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Local Solvers for High-Order Patch Smoothers via p-Multigrid

Michał Wichrowski

TL;DR

A vertex patch smoother where local problems are solved inexactly by a nested, matrix-free p-multigrid, creating a multigrid-within-multigrid framework that achieves robustness with respect to both polynomial degree $p$ and mesh refinement, even on heavily distorted meshes.

Abstract

I propose a vertex patch smoother where local problems are solved inexactly by a nested, matrix-free p-multigrid, creating a multigrid-within-multigrid framework. A single iteration of the local solver can be evaluated with $\mathcal{O}(p^{d+1})$ operations, and the approach is applicable to non-separable problems on unstructured meshes. Numerical experiments demonstrate limited sensitivity to geometric distortion and high-contrast coefficients. When used in a global geometric multigrid solver, the method achieves robustness with respect to both polynomial degree $p$ and mesh refinement, even on heavily distorted meshes.

Local Solvers for High-Order Patch Smoothers via p-Multigrid

TL;DR

A vertex patch smoother where local problems are solved inexactly by a nested, matrix-free p-multigrid, creating a multigrid-within-multigrid framework that achieves robustness with respect to both polynomial degree and mesh refinement, even on heavily distorted meshes.

Abstract

I propose a vertex patch smoother where local problems are solved inexactly by a nested, matrix-free p-multigrid, creating a multigrid-within-multigrid framework. A single iteration of the local solver can be evaluated with operations, and the approach is applicable to non-separable problems on unstructured meshes. Numerical experiments demonstrate limited sensitivity to geometric distortion and high-contrast coefficients. When used in a global geometric multigrid solver, the method achieves robustness with respect to both polynomial degree and mesh refinement, even on heavily distorted meshes.
Paper Structure (16 sections, 9 equations, 7 figures, 5 tables, 2 algorithms)

This paper contains 16 sections, 9 equations, 7 figures, 5 tables, 2 algorithms.

Table of Contents

  1. Introduction
  2. Geometric finite element multigrid methods
  3. Vertex patch smoothers
  4. Local solvers via p-multigrid
  5. p-multigrid components
  6. Transfer operators
  7. Level operators
  8. Coarse solver
  9. Smoothers
  10. Numerical validation on a single patch
  11. Baseline Performance
  12. Resilience to Geometric Distortion
  13. Resilience to Variable Coefficients
  14. Summary
  15. Application as a smoother in geometric multigrid
  16. ...and 1 more sections

Figures (7)

  • Figure 1: Workflow in smoother application on a single patch
  • Figure 1: Deformed structured (top row) and unstructured (bottom row) patches in 2D. The distortion parameter $\delta$ is $0\%$, $10\%$, $25\%$, and $35\%$ (left to right).
  • Figure 1: Meshes used in the numerical experiments.
  • Figure 2: Number of CG iterations versus polynomial degree $p$ for solving the local patch problem on a Cartesian patch in 2D and 3D. The CG method is preconditioned with a single p-multigrid V-cycle; the dashed line denotes results for a non-structured (simplicial) patch.
  • Figure 3: Number of CG iterations vs. mesh distortion for the Cartesian-reinforced smoother. The CG solver is preconditioned with a single p-multigrid V-cycle applied as the Cartesian-reinforced patch solver. Each point reports the average number of CG iterations required to reduce the relative residual to $10^{-8}$. The left and right subplots show results for 2D and 3D vertex patches, respectively; the horizontal axis denotes the mesh distortion parameter used to deform the patches.
  • ...and 2 more figures