一种直接不连续伽辽金方法用于分数对流-扩散和薛定谔类型方程
A direct discontinuous Galerkin method for fractional convection-diffusion and Schrödinger type equations
AI总结：
本文提出一种直接不连续伽辽金方法求解含分数拉普拉斯算子的分数对流-扩散和薛定谔类型方程，通过将分数算子分解为二阶导数和分数积分，实现高阶收敛性与稳定性。
摘要：
本文开发了一种直接不连续伽辽金（DDG）有限元方法，用于求解具有分数拉普拉斯算子的分数对流-扩散和薛定谔类型方程。分数算子阶数α（1<α<2）被表示为二阶导数与阶数为2-α的分数积分的组合。这些问题被表述为抛物型方程和低阶积分方程的系统。这允许我们应用基于直接弱形式的DDG方法，每个计算单元中对分数对流-扩散和薛定谔类型方程的解进行处理，让单元通过数值通量（∂_x u）*进行通信。此外，我们证明了对于一般分数对流-扩散和薛定谔问题，方法具有稳定性以及最优收敛阶O(h^{N+1})，其中h和N分别为空间步长和多项式次数。DDG方法具有更易实现的公式和实现优势以及高阶精度。最后，数值实验验证了该方法的理论结果。
英文摘要：
A direct discontinuous Galerkin (DDG) finite element method is developed for solving fractional convection-diffusion and Schrödinger type equations with a fractional Laplacian operator of order $α$ $(1<α<2)$. The fractional operator of order $α$ is expressed as a composite of second order derivative and a fractional integral of order $2-α$. These problems have been expressed as a system of parabolic equation and low order integral equation. This allows us to apply the DDG method which is based on the direct weak formulation for solutions of fractional convection-diffusion and Schrödinger type equations in each computational cell, letting cells communicate via the numerical flux $(\partial_{x}u)^{*}$ only. Moreover, we prove stability and optimal order of convergence $O(h^{N+1})$ for the general fractional convection-diffusion and Schrödinger problems where $h$, $N$ are the space step size and polynomial degree. The DDG method has the advantage of easier formulation and implementation as well as the high order accuracy. Finally, numerical experiments confirm the theoretical results of the method.
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