Robust Algebraic multilevel preconditioning in $H(\mathrm{curl})$ and $H(\mathrm{div})$

TL;DR

This paper introduces a robust algebraic multilevel iteration method tailored for solving linear systems in $H( ext{curl})$ and $H( ext{div})$ spaces, demonstrating optimal complexity and robustness even with parameter jumps.

Contribution

The paper develops a new algebraic multilevel preconditioning method specifically for $H( ext{curl})$ and $H( ext{div})$ discretizations, with explicit recursion formulas and proven robustness.

Findings

Method is robust with respect to problem parameters.

Achieves optimal order complexity.

Numerical results confirm effectiveness even with parameter jumps.

Abstract

An algebraic multilevel iteration method for solving system of linear algebraic equations arising in $H(\mathrm{curl})$ and $H(\mathrm{div})$ spaces are presented. The algorithm is developed for the discrete problem obtained by using the space of lowest order Nedelec and Raviart-Thomas-Nedelec elements. The theoretical analysis of the method is based only on some algebraic sequences and generalized eigenvalues of local (element-wise) problems. In the hierarchical basis framework, explicit recursion formulae are derived to compute the element matrices and the constant $\gamma$ (which measures the quality of the space splitting) at any given level. It is proved that the proposed method is robust with respect to the problem parameters, and is of optimal order complexity. Supporting numerical results, including the case when the parameters have jumps, are also presented.