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Mixed finite element methods for addressing multi-species diffusion using the Maxwell-Stefan equations

机译：使用Maxwell-Stefan方程解决多物种扩散的混合有限元方法

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摘要

The Maxwell-Stefan equations are a system of nonlinear partial differential equations that describe the diffusion of multiple chemical species in a container. These equations are of particular interest for their applications to biology and chemical engineering. The nonlinearity and coupled nature of the equations involving many variables rule out analytical solutions, so numerical methods are often used. In the literature the system is inverted to write fluxes as functions of the species gradient before any numerical method is applied. In this paper it is shown that employing a mixed finite element method makes the inversion unnecessary, allowing the numerical solution of Maxwell-Stefan equations in their primitive form. A mixed variational formulation is derived in the general n-ary case. A priori error estimates between the finite element and exact solutions are obtained. The order of convergence of the method is then verified and compared with standard methods using a manufactured solution. Finally, the solution is computed for a test case from the literature involving the diffusion of three species and compared to solutions from other methods.
机译：Maxwell-Stefan方程是一个非线性偏微分方程组，用于描述容器中多种化学物质的扩散。这些方程式特别适用于它们在生物学和化学工程中的应用。涉及许多变量的方程的非线性和耦合性质排除了解析解，因此经常使用数值方法。在文献中，在应用任何数值方法之前，将系统反转以将通量写入物种梯度函数。本文表明，采用混合有限元方法使反演变得不必要，从而允许以原始形式对Maxwell-Stefan方程进行数值求解。在一般的n元情况下，得出混合的变式公式。获得了有限元和精确解之间的先验误差估计。然后验证该方法的收敛顺序，并使用制造的解决方案将其与标准方法进行比较。最后，针对涉及三个物种扩散的文献中的测试案例计算出解决方案，并将其与其他方法的解决方案进行比较。

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