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We present quasi-diagonal preconditioners for piecewise polynomialdiscretizations of pseudodifferential operators of order minus two in any spacedimension. Here, quasi-diagonal means diagonal up to a sparse transformation.Considering shape regular simplicial meshes and arbitrary fixed polynomialdegrees, we prove, for dimensions larger than one, that our preconditioners areasymptotically optimal. Numerical experiments in two, three and four dimensionsconfirm our results. For each dimension, we report on condition numbers forpiecewise constant and piecewise linear polynomials.

Optimal quasi-diagonal preconditioners for pseudodifferential operators of order minus two