hp-FEM for singularly perturbed reaction-diffusion equations in curvilinear polygons

The hp-version of the finite element method (hp-FEM) is applied to a singularly perturbed reaction-diffusion model problem of the form -ε²Δu + u = f in curvilinear polygonal domains Ω. For piecewise analytic input data it is shown that the hp-FEM can lead to robust exponential convergence on appropriately designed meshes, i.e., the rate of convergence is exponential and does not deteriorate as the perturbation parameter ε tends to zero. These meshes consist of one layer of needle elements of width O(pε) at the boundary (here, p is the polynomial degree employed) and geometric refinement toward the vertices of the domain.

The key ingredients of the convergence analysis will be presented. It is based on new, sharp analytic regularity results for the solution u in which the critical parameters (ε, distance to the boundary δΩ, distance to the vertices of Ω) enter explicitly.

Numerical examples will illustrate the robust exponential convergence of the hp-FEM. Additionally, the effect of numerical quadrature is considered, particularly for quadrature rules with 'minimal' number of points as used in spectral methods.