hp FEM for ReactionDiusion Equations I Robust Exponential Convergence

A singularly perturbed reactiondiusion equation in two dimensions is considered We assume analyticity of the input data ie the boundary of the domain is an analytic curve and the right hand side is analytic We show that the hp version of the nite element method leads to robust exponential convergence provided that one layer of needle elements of width Op is inserted near the domain boundary that is the rate of convergence is O expbp and independent of the perturbation parameter Additionally we show that the use of numerical quadrature for the evaluation of the stiness matrix and the load vector retains the exponential rate of convergence In particular the Spectral Element Method based on the use of a GaussLobatto quadrature rule with p p points yields robust exponential convergence Keywords boundary layer singularly perturbed problem asymptotic expansions error bounds Many boundary value problems BVPs arising in mechanics depend on a small or large paramete