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Additive Schwarz methods for fourth-order variational inequalities

Jongho Park

TL;DR

It is proved that the two-level method is scalable in the sense that the convergence rate of the method depends on H/h and H/δ only, where h and H are the typical diameters of an element and a subdomain, respectively, and δ measures the overlap among the subdomains.

Abstract

Fourth-order variational inequalities are encountered in various scientific and engineering disciplines, including elliptic optimal control problems and plate obstacle problems. In this paper, we consider additive Schwarz methods for solving fourth-order variational inequalities. Based on a unified framework of various finite element methods for fourth-order variational inequalities, we develop one- and two-level additive Schwarz methods. We prove that the two-level method is scalable in the sense that the convergence rate of the method depends on $H/h$ and $H/δ$ only, where $h$ and $H$ are the typical diameters of an element and a subdomain, respectively, and $δ$ measures the overlap among the subdomains. This proof relies on a new nonlinear positivity-preserving coarse interpolation operator, the construction of which was previously unknown. To the best of our knowledge, this analysis represents the first investigation into the scalability of the two-level additive Schwarz method for fourth-order variational inequalities. Our theoretical results are verified by numerical experiments.

Additive Schwarz methods for fourth-order variational inequalities

TL;DR

It is proved that the two-level method is scalable in the sense that the convergence rate of the method depends on H/h and H/δ only, where h and H are the typical diameters of an element and a subdomain, respectively, and δ measures the overlap among the subdomains.

Abstract

Fourth-order variational inequalities are encountered in various scientific and engineering disciplines, including elliptic optimal control problems and plate obstacle problems. In this paper, we consider additive Schwarz methods for solving fourth-order variational inequalities. Based on a unified framework of various finite element methods for fourth-order variational inequalities, we develop one- and two-level additive Schwarz methods. We prove that the two-level method is scalable in the sense that the convergence rate of the method depends on and only, where and are the typical diameters of an element and a subdomain, respectively, and measures the overlap among the subdomains. This proof relies on a new nonlinear positivity-preserving coarse interpolation operator, the construction of which was previously unknown. To the best of our knowledge, this analysis represents the first investigation into the scalability of the two-level additive Schwarz method for fourth-order variational inequalities. Our theoretical results are verified by numerical experiments.
Paper Structure (18 sections, 10 theorems, 75 equations, 2 figures, 1 algorithm)

This paper contains 18 sections, 10 theorems, 75 equations, 2 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Finite element discretizations
  4. Additive Schwarz method
  5. One-level method
  6. Domain decomposition
  7. Convergence analysis
  8. Two-level method
  9. Coarse space
  10. Convergence analysis
  11. Local and coarse problems
  12. Local problems
  13. Coarse problems
  14. Numerical experiments
  15. Displacement obstacle problem of clamped plates
  16. ...and 3 more sections

Key Result

Theorem 2.4

\newlabelThm:conv0 Assume that Assumptions Ass:FEM and Ass:stable holds. In Algorithm Alg:ASM, we have $\{ u^{(n)} \} \subset K_h$ and

Figures (2)

  • Figure 1: Decay of the relative energy error $\frac{F_h (u^{(n)}) - F_h(u_h)}{| F_h (u_h) |}$ in the one- and two-level additive Schwarz methods for the displacement obstacle problem of clamped plates \ref{['PL']}. The subdomain size $H$ and the mesh size $h$ vary satisfying $H/h = 2^3$.
  • Figure 2: Decay of the relative energy error $\frac{F_h (u^{(n)}) - F_h(u_h)}{| F_h (u_h) |}$ in the one- and two-level additive Schwarz methods for the elliptic distributed optimal control problem \ref{['OC']}. The subdomain size $H$ and the mesh size $h$ vary satisfying $H/h = 2^3$.

Theorems & Definitions (25)

  • Remark 2.2
  • Theorem 2.4
  • Proof 1
  • Remark 2.5
  • Theorem 3.2
  • Proof 2
  • Proposition 4.3
  • Proof 3
  • Lemma 4.4
  • Remark 4.5
  • ...and 15 more