Finite-element methods for singularly perturbed high-order elliptic two-point boundary value problems. II: convection-diffusion-type problems

摘要

We consider singularly perturbed high-order elliptic two-point boundary value problems of convection-diffusion type. Under suitable hypotheses, the coercivity of the associated bilinear form is proved and a representation result for the solutions of such problems is given. A family of Galerkin finite-element methods based on piecewise polynomial test/trial functions on a Shishkin mesh is constructed and proved to be convergent, uniformly in the perturbation para-meter, in energy and WL norms. Numerical results are presented for a second-order problem and fourth-order problems.