On the robust exponential convergence of hp finite element methods for problems with boundary layers

Abstract

The hp version of the finite element method for a one-dimensional, singularly perturbed elliptic-elliptic model problem with analytic input data is considered. It is shown that the use of piecewise polynomials of degree p on a mesh consisting of three suitably chosen elements leads to robust exponential convergence, i.e., the exponential rate of convergence depends only on the input data and is independent of the perturbation parameter.

An hp finite element method for convection-diffusion problems in one dimension

“…These regularity results are necessary for the proof of robust exponential convergence of the hp FEM obtained in the present paper. Although regularity results related to the ones presented here can be found in the literature (e.g., in the books by Roos et al (1996), Morton (1995)), the speci"c derivative bounds seem to be new (see also Melenk (1997) for the related case of a reaction-diffusion equation). The solution u ε of (1.1), (1.2) is analytic on Ω; however, for small values of ε, it exhibits a boundary layer at the out#ow boundary.…”

“…The proof is very similar to that of Theorem 16 of Melenk (1997). Theorem 3.3 together with (3.22) shows that for analytic input data robust exponential convergence can be achieved by the FE scheme (3.4) provided the space S p 0 is designed properly (i.e.…”

“…In most PG methods advocated in practice, the trial spaces consist of piecewise polynomials of a "xed, low degree p and accuracy is obtained by decreasing the mesh width h. Such h versions, given stability of the method, can only lead to robust algebraic rates of convergence of the form O(N −q ) (uniformly in ε) for some q > 0 where N stands for the number of degrees of freedom. Recently , and Melenk (1997) demonstrated for the related reaction-diffusion equation, which also exhibits sharp boundary layers, that robust exponential convergence of the hp FEM can be obtained, i.e., the error is O(exp(−bN )) for some b > 0 independent of ε. In the present paper it will also be shown that robust exponential approximability can be achieved by piecewise polynomials of increasingly higher degree on appropriate meshes for the solutions of convection-diffusion equations.…”

“…(1.17) Theorem 1.1 can be proved using the same techniques that are employed by Melenk (1997). The lemmata that allow for the use of these techniques are collected in the Appendix; the reader is referred to Melenk & Schwab (1997) where the arguments are worked out in detail.…”

“…These`minimal' meshes T κ,ε depend on the polynomial degree p. In practice, it may be more convenient to "x a mesh T and then increase p until the desired accuracy is reached. For example, piecewise polynomials on a mesh which is graded geometrically towards the out#ow boundary have approximation properties similar to the minimal meshes T κ,ε provided that the smallest element of the geometric mesh is of size O(ε) (see Melenk (1997)). …”