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

TL;DR

This paper introduces a preconditioning approach for 3d-1d coupled elliptic systems using fractional Laplacian-based norms, enabling efficient multilevel algorithms for complex multiscale problems.

Contribution

It proposes a novel preconditioning method employing fractional Sobolev norms for 3d-1d coupled systems, improving computational efficiency.

Findings

Negative fractional Sobolev norms effectively precondition the Schur complement.

The proposed block diagonal preconditioner demonstrates robustness.

Numerical experiments confirm the method's efficiency.

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.