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Schwarz Methods for Nonlocal Problems

Matthias Schuster, Christian Vollmann, Volker Schulz

Abstract

The first domain decomposition methods for partial differential equations were already developed in 1870 by H. A. Schwarz. Here we consider a nonlocal Dirichlet problem with variable coefficients, where a nonlocal diffusion operator is used. We find that domain decomposition methods like the so-called Schwarz methods seem to be a natural way to solve these nonlocal problems. In this work we show the convergence for nonlocal problems, where specific symmetric kernels are employed, and present the implementation of the multiplicative and additive Schwarz algorithms in the above mentioned nonlocal setting.

Schwarz Methods for Nonlocal Problems

Abstract

The first domain decomposition methods for partial differential equations were already developed in 1870 by H. A. Schwarz. Here we consider a nonlocal Dirichlet problem with variable coefficients, where a nonlocal diffusion operator is used. We find that domain decomposition methods like the so-called Schwarz methods seem to be a natural way to solve these nonlocal problems. In this work we show the convergence for nonlocal problems, where specific symmetric kernels are employed, and present the implementation of the multiplicative and additive Schwarz algorithms in the above mentioned nonlocal setting.
Paper Structure (21 sections, 11 theorems, 86 equations, 8 figures, 3 tables, 5 algorithms)

This paper contains 21 sections, 11 theorems, 86 equations, 8 figures, 3 tables, 5 algorithms.

Table of Contents

  1. Introduction
  2. Problem Formulation
  3. Nonlocal Dirichlet Problems
  4. Nonlocal Problems with Robin Boundary Conditions
  5. Schwarz Methods
  6. Schwarz Methods for Nonlocal Dirichlet Problems
  7. Schwarz Methods for Nonlocal Problems with Neumann Boundary Conditions
  8. Well-posedness of the Multiplicative Schwarz Method
  9. Well-posedness of the Multiplicative Schwarz Method for Nonlocal Dirichlet Problems
  10. Well-posedness of the Multiplicative Schwarz Method for Nonlocal Problems with Neumann Boundary Conditions
  11. Schwarz Methods for the Finite Element Formulation
  12. Finite Element Approximation
  13. Splitting of Inner Boundary Basis Functions
  14. Formulation of the Schwarz Methods
  15. Patch Tests
  16. ...and 6 more sections

Key Result

Theorem 2.4

The space $\mathcal{V}_c(\Omega, \mathcal{I}^N, \mathcal{I}^D)$ is complete regarding $||\cdot||_{\mathcal{V}(\Omega, \mathcal{I}^N, \mathcal{I}^D)}$. As a consequence, $V_c(\Omega, \mathcal{I}^N, \mathcal{I}^D)$ is a Hilbert space with respect to $||\cdot||_{V(\Omega, \mathcal{I}^N, \mathcal{I}^D)}

Figures (8)

  • Figure 7.1: The domain $\Omega = (0,1)^2$ is divided into $\Omega_1$, $\Omega_2$ and $\Omega_3$. The nonlocal boundary $\mathcal{I}$ is depicted in red.
  • Figure 7.2: Here, we can see the residual error of the multiplicative and additive Schwarz method for the nonlocal Dirichlet problem with a singular symmetric kernel regarding different choices for the mesh resolution $h$. In all cases the error decreases in a linear fashion, which we expected at least for the multiplicative Schwarz version due to Remark \ref{['remark:conv_discrete_schwarz']}. Moreover, the additive version needs roughly twice as many iterations as the multiplicative Schwarz algorithm.
  • Figure 7.3: In this example $\Omega$ is divided in two subdomains $\Omega_1$ and $\Omega_2$ and the nonlocal boundary $\mathcal{I}$ is decomposed in $\mathcal{I}_1^N$ and $\mathcal{I}_2^N$.
  • Figure 7.4: In this picture the residual error regarding the multiplicative Schwarz method for the nonlocal Problem with Neumann boundary condition w.r.t. a selection of mesh parameter $h$ is depicted. Again, we can observe a linear decrease in the residual error.
  • Figure 7.5: Here, we see the decomposition of $\Omega$ in a turquoise area $\Omega_1$ and in a gray domain $\Omega_2$ that we use for the patch test and for testing the (preconditioned) GMRES in Chapter \ref{['chap:preconditioned_gmres']}. The nonlocal boundary $\mathcal{I}$ is again depicted in red.
  • ...and 3 more figures

Theorems & Definitions (30)

  • Definition 2.1
  • Remark 2.2
  • Definition 2.3
  • Theorem 2.4
  • proof
  • Corollary 2.5
  • Remark 3.1
  • Remark 3.2
  • Theorem 4.1
  • Remark 4.2
  • ...and 20 more