Iterative strong coupling of dimensionally heterogeneous models

Abstract

SUMMARYIn this article we address decomposition strategies especially tailored to perform strong coupling of dimensionally heterogeneous models, under the hypothesis that one wants to solve each submodel separately and implement the interaction between subdomains by boundary conditions alone. The novel methodology takes full advantage of the small number of interface unknowns in this kind of problems. Existing algorithms can be viewed as variants of the 'natural' staggered algorithm in which each domain transfers function values to the other, and receives fluxes (or forces), and vice versa. This natural algorithm is known as Dirichlet-to-Neumann in the Domain Decomposition literature. Essentially, we propose a framework in which this algorithm is equivalent to applying Gauss-Seidel iterations to a suitably defined (linear or nonlinear) system of equations. It is then immediate to switch to other iterative solvers such as GMRES or other Krylov-based method, which we assess through numerical experiments showing the significant gain that can be achieved. Indeed, the benefit is that an extremely flexible, automatic coupling strategy can be developed, which in addition leads to iterative procedures that are parameter-free and rapidly converging. Further, in linear problems they have the finite termination property.

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