Multiscale approximation and two-grid preconditioner for extremely anisotropic heat flow

TL;DR

This paper develops a spectral multiscale method and a two-grid preconditioner to efficiently solve extremely anisotropic heat flow problems in magnetized plasmas, achieving high accuracy and convergence independent of anisotropy levels.

Contribution

It introduces a novel spectral multiscale basis for coarse grid approximation and a two-level spectral preconditioner that are highly effective for extreme anisotropy.

Findings

Achieves accurate solutions for anisotropy ratios up to 10^12.

Reduces system size significantly while maintaining accuracy.

Converges in O(1) iterations regardless of anisotropy.

Abstract

We consider anisotropic heat flow with extreme anisotropy, as arises in magnetized plasmas for fusion applications. Such problems pose significant challenges in both obtaining an accurate approximation as well in the construction of an efficient solver. In both cases, the underlying difficulty is in forming an accurate approximation of temperature fields that follow the direction of complex, non-grid-aligned magnetic fields. In this work, we construct a highly accurate coarse grid approximation using spectral multiscale basis functions based on local anisotropic normalized Laplacians. We show that the local generalized spectral problems yield local modes that align with magnetic fields, and provide an excellent coarse-grid approximation of the problem. We then utilize this spectral coarse space as an approximation in itself, and as the coarse-grid in a two-level spectral preconditioner. Numerical results are presented for several magnetic field distributions and anisotropy ratios up to $10^{12}$, showing highly accurate results with a large system size reduction, and two-grid preconditioning that converges in $O(1)$ iterations, independent of anisotropy.