An efficient weak Galerkin FEM for third-order singularly perturbed convection-diffusion differential equations on layer-adapted meshes

Abstract

In this article, we study the weak Galerkin finite element method to solve a class of a third order singularly perturbed convection-diffusion differential equations. Using some knowledge on the exact solution, we prove a robust uniform convergence of order $O\left(N^{-(k-1/2)}\right)$ on the layer-adapted meshes including Bakhvalov-Shishkin type, and Bakhvalov-type and almost optimal uniform error estimates of order $O\left({(N^{-1}\ln N)}^{(k-1/2)}\right)$ on Shishkin-type mesh with respect to the perturbation parameter in the energy norm using high-order piecewise discontinuous polynomials of degree k. Here N is the number mesh intervals. We conduct numerical examples to support our theoretical results.