|Authors||M. Kuchta, K. Mardal and M. Mortensen|
|Title||Preconditioning trace coupled 3d-1d systems using fractional Laplacian|
|Project(s)||No Simula project|
|Publication Type||Journal Article|
|Journal||Numerical Methods for Partial Differential Equations|
|Keywords||Lagrange multipliers, preconditioning, saddle-point problem, trace|
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.