In this paper, we construct a quadrature scheme to numerically solve the nonlocal diffusion equation $(\mathcal{A}^\alpha+b\mathcal{I})u=f$ with $\mathcal{A}^\alpha$ the $\alpha$-th power of the regularly accretive operator $\mathcal{A}$. Rigorous error analysis is carried out and sharp error bounds (up to some negligible constants) are obtained. The error estimates include a wide range of cases in which the regularity index and spectral angle of $\mathcal{A}$, the smoothness of $f$, the size of $b$ and $\alpha$ are all involved. The quadrature scheme is exponentially convergent with respect to the step size and is root-exponentially convergent with respect to the number of solves. Some numerical tests are presented in the last section to verify the sharpness of our estimates. Furthermore, both the scheme and the error bounds can be utilized directly to solve and analyze time-dependent problems.

翻译：本文构造了一种求积格式，用于数值求解非局部扩散方程$(\mathcal{A}^\alpha+b\mathcal{I})u=f$，其中$\mathcal{A}^\alpha$是正则增生算子$\mathcal{A}$的$\alpha$次幂。我们进行了严格的误差分析，并获得了精确的误差界（忽略可忽略的常数）。误差估计涵盖了多种情况，包括$\mathcal{A}$的正则性指标和谱角、$f$的光滑性、$b$与$\alpha$的取值等参数。该求积格式关于步长呈指数收敛，关于求解次数呈根指数收敛。最后一节通过数值试验验证了估计的精确性。此外，该格式与误差界可直接用于求解和分析时间依赖问题。