The error of approximation of a function using Poly-Sinc approximation was shown to have a convergence rate of exponential order over a global partition. However, it was assumed that the function values at the interpolation points were known. In this paper, we extend our work on poly-Sinc-based discontinuous Galerkin approximation, in which the function values at the interpolation points are replaced by unknown coefficients and solved by the discontinuous Galerkin method. To deal with functions having a singularity at an endpoint, we use Sinc partitions in our discontinuous Galerkin approximation. For such functions, we show that, by using a weighted $L^2$ norm over a Sinc partition in which the weight function is the reciprocal of the asymptotic density of Sinc points in that partition and computing the ${\ell }_2$ norm over all Sinc partitions constituting the global partition except the partition containing the singularity, the error of approximation between the exact solution of the ordinary differential equation and its poly-Sinc-based discontinuous Galerkin approximation has a convergence rate of exponential order similar to the one over the global partition. The numerical results are in agreement with our theoretical derivations of the approximation error. We compare our poly-Sinc-based discontinuous Galerkin method with the poly-Sinc collocation method. Our method shows better performance in the $L^2$ norm.

结果表明，使用Poly-Sinc逼近函数的逼近误差在全局分区上具有指数级的收敛速度。但是，假定插值点处的函数值是已知的。在本文中，我们扩展了基于聚Sinc的不连续Galerkin逼近的工作，其中插值点处的函数值被未知系数替换，并通过不连续Galerkin方法求解。为了处理在端点处具有奇点的函数，我们在不连续的Galerkin近似中使用Sinc分区。对于此类功能，我们通过使用加权${\mathrm{大号}}^{\mathrm{2个}}$ Sinc分区上的范数，其中权重函数是该分区中Sinc点的渐近密度的倒数，并计算 ${\ell }_{\mathrm{2个}}$除了包含奇异性的分区之外，构成全局分区的所有Sinc分区的范数，常微分方程的精确解与其基于多辛克的不连续Galerkin逼近之间的逼近误差具有与一个相似的指数级收敛速度。在全局分区上。数值结果与我们对近似误差的理论推导一致。我们将基于聚Sinc的不连续Galerkin方法与基于聚Sinc的搭配方法进行了比较。我们的方法在${\mathrm{大号}}^{\mathrm{2个}}$ 规范。