Numerical Integration of the Function with Twin Boundary Layers

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 [see formula in PDF] 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.