遗传指数拟合方法求解多维漂移扩散方程
Genetic Exponentially Fitted Method for Solving Multi-dimensional Drift-diffusion Equations
AI总结：
提出一种基于指数拟合基函数的有限元方法，用于求解多维漂移扩散方程，通过利用连续分段多项式空间与无散向量空间的一一对应构造高阶非协调基函数，并证明一阶收敛性。
摘要：
本文提出了一种通用方法，用于开发高阶指数拟合基函数，以对多维漂移扩散方程进行有限元逼近，从而模拟生物分子电扩散过程。这类方法对于实现数值稳定性和效率非常理想。我们发现，通过利用次数为$k+1$的连续分段多项式空间与次数为$k$的无散向量空间之间的一一对应关系，可以构造出高阶二维指数拟合基函数，这些基函数在选定节点集上严格插值，但通常在边上不连续，从而张成非协调有限元空间。对于由无散Raviart-Thomas空间$RT_0^0$在两个不同节点集上构造的方法，证明了其一阶收敛性。
英文摘要：
A general approach was proposed in this article to develop high-order exponentially fitted basis functions for finite element approximations of multi-dimensional drift-diffusion equations for modeling biomolecular electrodiffusion processes. Such methods are highly desirable for achieving numerical stability and efficiency. We found that by utilizing the one-one correspondence between continuous piecewise polynomial space of degree $k+1$ and the divergence-free vector space of degree $k$, one can construct high-order 2-D exponentially fitted basis functions that are strictly interpolative at a selected node set but are discontinuous on edges in general, spanning nonconforming finite element spaces. First order convergence was proved for the methods constructed from divergence-free Raviart-Thomas space $RT_0^0$ at two different node sets
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