Solvers for Multiphysics Problems in Brain Biomechanics

Ana Budisa
Simula Research Laboratory, Oslo, Norway
Monday 28th November 2022
Aula Saleri - MOX - Dipartimento di Matematica
We are interested in reliable simulations of some biophysical processes in the brain, such as blood flow and metabolic waste clearance. Modeling those processes results in interface-driven multiphysics problems that can be coupled across dimensions. However, the complexity of the interface coupling often deteriorates the performance of standard methods to finding the numerical solution. Therefore, we derive preconditioners and solution techniques which target specifically such multiphysics problems for order optimal solving performance. The coupling that enforces constraints on the interface can primarily be imposed in two ways – with or without a Lagrange multiplier. In the former, the well-posedness of the system of equations is given in fractional Sobolev spaces weighted by material parameters. Therefore, the robust preconditioners for the interface problems are represented as a sum of fractional Laplacians that can include both negative and positive fractionalities. To handle fractional operators numerically, we implement methods based on the rational approximation. For the systems without a Lagrange multiplier, we focus on solvers based on algebraic multigrid method with custom smoothers that preserve the coupling information on each coarse level. We prove that, for the two-level setting, we obtain convergence that is independent of the mesh and material parameters. In both approaches, we show parameter-independence and scalability with regards to number of the degrees of freedom of the system. This is demonstrated on several numerical examples on realistic geometries, such as the viscous-porous flow coupling of the cerebrospinal fluid or the mixeddimensional model of flow in vascularized brain tissue. Contatto: