An Implicit Discontinuous Galerkin Scheme for Solving the Cardiovascular Drug-eluting Stents' Model

Abstract

This article proposes an implicit discontinuous Galerkin scheme for solving the 2D model of drug release from cardiovascular drug-eluting stents, given the binding of the drug that is saturated and reversible. The scheme uses discontinuous Galerkin discretization based on bubble modal basis functions in space and the backward Euler discretization in time. For this purpose, we have divided each domain in space into several sub-domains. In each sub-domain, we have used a bubble modal basis as the local bases. These are more efficiently matched in each sub-domain than the global bases, especially in the discontinuous points of the domain. The existence and uniqueness of the discontinuous Galerkin solution are examined. The consistency, stability, and convergence of the proposed scheme are thoroughly provided. Under the stability conditions, the domains of penalty parameters and time steps are determined such that the scheme remains stable. The appropriate range of changes for the velocity, $\mathbf{u}_w$, is obtained, regarding the convergence analysis. Numerical results show high performance due to the accuracy and efficiency of the proposed scheme.