标题： 正则化椭圆问题的先验误差估计 显示英文标题

标题： A priori error estimates of regularized elliptic problems

摘要： 狄拉克δ分布的近似常用于通过卷积创建一系列光滑函数，以逼近非光滑（广义）函数。 在本工作中，我们展示了这种近似过程在标准Sobolev范数下的先验收敛速率，在对狄拉克δ分布近似的要求最小正则性条件下。 将这些估计应用于具有奇异支撑强迫项的椭圆问题的数值解，使我们能够为相应的正则化问题提供精确的$H^1$和$L^2$误差估计。 作为应用，我们展示了正则化浸入界面方法的有限元近似结果与非正则化方法具有相同的收敛速率，只要将狄拉克δ近似的支撑设置为网格尺寸的倍数，而实现复杂度仅为一小部分。 提供了数值实验来支持我们的理论。 显示英文摘要

摘要： Approximations of the Dirac delta distribution are commonly used to create sequences of smooth functions approximating nonsmooth (generalized) functions, via convolution. In this work, we show a priori rates of convergence of this approximation process in standard Sobolev norms, with minimal regularity assumptions on the approximation of the Dirac delta distribution. The application of these estimates to the numerical solution of elliptic problems with singularly supported forcing terms allows us to provide sharp $H^1$ and $L^2$ error estimates for the corresponding regularized problem. As an application, we show how finite element approximations of a regularized immersed interface method result in the same rates of convergence of its non-regularized counterpart, provided that the support of the Dirac delta approximation is set to a multiple of the mesh size, at a fraction of the implementation complexity. Numerical experiments are provided to support our theories.