Mathematics

CLC number: O241

Numerical Integration of the Function with Twin Boundary Layers

带有双边界层函数的数值积分

School of Mathematical Sciences, Suzhou University of Science and Technology, Suzhou 215009, Jiangsu, China

^† Corresponding author. E-mail: ycheng@usts.edu.cn

Received: 28 May 2024

Abstract

Traditional numerical integration requires sufficient smoothness of the integrand to achieve high-order algebraic accuracy. If the function has a boundary layer with large gradient, the composite integration formula on the uniform mesh will produce very large integration errors. In this paper, we study the Newton-Cotes formula based on Lagrange interpolation functions, local $L^{\mathrm{2}}$ projection approximating the integrand, and Gauss integration. On the Shishkin mesh, we establish an optimal-order integration error estimate uniformly in the perturbation parameter. The convergence rate is the same as that for the smooth function. Numerical experiments confirm the sharpness of our theoretical results.

摘要

传统的数值积分需要被积函数具有较高的光滑性以获得高阶代数精度。当函数具有大梯度变化的双边界层时，基于均匀网格的复化积分公式将产生较大的积分误差。本文研究三类数值积分公式，即基于拉格朗日基函数的牛顿-科特斯公式、被积函数基于局部L²投影近似以及经典的高斯积分。借助于分片一致的Shishkin网格，证明了与小参数无关的最优阶积分误差估计，该收敛阶与光滑函数数值积分收敛阶相同。数值实验证实了我们的理论结果。