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On the role of relaxation and acceleration in the non-overlapping Schwarz alternating method for coupling

Giulia Sambataro, Irina Tezaur

Abstract

The purpose of this paper is to study the influence of relaxation and acceleration techniques on the convergence behavior of the non-overlapping Schwarz algorithm with alternating Dirichlet-Neumann transmission conditions in the context of domain decomposition- (DD-) based coupling. After demonstrating that the multiplicative Schwarz scheme can be formulated as a fixed-point iteration, we explore, both theoretically and numerically, two promising techniques for speeding up the method: (i) Aitken acceleration and (ii) Anderson acceleration. In the process, we derive a robust and efficient adaptive variant of Anderson acceleration, termed "Anderson with memory adaptation". We compare the proposed acceleration strategies to the well-known classical relaxed Dirichlet-Neumann Schwarz alternating method. Our results suggest that, while Aitken-accelerated Schwarz is the best approach in terms efficiency and robustness when considering two sub-domain DDs, Anderson-accelerated Schwarz is the method of choice in larger multi-domain setting.

On the role of relaxation and acceleration in the non-overlapping Schwarz alternating method for coupling

Abstract

The purpose of this paper is to study the influence of relaxation and acceleration techniques on the convergence behavior of the non-overlapping Schwarz algorithm with alternating Dirichlet-Neumann transmission conditions in the context of domain decomposition- (DD-) based coupling. After demonstrating that the multiplicative Schwarz scheme can be formulated as a fixed-point iteration, we explore, both theoretically and numerically, two promising techniques for speeding up the method: (i) Aitken acceleration and (ii) Anderson acceleration. In the process, we derive a robust and efficient adaptive variant of Anderson acceleration, termed "Anderson with memory adaptation". We compare the proposed acceleration strategies to the well-known classical relaxed Dirichlet-Neumann Schwarz alternating method. Our results suggest that, while Aitken-accelerated Schwarz is the best approach in terms efficiency and robustness when considering two sub-domain DDs, Anderson-accelerated Schwarz is the method of choice in larger multi-domain setting.
Paper Structure (26 sections, 3 theorems, 49 equations, 12 figures, 4 tables, 4 algorithms)

This paper contains 26 sections, 3 theorems, 49 equations, 12 figures, 4 tables, 4 algorithms.

Table of Contents

  1. Introduction
  2. Non-overlapping Schwarz for DD-based coupling
  3. Problem formulation
  4. Dirichlet-Neumann Schwarz scheme
  5. Reformulation of multiplicative Schwarz as a fixed-point iteration method
  6. Schwarz acceleration techniques
  7. Aitken acceleration
  8. Analysis of the Aitken-accelerated Schwarz scheme
  9. Anderson acceleration
  10. Comparisons to and connections with Aitken acceleration
  11. Analysis of the Anderson-accelerated Schwarz scheme
  12. Anderson with memory adaptation
  13. Numerical results
  14. One-dimensional Laplace equation
  15. Schwarz with classical relaxation
  16. ...and 11 more sections

Key Result

Theorem 1

Suppose $d=1$. Let $g^{(k+1)}=T(g^{(k)})$ be the fixed-point iteration of an unrelaxed Schwarz scheme (line line:fp-iter in Algorithm alg:DN_relax_fixed_point with $\rho =1$). Assuming that the unrelaxed Schwarz method converges linearly ($p=1$), the Aitken-accelerated Schwarz algorithm (Algorithm a

Figures (12)

  • Figure 1: (a) Global domain. (b) Sketch of a non-overlapping decomposition of the domain $\Omega$ into $\Omega_1$ and $\Omega_2$ with interface $\Gamma$.
  • Figure 2: 1D Laplace equation. (a) Classical relaxation-accelerated Schwarz solutions with the optimal choice of $\rho=\bar{x}$. (b) Aitken accelerated Schwarz solutions with $\rho^{(1)}=1$. (c) Unrelaxed Anderson-accelerated Schwarz solutions. The middle and right sub-plots in (c) also show the values of $\alpha_0$ and $\alpha_1$ in histogram form for $k=2$ and $k=3$, respectively. All subplots are for the specific case of $\bar{x} = 0.7$.
  • Figure 3: 2D nonlinear elasticity. (a) The global domain. (b) The global displacement magnitude solution. (c) Non-overlapping partition with $N_{\text{dd}} = 2$.
  • Figure 4: 2D nonlinear elasticity with $N_{\text{dd}} = 2$. Convergence of the Dirichlet-Neumann Schwarz alternating method with classical relaxation for different values of $\rho$. The left panel is showing convergence of $e_{\text{abs}}^{(k)}$, whereas the right panel shows convergence of $e_{\text{rel}}^{(k)}$.
  • Figure 5: 2D nonlinear elasticity with $N_{\text{dd}} = 2$. Sub-plots (a) and (b) depict the convergence of Aitken-accelerated Schwarz for five different choices of $\rho^{(1)}$ and two choices of $N_0$. Sub-plots (c) and (d) depict the values of the Aitken parameter $\rho^{(k)}$ as a function of the Schwarz iteration $k$, again for two choice of $N_0$. The parameter $N_0$ is set to $10$ in subfigures (a) and (c), and to $2$ in subfigures (b) and (d).
  • ...and 7 more figures

Theorems & Definitions (16)

  • Remark 1
  • Remark 2
  • Remark 3
  • Remark 4
  • Remark 5
  • Theorem 1
  • proof
  • Remark 6
  • Remark 7
  • Theorem 2
  • ...and 6 more