分数阶Laplacian算子的局部能量估计
Local energy estimates for the fractional Laplacian
摘要：
$s \in (0,1)$阶的积分分数拉普拉斯算子是一个非局部算子。众所周知，解决方案的狄利克雷问题涉及这样一个运营商表现出代数边界奇异性，无论域的正则性。这反过来又恶化了解的全局正则性，从而降低了数值解的整体收敛速度。对于有限元离散，我们得到的局部误差估计的$H^s$ -离散和显示最佳的收敛速度在内部的域仅假设网格是形状规则的。这些估计量化了减少的近似误差集中在域的边界附近的事实。我们用几个数值例子来说明我们的理论结果。
英文摘要：
The integral fractional Laplacian of order $s \in (0,1)$ is a nonlocal operator. It is known that solutions to the Dirichlet problem involving such an operator exhibit an algebraic boundary singularity regardless of the domain regularity. This, in turn, deteriorates the global regularity of solutions and as a result the global convergence rate of the numerical solutions. For finite element discretizations, we derive local error estimates in the $H^s$-seminorm and show optimal convergence rates in the interior of the domain by only assuming meshes to be shape-regular. These estimates quantify the fact that the reduced approximation error is concentrated near the boundary of the domain. We illustrate our theoretical results with several numerical examples.
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