Ultrasound enhanced drug transport is a multiphysics problem that involves acoustic waves propagation, bioheat transfer and drug transport. The numerical modeling of this problem requires the solution of a coupled system of partial differential equations. A wave-type equation for acoustic pressure and two nonlinear parabolic-type equations: a diffusion-reaction equation for bioheat transfer and a convection-diffusion-reaction equation for drug transport.

In this paper we focus on the numerical analysis of such coupled system. We propose and derive convergence estimates for a piecewise linear finite element method (FEM) with quadrature. We prove that the FEM is second order convergent for concentration with respect to a discrete L²-norm. Since concentration depends on the gradient of acoustic pressure, this result shows that the FEM is superconvergent. In fact, piecewise linear FEM have optimal order one in the H¹-norm then, the optimal convergence rate for concentration in a L²-norm should be at most one. Numerical results backing the theoretical findings are included.