2018 Nov 16, Luka Grubišić, University of Zagreb, Constrained PDE models on metric graphs: should we let the linear algebra solver do the heavy lifting?

Luka Grubišić, University of Zagreb
Constrained PDE models on metric graphs: should we let the linear algebra solver do the heavy lifting?  
In this talk we present two 1D models of an endovascular stent. Endovascular stents are biomedical devices made of struts used for treating arterial stenosis. The state of the system in both models is described by a vector valued function on a metric graph which satisfies a system of ODEs and a set of algebraic constraints. Both models are obtained by Γ-convergence from 3D nonlinear elasticity. As a result of asymptotic analysis, solutions are contained in a set of functions which are constrained by a set of algebraic constraints in the nodes of the graph and by requiring that the middle line of a strut does not extend. In the first model we place all constraints in the variational product space and build a finite element approximation there, whereas in the second model we study the problem in a large “free” product space and leave all of the constraints as a part of the system matrix to be removed by a linear algebra solver in a search of the solution. We will present convergence results for both models, but the much more puzzling question is which of the models will yield more efficient numerical methods. Namely, the second model yields a system matrix which is more than three times larger than in the first model. We present results of empirical comparison of the solution methods. We will further also study properties of the eigenvalue and the dynamical problem on a metric graph and discuss the solution methods and their efficiency. Finally, we will present validation experiments for the method by comparing it empirically to the 3D model solved by the standard legacy finite element code. This is a joint work with M. Ljulj, V. Mehrmann and J. Tambaca.