshes and arbitrary fixed polynomial degrees, we prove, for dimensions larger than one, that our preconditioners are asymptotically optimal. Numerical experiments in two, three and four dimensions confirm our results. For each dimension, we report on condition numbers for piecewise constant and piecewise linear polynomials.

MSC:

65N38 Boundary element methods for boundary value problems involving PDEs
35J25 Boundary value problems for second-order elliptic equations
65F35 Numerical computation of matrix norms, conditioning, scaling
65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
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