@article {doi:10.1002/num.22304,
	title = {Preconditioning trace coupled 3d-1d systems using fractional Laplacian},
	journal = {Numerical Methods for Partial Differential Equations},
	volume = {35},
	number = {1},
	pages = {375-393},
	publisher = { Wiley},
	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 standard 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 system. The norms are employed to construct a robust block diagonal preconditioner for the coupled problem.},
	keywords = {Lagrange multipliers, preconditioning, saddle-point problem, trace},
	doi = {10.1002/num.22304},
	url = {https://onlinelibrary.wiley.com/doi/abs/10.1002/num.22304},
	author = {Kuchta, Miroslav and Mardal, Kent-Andre and Mortensen, Mikael}
}
