分数拉普拉斯算子的数值均质化
Numerical Homogenization of Heterogeneous Fractional Laplacians
AI总结：
本文提出一种多尺度方法求解具有异质扩散系数的分数拉普拉斯算子，通过子网格修正和准插值算子构建细尺度空间，实现计算高效且收敛率最优。
摘要：
本文开发了一种数值多尺度方法，用于求解具有异质扩散系数的分数拉普拉斯算子。当系数异质时，会增加计算成本。分数拉普拉斯算子在标准形式下是非局部算子，但Caffarelli-Silvestre扩展允许方程局部化。这增加了额外的空间维度和依赖分数阶的奇异/退化系数。通过子网格修正方法，我们修正基函数以自然加权Sobolev空间，并证明这些修正可以截断以设计计算高效且收敛率最优的方案。该方法的关键是使用准插值算子构建细尺度空间。由于扩展问题在临界边界上的解是主要关注点，我们构造了一个具有d和d+1维子集平均的投影准插值算子。我们证明该算子在加权Sobolev空间中满足局部稳定性和局部逼近性质。进一步，我们展示在足够光滑的力作用下，可以得到更高的收敛率，并利用临界边界上的全局L^2投影。我们还通过一些数值示例，利用二维2+1维中的投影准插值算子，演示了方法的收敛率和有效性。
英文摘要：
In this paper, we develop a numerical multiscale method to solve the fractional Laplacian with a heterogeneous diffusion coefficient. When the coefficient is heterogeneous, this adds to the computational costs. Moreover, the fractional Laplacian is a nonlocal operator in its standard form, however the Caffarelli-Silvestre extension allows for a localization of the equations. This adds a complexity of an extra spacial dimension and a singular/degenerate coefficient depending on the fractional order. Using a sub-grid correction method, we correct the basis functions in a natural weighted Sobolev space and show that these corrections are able to be truncated to design a computationally efficient scheme with optimal convergence rates. A key ingredient of this method is the use of quasi-interpolation operators to construct the fine scale spaces. Since the solution of the extended problem on the critical boundary is of main interest, we construct a projective quasi-interpolation that has both $d$ and $d+1$ dimensional averages over subsets in the spirit of the Scott-Zhang operator. We show that this operator satisfies local stability and local approximation properties in weighted Sobolev spaces. We further show that we can obtain a greater rate of convergence for sufficient smooth forces, and utilizing a global $L^2$ projection on the critical boundary. We present some numerical examples, utilizing our projective quasi-interpolation in dimension $2+1$ for analytic and heterogeneous cases to demonstrate the rates and effectiveness of the method.
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