Preconditioning trace coupled 3$d$-1$d$ systems using fractional Laplacian
This work addresses the challenge of preconditioning multiscale problems with different dimensionalities, which is relevant for computational mechanics and multiphysics simulations.
The paper develops a robust block diagonal preconditioner for 3d-1d coupled elliptic systems using fractional Laplacian norms, enabling the use of standard multilevel algorithms for the 3d domain. Numerical experiments demonstrate the effectiveness of the preconditioner.
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.