We present convergence theory for corrected quadrature rules on uniform Cartesian grids for functions with a point singularity. We begin by deriving an error estimate for the punctured trapezoidal rule, and then derive error expansions. We define the corrected trapezoidal rules, based on the punctured trapezoidal rule, where the weights for the nodes close to the singularity are judiciously corrected based on these expansions. Then we define the composite corrected trapezoidal rules for a larger family of functions using series expansions around the point singularity and applying corrected trapezoidal rules appropriately. We prove that we can achieve high order accuracy by using a sufficient number of correction nodes around the point singularity and of expansion terms.

翻译：我们提出了关于统一的笛卡尔格网格中具有点奇特功能的校正二次曲线规则的趋同理论。 我们首先对被刺穿的捕捉和分裂规则得出错误估计,然后得出错误扩展。 我们根据被刺穿的捕捉和分裂规则界定了被纠正的捕捉和分裂规则,在这些扩展的基础上,接近奇特的结点的结点的权重得到了明智的纠正。 然后我们用围绕点奇特和扩张条件的系列扩展来界定一个更大的函数大家庭的经纠正的复合孔虫和分裂规则。 我们证明,我们可以通过在点奇特和扩张条件周围使用足够数量的校正节点来达到高度的准确性。