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arXiv · 1612.03574

Preconditioning trace coupled 3$d$-1$d$ systems using fractional Laplacian

Miroslav Kuchta·Kent-Andre Mardal·Mikael Mortensen
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Abstract

Multiscale or multiphysics problems often involve coupling of partial differential equations posed on domains of different dimensionality. In this work we consider a simplified model problem of a 3d-1d coupling and the main objective is to construct algorithms that may utilize stan- dard multilevel algorithms for the 3d domain, which has the dominating computational complexity. Preconditioning for a system of two elliptic problems posed, respectively, in a three dimensional domain and an embedded one dimensional curve and coupled by the trace constraint is discussed. Investigating numerically the properties of the well-defined discrete trace operator, it is found that negative fractional Sobolev norms are suitable preconditioners for the Schur complement of the sys- tem. The norms are employed to construct a robust block diagonal preconditioner for the coupled problem.

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Miroslav Kuchta, Kent-Andre Mardal, Mikael Mortensen. 2016-12-12. Preconditioning trace coupled 3$d$-1$d$ systems using fractional Laplacian. https://doi.org/10.1002/num.22304

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