Optimal quasi-diagonal preconditioners for pseudodifferential operators of order minus two
Provides optimal preconditioners for a class of operators, benefiting numerical simulations in computational science and engineering, though the problem is specialized.
The paper introduces quasi-diagonal preconditioners for pseudodifferential operators of order minus two, proving asymptotic optimality for shape regular simplicial meshes in dimensions >1, with numerical validation in 2D, 3D, and 4D showing improved condition numbers.
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.