分数拉普拉斯算子的正则性理论与高阶数值方法
Regularity theory and high order numerical methods for the (1d)-Fractional Laplacian
AI总结：
本文研究了一维分数拉普拉斯边界值问题的正则性理论及高阶数值方法，通过分解解为边缘奇异权重与正则未知数的乘积，获得解的正则性特征，并提出基于切比雪夫多项式的Nyström方法，具有高精度和高效性。
摘要：
本文提出了针对一维分数拉普拉斯边界值问题的正则性结果及相关高阶数值方法。基于将解分解为特定边缘奇异权重ω与正则未知数的乘积，通过右端项的光滑性来表征解的正则性。特别地，对于右端项在伯恩斯坦椭圆内解析的情况，正则未知数在相同椭圆内也解析。此外，本文还给出了一个精确的Sobolev正则性结果，通过引入的加权Sobolev空间完全刻画了分数拉普拉斯算子的像空间。本文的理论依赖于对特定加权积分算子的完整特征分解，基于切比雪夫多项式基底。所提出的基于切比雪夫的Nyström数值方法对于分数拉普拉斯Dirichlet问题显著优于其他算法。本文提出的精确误差估计表明，该算法具有谱精度，其收敛率仅取决于右端项的光滑性。特别是，对于解析（或无限光滑）右端项，收敛速度呈指数级（或比网格尺寸的任意幂更快）。算法的性质通过各种数值结果进行了展示。
英文摘要：
This paper presents regularity results and associated high-order numerical methods for one-dimensional Fractional-Laplacian boundary-value problems. On the basis of a factorization of solutions as a product of a certain edge-singular weight $ω$ times a "regular" unknown, a characterization of the regularity of solutions is obtained in terms of the smoothness of the corresponding right-hand sides. In particular, for right-hand sides which are analytic in a Bernstein Ellipse, analyticity in the same Bernstein Ellipse is obtained for the "regular" unknown. Moreover, a sharp Sobolev regularity result is presented which completely characterizes the co-domain of the Fractional-Laplacian operator in terms of certain weighted Sobolev spaces introduced in (Babuška and Guo, SIAM J. Numer. Anal. 2002). The present theoretical treatment relies on a full eigendecomposition for a certain weighted integral operator in terms of the Gegenbauer polynomial basis. The proposed Gegenbauer-based Nyström numerical method for the Fractional-Laplacian Dirichlet problem, further, is significantly more accurate and efficient than other algorithms considered previously. The sharp error estimates presented in this paper indicate that the proposed algorithm is spectrally accurate, with convergence rates that only depend on the smoothness of the right-hand side. In particular, convergence is exponentially fast (resp. faster than any power of the mesh-size) for analytic (resp. in nitely smooth) right-hand sides. The properties of the algorithm are illustrated with a variety of numerical results.
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