Preconditioning trace coupled 3$d$-1$d$ systems using fractional Laplacian
Preprint
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.Keywords
All Related Versions
This publication has 22 references indexed in Scilit:
- Optimal solvers for linear systems with fractional powers of sparse SPD matricesNumerical Linear Algebra with Applications, 2018
- Numerical approximation of fractional powers of elliptic operatorsMathematics of Computation, 2015
- A computational model of drug delivery through microcirculation to compare different tumor treatmentsInternational Journal for Numerical Methods in Biomedical Engineering, 2014
- Cerebral Microcirculation and Oxygen Tension in the Human Secondary CortexAnnals of Biomedical Engineering, 2013
- Modeling Blood Flow Circulation in Intracranial Arterial Networks: A Comparative 3D/1D Simulation StudyAnnals of Biomedical Engineering, 2010
- Preconditioning discretizations of systems of partial differential equationsNumerical Linear Algebra with Applications, 2010
- Vascular Graph Model to Simulate the Cerebral Blood Flow in Realistic Vascular NetworksJournal of Cerebral Blood Flow & Metabolism, 2009
- Numerical solution of saddle point problemsActa Numerica, 2005
- Appending boundary conditions by Lagrange multipliers: Analysis of the LBB conditionNumerische Mathematik, 2001
- The finite element method with Lagrangian multipliersNumerische Mathematik, 1973