On preconditioning saddle point systems with trace constraints coupling 3$d$ and 1$d$ domains -- applications to matching and nonmatching FEM discretizations

摘要

Multiscale or multiphysics problems often involve the coupling of partialdifferential equations posed on domains of different dimensionality. In thiswork we consider a simplified model problem of a 3$d$-1$d$ coupling and themain motivation is to construct algorithms that may utilize standard multilevelalgorithms for the 3$d$ domain, which has the dominating computationalcomplexity. Preconditioning for a system of two elliptic problems posed,respectively, in a three dimensional domain and an embedded one dimensionalcurve and coupled by the trace constraint is discussed. Investigatingnumerically the properties of the well-defined discrete trace operator, it isfound that negative fractional Sobolev norms are suitable preconditioners forthe Schur complement. The norms are employed to construct a robust blockdiagonal preconditioner for the coupled problem.