Galerkin methods based on hermite splines for singular perturbation problems

摘要

We develop Galerkin methods for solving the singularly perturbed two-point boundary value problem of high-order elliptic differential equations. These methods are based on Hermite splines with knots adapted to the singular behavior of the solution of the problem. We prove an optimal order of uniform convergence for the method with respect to the perturbation parameter. Specifically, we present a sufficient condition on the mesh of grid points that ensures the corresponding approximate solution has the optimal order of uniform convergence in the energy norm. We also construct optimal meshes that satisfy the sufficient condition. Numerical examples are presented to illustrate the method and the corresponding theoretical estimates.