Galerkin finite element methods for convection-diffusion problems with exponential layers on Shishkin triangular meshes and hybrid meshes

In this work, we provide a convergence analysis for a Galerkin finite element method on a Shishkin triangular mesh and a hybrid mesh for a singularly perturbed convection-diffusion equation. The hybrid mesh replaces the triangles of the Shishkin mesh by rectangles in the layer regions. The supercloseness results are established that the computed solution converges to the interpolant of the true solution with 3/2 order and 2 order (up to a logarithmic factor) on the two kinds of mesh, respectively. These convergence rates are uniformly valid with respect to the diffusion parameter and imply that the hybrid mesh is superior to the Shishkin triangular mesh. Numerical experiments illustrate these theoretical results.