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

The hp-version of the finite element method (hp-FEM) is appl

ied to

a singularly perturbed reaction-diffusion model problem of the

form

C="melenk/img1.gif"> in curvilinear polygonal

domains

. 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

TOM ALT="tex2html_wrap_inline20" SRC="melenk/img3.gif"> tends to zero.

These meshes consist of one layer of needle elements of width

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 (

"tex2html_wrap_inline20" SRC="melenk/img3.gif">, distance to the

boundary

, distance to the vertices of

BOTTOM ALT="tex2html_wrap_inline16" SRC="melenk/img2.gif">) 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.