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Transmission problems and domain decompositions for non-autonomous parabolic equations on evolving domains

Amal Alphonse, Ana Djurdjevac, Emil Engström, Eskil Hansen

Abstract

Parabolic equations on evolving domains model a multitude of applications including various industrial processes such as the molding of heated materials. Such equations are numerically challenging as they require large-scale computations and the usage of parallel hardware. Domain decomposition is a common choice of numerical method for stationary domains, as it gives rise to parallel discretizations. In this study, we introduce a variational framework that extends the use of such methods to evolving domains. In particular, we prove that transmission problems on evolving domains are well posed and equivalent to the corresponding parabolic problems. This in turn implies that the standard non-overlapping domain decompositions, including the Robin-Robin method, become well defined approximations. Furthermore, we prove the convergence of the Robin--Robin method. The framework is based on a generalization of fractional Sobolev-Bochner spaces on evolving domains, time-dependent Steklov-Poincaré operators, and elements of the approximation theory for monotone maps.

Transmission problems and domain decompositions for non-autonomous parabolic equations on evolving domains

Abstract

Parabolic equations on evolving domains model a multitude of applications including various industrial processes such as the molding of heated materials. Such equations are numerically challenging as they require large-scale computations and the usage of parallel hardware. Domain decomposition is a common choice of numerical method for stationary domains, as it gives rise to parallel discretizations. In this study, we introduce a variational framework that extends the use of such methods to evolving domains. In particular, we prove that transmission problems on evolving domains are well posed and equivalent to the corresponding parabolic problems. This in turn implies that the standard non-overlapping domain decompositions, including the Robin-Robin method, become well defined approximations. Furthermore, we prove the convergence of the Robin--Robin method. The framework is based on a generalization of fractional Sobolev-Bochner spaces on evolving domains, time-dependent Steklov-Poincaré operators, and elements of the approximation theory for monotone maps.

Paper Structure

This paper contains 10 sections, 36 theorems, 186 equations, 2 figures, 1 table.

Table of Contents

  1. Introduction
  2. Evolving domains
  3. Abstract time-evolving function spaces
  4. Compatibility for spaces defined on evolving domains
  5. Spaces on the interior
  6. Spaces on the boundary
  7. Weak formulation and existence of solutions
  8. Temporal H½̂-setting for evolving domains
  9. Transmission problems on evolving domains
  10. Steklov--Poincaré operators and convergence of the Robin--Robin scheme on evolving domains

Key Result

Lemma 3.3

If $(X,\phi)$ is compatible and $Y\bigl(\mathbb{R}_+; X(0)\bigr)$ is a separable Hilbert space, then $Y_X(\mathbb{R}_+)$ is a separable Hilbert space.

Figures (2)

  • Figure 1: Left image: sketch of a steel beam $\Omega$ at time $t$ being reshaped via hot rolling and thereafter solidified by water cooling. Right image: Decomposition of the steel beam at time $t$, where $\Omega(t)$ is decomposed into $\Omega_1(t)$ and $\Omega_2(t)=\Omega_{2,1}(t)\cup \Omega_{2,2}(t)$.
  • Figure 2: An example of an evolving domain decomposition with an interior domain $\Omega_1$ and an exterior domain $\Omega_2$.

Theorems & Definitions (71)

  • Definition 3.1
  • Definition 3.2
  • Lemma 3.3
  • proof
  • Definition 4.1
  • Lemma 4.2
  • proof
  • Lemma 4.3
  • proof
  • Lemma 4.4
  • ...and 61 more