Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

An abstract framework for heterogeneous coupling: stability, approximation and preconditioning

Silvia Bertoluzza, Erik Burman

TL;DR

This work develops an abstract, solver-agnostic framework for heterogeneous coupling inspired by FETI, reducing the global problem to a Schur-type system on Lagrange multipliers and enabling black-box local solvers for subproblems. It provides a rigorous continuous well-posedness theory under inf-sup conditions and a finite kernel $Z=\ker A$, then extends to discretization with controlled consistency and optional stabilization when coupling meshes do not align. A comprehensive preconditioning strategy for the Schur system is constructed, combining a canonical mapping with local preconditioners to yield favorable spectral properties, including a bound $\kappa(\widehat{M}^{-1}\widehat{S}_h) \lesssim |\,\log(H/\delta)\|^2$ under localization assumptions. The framework is demonstrated on domain decomposition examples with Neumann and Dirichlet couplings and is applicable to FEM/BEM and multiphysics problems, offering flexible, scalable, and robust code-coupling capabilities with provable stability. The approach thus enables integration of heterogeneous subproblem solvers while maintaining theoretical guarantees and practical efficiency through sophisticated preconditioning.

Abstract

We consider heterogeneous coupling problems on an abstract level, establishing fundamental principles of domain decomposition agnostic to the solvers of the local subproblems. Introducing a coupling framework reminiscent of FETI methods, but here on abstract form, we establish conditions for stability and minimal requirements for well-posedness on the continuous level, as well as conditions on local solvers for the approximation of subproblems. We then discuss stability of the resulting Lagrange multiplier methods and show stability under a mesh condition between the local discretizations and the mortar space. If this condition is not satisfied we show how a stabilization, acting only on the multiplier can be used to achieve stability. The design of preconditioners of the Schur complement system is discussed in the unstabilized case. Finally we discuss some applications that enter the framework.

An abstract framework for heterogeneous coupling: stability, approximation and preconditioning

TL;DR

This work develops an abstract, solver-agnostic framework for heterogeneous coupling inspired by FETI, reducing the global problem to a Schur-type system on Lagrange multipliers and enabling black-box local solvers for subproblems. It provides a rigorous continuous well-posedness theory under inf-sup conditions and a finite kernel , then extends to discretization with controlled consistency and optional stabilization when coupling meshes do not align. A comprehensive preconditioning strategy for the Schur system is constructed, combining a canonical mapping with local preconditioners to yield favorable spectral properties, including a bound under localization assumptions. The framework is demonstrated on domain decomposition examples with Neumann and Dirichlet couplings and is applicable to FEM/BEM and multiphysics problems, offering flexible, scalable, and robust code-coupling capabilities with provable stability. The approach thus enables integration of heterogeneous subproblem solvers while maintaining theoretical guarantees and practical efficiency through sophisticated preconditioning.

Abstract

We consider heterogeneous coupling problems on an abstract level, establishing fundamental principles of domain decomposition agnostic to the solvers of the local subproblems. Introducing a coupling framework reminiscent of FETI methods, but here on abstract form, we establish conditions for stability and minimal requirements for well-posedness on the continuous level, as well as conditions on local solvers for the approximation of subproblems. We then discuss stability of the resulting Lagrange multiplier methods and show stability under a mesh condition between the local discretizations and the mortar space. If this condition is not satisfied we show how a stabilization, acting only on the multiplier can be used to achieve stability. The design of preconditioners of the Schur complement system is discussed in the unstabilized case. Finally we discuss some applications that enter the framework.
Paper Structure (13 sections, 16 theorems, 204 equations)

This paper contains 13 sections, 16 theorems, 204 equations.

Table of Contents

  1. Introduction
  2. Problem setting
  3. Discretization
  4. Including stabilization
  5. Solving the resulting discrete problem
  6. Construction of the preconditioner $\widehat{M}^{-1}$
  7. Efficiency of the preconditioner
  8. The solution recipe
  9. Examples
  10. Domain decomposition with Neumann coupling
  11. Domain decomposition with Dirichlet coupling
  12. Proof of Theorem \ref{['thm:preconditioning']}
  13. Proof of Lemma \ref{['lem:minusprec']}

Key Result

Theorem 2.5

The following inf-sup condition holds Moreover the operator $s$ satisfies and

Theorems & Definitions (39)

  • Remark 2.2
  • Theorem 2.5
  • proof
  • Corollary 2.6
  • Lemma 3.6
  • proof
  • Remark 3.7
  • Theorem 3.9
  • proof
  • Proposition 3.10
  • ...and 29 more