This research demonstrates improved accuracy in numerical integration for functions with boundary layers, suggesting effective methods for challenging integrands.
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 L2 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.
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