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

Maria Vasilyeva, Golo A. Wimmer, Ben S. Southworth

TL;DR

Extreme anisotropy in heat diffusion along magnetic field lines poses accuracy and solver challenges on non-grid-aligned meshes. The authors introduce a generalized spectral multiscale coarse space built from local eigenproblems of an anisotropic graph Laplacian and integrate it into a two-grid preconditioner, achieving anisotropy-independent convergence. They prove convergence properties of the multiscale space and demonstrate large-scale reduction in degrees of freedom (up to $10^{12}$ anisotropy) with robust performance across several magnetic-field configurations. This approach offers a scalable, accurate framework for magnetic confinement fusion simulations and related anisotropic diffusion problems, with potential extensions to evolving fields and alternative discretizations.

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.

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

TL;DR

Extreme anisotropy in heat diffusion along magnetic field lines poses accuracy and solver challenges on non-grid-aligned meshes. The authors introduce a generalized spectral multiscale coarse space built from local eigenproblems of an anisotropic graph Laplacian and integrate it into a two-grid preconditioner, achieving anisotropy-independent convergence. They prove convergence properties of the multiscale space and demonstrate large-scale reduction in degrees of freedom (up to anisotropy) with robust performance across several magnetic-field configurations. This approach offers a scalable, accurate framework for magnetic confinement fusion simulations and related anisotropic diffusion problems, with potential extensions to evolving fields and alternative discretizations.

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 , showing highly accurate results with a large system size reduction, and two-grid preconditioning that converges in iterations, independent of anisotropy.

Paper Structure

This paper contains 13 sections, 62 equations, 4 figures, 6 tables.

Table of Contents

  1. Introduction
  2. Problem formulation
  3. Multiscale space approximation
  4. Spectral multiscale basis functions
  5. Coarse grid approximation
  6. Convergence of the multiscale space
  7. Two-grid multiscale preconditioner
  8. Numerical results
  9. Multiscale space approximation
  10. Two-grid multiscale preconditioner
  11. Conclusion
  12. Multiscale space
  13. Two-grid convergence

Figures (4)

  • Figure 1: Illustration of the $10 \times 10$ coarse grid $\mathcal{T}_H$ with local domain $\omega_i$ and linear partition of unity function $\chi_i$
  • Figure 2: Local domain with magnetic field and corresponding eigenvectors for three different anisotropy ratios.
  • Figure 3: Reference solutions with magnetic field lines at final time for Test 1, 2 and 3, $DOF_h=201,385$
  • Figure 4: Error vs $DOF_H$ for Tests 1, 2 and 3 (from left to right).