超声增强药物递送中非线性多物理场问题的间断伽辽金逼近
Discontinuous Galerkin approximation of a nonlinear multiphysics problem arising in ultrasound-enhanced drug delivery
AI总结：
针对超声增强药物递送，提出耦合Westervelt波动方程与对流扩散方程的数学模型，采用间断伽辽金方法离散，证明了半离散压力子问题的适定性与最优收敛率，并验证了数值实验。
摘要：
受超声增强药物递送模拟的启发，本文对捕捉超声波对药物扩散系数影响的数学模型进行了数值分析。所研究的系统由Westervelt波动方程（描述超声的非线性传播）与对流扩散方程（模拟药物浓度）耦合而成。特别地，药物递送通过压力依赖的扩散系数受超声影响。Westervelt方程补充了线性吸收边界条件，以减少计算域边界上的虚假反射。对于该多物理场系统的空间离散化，我们在单纯形网格上采用间断伽辽金方法。在精确压力和网格尺寸的适当假设下，我们首先建立了半离散压力子问题的适定性、非退化性和能量范数下的最优收敛率。然后利用半离散压力的小性，在精确浓度的适当正则性下，建立了波动-对流-扩散系统的适定性和收敛性。最后，通过数值实验说明了理论发现。
英文摘要：
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.
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