High-Order Finite Element Methods for Singularly-Perturbed Elliptic and ParabolicProblems

摘要

We develop a framework for applying high-order finite element methods tosingularly-perturbed elliptic and parabolic differential systems that utilizes special quadrature rules to confine spurious effects, such as excess diffusion and non-physical oscillations, to boundary and interior layers. This approach is more suited for use with adaptive mesh-refinement and order-variation techniques than other problem-dependent methods. Quadrature rules, developed for two-point convection-diffusion and reaction-diffusion problems, are used with finite element software to solve examples involving ordinary and partial differential equations. Numerical artifacts are confined to layers for all combinations of meshes, orders, and singular perturbation parameters that were tested. Radau or Lobatto quadrature used with the finite element method to solve, respectively, convection-and reaction-diffusion problems provide the benefits of the specialized quadrature formulas and are simpler to implement.