An $hp$ finite element method for a singularly perturbed reaction-convection-diffusion boundary value problem with two small parameters

Abstract: We consider a second order singularly perturbed boundary value problem, of reaction-convection-diffusion type with two small parameters, and the approximation of its solution by the $hp$ version of the Finite Element Method on the so-called {\emph{Spectral Boundary Layer}} mesh. We show that the method converges uniformly, with respect to both singular perturbation parameters, at an exponential rate when the error is measured in the energy norm. Numerical examples are also presented, which illustrate our theoretical findings.