论文标题
Lipschitz域上的分数椭圆问题：规律性和近似值
Fractional Elliptic Problems on Lipschitz Domains: Regularity and Approximation
论文作者
论文摘要
这项调查取决于Lipschitz域上线性和准线性分数椭圆形问题的规律性和近似之间的相互作用。对于线性的Dirichlet积分拉普拉斯式，在简短地回顾了Hölder的规律性和应用之后，我们讨论了BESOV空间及其Sobolev的新型最佳移动定理。这些结果扩展到有限地平线的问题，对随后的误差分析具有重要作用。此外，我们介绍了BESOV规律性到分数$ p $ laplacian的扩展，并回顾了分数最小图和粘性的规律性。我们使用连续的分段线性有限元元素离散这些问题，并得出线性问题的全局和局部错误估计，从而改善了准均匀和分级网格的一些现有错误估计。我们还提出了一个BPX预处理程序，该预处理相对于分数顺序和级别数量都很健壮。我们以分数准线性问题的离散化及其错误分析结束。我们通过几个启发性的数值实验来说明理论。
This survey hinges on the interplay between regularity and approximation for linear and quasi-linear fractional elliptic problems on Lipschitz domains. For the linear Dirichlet integral Laplacian, after briefly recalling Hölder regularity and applications, we discuss novel optimal shift theorems in Besov spaces and their Sobolev counterparts. These results extend to problems with finite horizon and are instrumental for the subsequent error analysis. Moreover, we dwell on extensions of Besov regularity to the fractional $p$-Laplacian, and review the regularity of fractional minimal graphs and stickiness. We discretize these problems using continuous piecewise linear finite elements and derive global and local error estimates for linear problems, thereby improving some existing error estimates for both quasi-uniform and graded meshes. We also present a BPX preconditioner which turns out to be robust with respect to both the fractional order and the number of levels. We conclude with the discretization of fractional quasi-linear problems and their error analysis. We illustrate the theory with several illuminating numerical experiments.
