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Nodal Coarsening and Sparse Ideal Interpolation for H(curl) Problems in Algebraic Multigrid

Taoli Shen, James Brannick, Robert Falgout, Karsten Kahl, Jacob Schroder

Abstract

We propose a sparse interpolation construction and a practical coarsening algorithm for the algebraic multigrid (AMG) method, tailored towards H(curl). Building on the generalized AMG framework, we introduce an interior/exterior splitting that yields both a refinement-based and a fully algebraic construction of the interpolation. The refinement-based approach follows geometric hierarchy, while the purely algebraic interpolation is constructed through a coarsening process that first coarsens a nodal dual problem and then builds coarse and fine variables using a matching algorithm. We establish the weak approximation property and the commuting relation under certain assumptions. Combined with matching block smoothers, the proposed interpolation yields an effective algebraic multilevel method. Numerical experiments show robustness under strong coefficient jumps, where the proposed methods substantially outperform standard geometric multigrid.

Nodal Coarsening and Sparse Ideal Interpolation for H(curl) Problems in Algebraic Multigrid

Abstract

We propose a sparse interpolation construction and a practical coarsening algorithm for the algebraic multigrid (AMG) method, tailored towards H(curl). Building on the generalized AMG framework, we introduce an interior/exterior splitting that yields both a refinement-based and a fully algebraic construction of the interpolation. The refinement-based approach follows geometric hierarchy, while the purely algebraic interpolation is constructed through a coarsening process that first coarsens a nodal dual problem and then builds coarse and fine variables using a matching algorithm. We establish the weak approximation property and the commuting relation under certain assumptions. Combined with matching block smoothers, the proposed interpolation yields an effective algebraic multilevel method. Numerical experiments show robustness under strong coefficient jumps, where the proposed methods substantially outperform standard geometric multigrid.
Paper Structure (10 sections, 3 theorems, 36 equations, 3 figures, 4 tables, 3 algorithms)

This paper contains 10 sections, 3 theorems, 36 equations, 3 figures, 4 tables, 3 algorithms.

Table of Contents

  1. Introduction
  2. The Curl-Curl Equation
  3. AMG interpolation for curl-curl
  4. Refinement-based Coarse Variables
  5. Sparse approximation of the ideal interpolation
  6. Fully Algebraic Coarsening
  7. Nodal coarsening
  8. Algebraic construction of coarse and fine variables
  9. Numerical Results
  10. Conclusions

Key Result

Lemma 1

Assume the stiffness interior block $A^s_{II}$ is invertible. Then

Figures (3)

  • Figure 1: Curl-curl on uniform quadrilateral mesh.
  • Figure 1: uniform meshes with Dirichlet boundary conditions
  • Figure 1: Unstructured Delaunay test. Left: piecewise-constant jump coefficients, $\mu=0.1$ in red and $\mu=10$ in blue. Right: finest level algebraic coarse variable pattern on $L=3$.

Theorems & Definitions (6)

  • Lemma 1
  • Proof 1
  • Theorem 2: Approximate harmonic property for $P_{\mathrm{app}}$
  • Proof 2
  • Theorem 3: Commuting property for $P_{\mathrm{app}}$
  • Proof 3