Discontinuous Galerkin approximation of a nonlinear multiphysics problem arising in ultrasound-enhanced drug delivery

TL;DR

This paper develops a discontinuous Galerkin numerical method for a complex multiphysics model combining nonlinear ultrasound wave propagation with drug diffusion, demonstrating theoretical convergence and validating results through simulations.

Contribution

It introduces a novel DG discretization for a coupled nonlinear wave and diffusion system, with proven stability and convergence under realistic assumptions.

Findings

Established well-posedness and optimal convergence for the pressure approximation.

Proved convergence of the coupled wave and diffusion system.

Validated theoretical results with numerical experiments.

Abstract

Motivated by simulations of ultrasound-enhanced drug delivery, this work presents the numerical analysis of a mathematical model that captures the influence of ultrasound waves on the diffusivity of the drug. The system under study consists of the Westervelt wave equation, accounting for the nonlinear propagation of ultrasound, coupled to a convection-diffusion equation modeling the drug concentration. In particular, drug delivery is affected by ultrasound through a pressure-dependent diffusion coefficient. The Westervelt equation is supplemented by linear absorbing boundary conditions as a means of reducing spurious reflections off the boundaries of computational domains. For spatial discretization of this multiphysics system, we employ a discontinuous Galerkin approach on simplicial meshes. Under suitable assumptions on the exact pressure and the mesh size, we first establish well-posedness, non-degeneracy, and optimal convergence rates in the energy norm for the semi-discrete pressure subproblem. The smallness of the semi-discrete pressure is then used to establish the well-posedness and convergence of the wave--convection-diffusion system under suitable regularity of the exact concentration. Finally, theoretical findings are illustrated through numerical experiments.