伪微分算子阶数为负二的最优拟对角预条件子
Optimal quasi-diagonal preconditioners for pseudodifferential operators of order minus two
AI总结：
本文提出用于伪微分算子阶数为负二的分段多项式离散化的拟对角预条件子，证明在高维情况下预条件子渐近最优，并通过数值实验验证结果。
摘要：
我们为任意空间维数中伪微分算子阶数为负二的分段多项式离散化提出拟对角预条件子。这里，拟对角意味着在稀疏变换下为对角形式。考虑形状正则的单纯形网格和任意固定多项式次数，我们证明在高维情况下，我们的预条件子渐近最优。二维、三维和四维中的数值实验证实了我们的结果。对于每个维度，我们报告了分段常数和分段线性多项式的情形下的条件数。
英文摘要：
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.
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