The main motivation of the present work is the numerical study of a system of Partial Differential Equations that governs drug transport, through a target tissue or organ, when enhanced by the simultaneous action of an electric field and a temperature rise. The electric field, while forcing charged drug molecules through the tissue or the organ, thus creating a convection field, also leads to a rise in temperature that affects drug diffusion. The differential system is composed by a nonlinear elliptic equation, describing the potential of the electric field, and by two parabolic equations: a diffusion–reaction equation for temperature and a convection–diffusion–reaction for drug concentration. The temperature and the concentration equations are coupled with the potential equation via a reaction term and the convection and diffusion terms respectively. As the parabolic equations depend directly on the potential and its gradient, the central question is the design and mathematical study of an accurate method for the elliptic equation and its gradient. We propose a finite difference method, which is equivalent to a fully discrete piecewise linear finite element method, with superconvergent/supercloseness properties. The method is second order convergent with respect to a $H^1$-discrete norm for the elliptic problem, and with respect to a $L^2$-discrete norm for the two parabolic problems. The stability properties of the method are also analyzed. Numerical experiments illustrating the drug transport for different electrical protocols are also included.

本工作的主要动机是对偏微分方程系统的数值研究，该系统控制电场和温度升高同时作用时通过目标组织或器官的药物传输。电场虽然迫使带电的药物分子穿过组织或器官，从而产生对流场，但也会导致温度升高，从而影响药物扩散。微分系统由描述电场电势的非线性椭圆方程和两个抛物线方程组成：温度的扩散反应方程和药物浓度的对流扩散反应。温度和浓度方程分别通过反应项以及对流和扩散项与电势方程耦合。由于抛物线方程直接取决于电势及其梯度，因此核心问题是椭圆方程及其梯度的精确方法的设计和数学研究。我们提出了一种有限差分方法，该方法等效于具有超收敛/超闭合特性的完全离散的分段线性有限元方法。该方法相对于 这等效于具有超收敛/超闭合特性的完全离散的分段线性有限元方法。该方法相对于 这等效于具有超收敛/超闭合特性的完全离散的分段线性有限元方法。该方法相对于$H^{\mathrm{1个}}$椭圆问题的离散规范，关于 ${\mathrm{大号}}^2$两个抛物线问题的离散范数。还分析了该方法的稳定性。还包括数值实验，说明了针对不同电方案的药物转运。