Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

An abstract approach to the Robin-Robin method

Emil Engström, Eskil Hansen

TL;DR

The paper extends an abstract Robin–Robin domain decomposition framework to initial-boundary value problems for linear and parabolic equations on Lipschitz domains. By formulating the interface problem via Steklov–Poincaré operators and employing a temporal $H^{1/2}$–based setting, it proves bijectivity and monotonicity of the interface operators using Banach–Nečas–Babuška, and establishes a well-posed parabolic weak formulation. It then shows the transmission problem is equivalent to a Steklov–Poincaré equation and demonstrates convergence of the Robin–Robin iterates to the coupled solution in $L^2(\\b R^+, V_i)$. These results enable a unified, rigorous treatment of linear and nonlinear elliptic and parabolic problems within the same abstract framework, with no additional regularity assumptions on the subdomain boundaries beyond Lipschitz continuity.

Abstract

Recently, their has been development of an abstract approach to the Robin--Robin method, enabling the treatment of linear and nonlinear elliptic and parabolic equations on Lipschitz domains within one framework. However, previously this setting has not been applicable to initial-boundary value problems. The aim of this short note is therefore to demonstrate that this general framework can be applied to such problems as well.

An abstract approach to the Robin-Robin method

TL;DR

The paper extends an abstract Robin–Robin domain decomposition framework to initial-boundary value problems for linear and parabolic equations on Lipschitz domains. By formulating the interface problem via Steklov–Poincaré operators and employing a temporal –based setting, it proves bijectivity and monotonicity of the interface operators using Banach–Nečas–Babuška, and establishes a well-posed parabolic weak formulation. It then shows the transmission problem is equivalent to a Steklov–Poincaré equation and demonstrates convergence of the Robin–Robin iterates to the coupled solution in . These results enable a unified, rigorous treatment of linear and nonlinear elliptic and parabolic problems within the same abstract framework, with no additional regularity assumptions on the subdomain boundaries beyond Lipschitz continuity.

Abstract

Recently, their has been development of an abstract approach to the Robin--Robin method, enabling the treatment of linear and nonlinear elliptic and parabolic equations on Lipschitz domains within one framework. However, previously this setting has not been applicable to initial-boundary value problems. The aim of this short note is therefore to demonstrate that this general framework can be applied to such problems as well.
Paper Structure (4 sections, 8 theorems, 29 equations)

This paper contains 4 sections, 8 theorems, 29 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Weak formulations of parabolic equations
  4. Transmission problem and Steklov--Poincaré operators

Key Result

lemma 1

The set $\mathcal{D}$ is dense in $H^{1/2}\bigl(\mathbb{R}^+, L^2(\Omega)\bigr)$, $H^{1/2}_{00}\bigl(\mathbb{R}^+, L^2(\Omega)\bigr)$, and $L^2\bigl(\mathbb{R}^+, V\bigr)$. The set $\mathcal{D}_i$ is dense in $H^{1/2}\bigl(\mathbb{R}^+, L^2(\Omega_i)\bigr)$, $H^{1/2}_{00}\bigl(\mathbb{R}^+, L^2(\Ome

Theorems & Definitions (12)

  • lemma 1
  • proof
  • lemma 2
  • remark 1
  • lemma 3
  • proof
  • lemma 4
  • lemma 5
  • lemma 6
  • theorem 1
  • ...and 2 more