Immersed finite element methods for the multi-layer porous wall model

Abstract

The dissertation is concerned with the multi-layer porous wall model which is proposed to simulate the drug transfer mechanism in the arterial wall when treat with the cardiovascular diseases. It is an interface problem with two types of interface points: the imperfect contact interface point at the first layer and the rough coefficient interface points at other layers. We firstly consider the linear and quadratic immersed finite element (IFE) methods to solve the steady-state problem. Then, we investigate fundamental properties of these IFE spaces. Through interpolation error analysis, we prove that these IFE spaces have optimal approximation capabilities. In addition, we get the optimal convergence rate by using both linear and quadratic IFE methods to solve the multi-layer porous wall model. Furthermore, we analyze the long time stability and the asymptotic behavior of the IFE method for the multi-layer porous wall model for the drug-eluting stents (DES). With the help of the IFE methods for the spatial descretization, and the implicit Euler scheme for the temporal discretization, respectively, we deduce the global stability of fully discrete solution. Then, we investigate the asymptotic behavior of the discrete scheme which reveals that the multi-layer porous wall model converges to the corresponding elliptic equation if the body force approaches to a steady-state. In addition, we use these IFE spaces to solve the unsteady problem. We prove that the backward Euler scheme has the optimal convergence rate in both the L² and H¹ norms. We also do some numerical experiments to verify the theoretical results. In the last part, some conclusions and future work plans are given.