Preconditioning trace coupled 3$d$-1$d$ systems using fractional Laplacian
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.
Cite
@article{arxiv.1612.03574,
title = {Preconditioning trace coupled 3$d$-1$d$ systems using fractional Laplacian},
author = {Miroslav Kuchta and Kent-Andre Mardal and Mikael Mortensen},
journal= {arXiv preprint arXiv:1612.03574},
year = {2018}
}