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A computational study of algebraic coarse spaces for two-level overlapping additive Schwarz preconditioners

Filipe A. C. S. Alves, Alexander Heinlein, Hadi Hajibeygi

TL;DR

This work develops an entirely algebraic framework for coarse spaces in two-level overlapping additive Schwarz preconditioners, focusing on energy-minimizing coarse-basis functions. It compares the algebraic MsFEM-based AMS approach with the classical GDSW and RGDSW coarse spaces, showing AMS to be robust and scalable for high-contrast diffusion when coarse-space interfaces connect to high-coefficient inclusions. The authors derive the AMS prolongation operator and highlight its vertex-centered nature with interface values computed recursively on edges and faces, providing numerical evidence that AMS often outperforms RGDSW and matches or surpasses GDSW across scenarios. The study delivers a practical, discretization-agnostic coarse-space construction within a unified framework for evaluating preconditioners in elliptic PDEs.

Abstract

The two-level overlapping additive Schwarz method offers a robust and scalable preconditioner for various linear systems resulting from elliptic problems. One of the key to these properties is the construction of the coarse space used to solve a global coupling problem, which traditionally requires information about the underlying discretization. An algebraic formulation of the coarse space reduces the complexity of its assembly. Furthermore, well-chosen coarse basis functions within this space can better represent changes in the problem's properties. Here we introduce an algebraic formulation of the multiscale finite element method (MsFEM) based on the algebraic multiscale solver (AMS) in the context of the two-level Schwarz method. We show how AMS is related to other energy-minimizing coarse spaces. Furthermore, we compare the AMS with other algebraic energy-minimizing spaces: the generalized Dryja-Smith-Widlund (GDSW), and the reduced dimension GDSW (RGDSW).

A computational study of algebraic coarse spaces for two-level overlapping additive Schwarz preconditioners

TL;DR

This work develops an entirely algebraic framework for coarse spaces in two-level overlapping additive Schwarz preconditioners, focusing on energy-minimizing coarse-basis functions. It compares the algebraic MsFEM-based AMS approach with the classical GDSW and RGDSW coarse spaces, showing AMS to be robust and scalable for high-contrast diffusion when coarse-space interfaces connect to high-coefficient inclusions. The authors derive the AMS prolongation operator and highlight its vertex-centered nature with interface values computed recursively on edges and faces, providing numerical evidence that AMS often outperforms RGDSW and matches or surpasses GDSW across scenarios. The study delivers a practical, discretization-agnostic coarse-space construction within a unified framework for evaluating preconditioners in elliptic PDEs.

Abstract

The two-level overlapping additive Schwarz method offers a robust and scalable preconditioner for various linear systems resulting from elliptic problems. One of the key to these properties is the construction of the coarse space used to solve a global coupling problem, which traditionally requires information about the underlying discretization. An algebraic formulation of the coarse space reduces the complexity of its assembly. Furthermore, well-chosen coarse basis functions within this space can better represent changes in the problem's properties. Here we introduce an algebraic formulation of the multiscale finite element method (MsFEM) based on the algebraic multiscale solver (AMS) in the context of the two-level Schwarz method. We show how AMS is related to other energy-minimizing coarse spaces. Furthermore, we compare the AMS with other algebraic energy-minimizing spaces: the generalized Dryja-Smith-Widlund (GDSW), and the reduced dimension GDSW (RGDSW).
Paper Structure (5 sections, 13 equations, 5 figures)

This paper contains 5 sections, 13 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Energy-minimizing coarse basis functions
  3. GDSW and RGDSW coarse basis functions
  4. AMS coarse basis functions
  5. Numerical examples

Figures (5)

  • Figure 1: Example of an interface partition of grid elements in two (left) and three (right) dimensions. The vertex, edge and face sub-components are marked in red, blue and yellow, respectively.
  • Figure 2: Discontinuous coefficient functions used in the examples. The dark blue regions correspond to the high coefficient inclusions with a value of $\alpha (x) = 10^8$, and $\alpha (x) = 1$ elsewhere. Illustration for $1/H = 4$.
  • Figure 3: Number of iterations (left) and condition number estimate (right) versus the number of subdomains for the coefficient function in \ref{['fig:coeff_functions']} (left).
  • Figure 4: Number of iterations (left) and condition number estimate (right) versus the number of subdomains for the coefficient function in \ref{['fig:coeff_functions']} (right).
  • Figure 5: Eigenvalues for each coarse space for the coefficient function in \ref{['fig:coeff_functions']} (right) and different subdomain sizes: left: $1/H = 8$; right: $1/H = 64$.