The $p$ and $hp$ versions of the finite element method for problems with boundary layers

Abstract

Abstract. We study the uniform approximation of boundary layer functions exp(−x/d) for x ∈ (0, 1), d ∈ (0, 1], by the p and hp versions of the finite element method. For the p version (with fixed mesh), we prove super-exponential convergence in the range p + 1/2 > e/(2d). We also establish, for this version, an overall convergence rate of O(p −1 √ ln p) in the energy norm error which is uniform in d, and show that this rate is sharp (up to the √ ln p term) when robust estimates uniform in d ∈ (0, 1] are considered. For the p version with variable mesh (i.e., the hp version), we show that exponential convergence, uniform in d ∈ (0, 1], is achieved by taking the first element at the boundary layer to be of size O(pd).Numerical experiments for a model elliptic singular perturbation problem show good agreement with our convergence estimates, even when few degrees of freedom are used and when d is as small as, e.g., 10 −8 . They also illustrate the superiority of the hp approach over other methods, including a low-order h version with optimal "exponential" mesh refinement.The estimates established in this paper are also applicable in the context of corresponding spectral element methods.

An a-posteriori error estimate forhp-adaptive DG methods for convection–diffusion problems on anisotropically refined meshes

“…Particularly in the anisotropic case, we notice that relative large polynomial degrees are applied near the boundaries. This is consistent with the theoretical results in [29] or [20, Section 3.4.6, page 118]. Indeed, since the solution is analytic and once the layers are resolved, p-refinement is the most effective refinement strategy.…”

“…They also overestimate the weighted L 2 -errors (which is not guaranteed by Theorem 3). On the basis of the a-priori analysis in [29] or [20, Section 3.4.6, page 118], we plot the errors against N 1 2 , where N is the number of degrees of freedom. In the asymptotic regime and in a semi-logarithmic scale, all the curves are roughly straight lines, indicating exponential convergence in N 1 2 .…”

On the influence of the geometry on skin effect in electromagnetism

“…, versus p in semilogarithmic coordinates, and in each case: σ = 5 with circles, σ = 20 with squares, and σ = 80 with diamonds. We refer to [19] for theoretical results of convergence for the p-version in presence of an exponentially decreasing boundary layer. In Figure 9 we use mesh M 3 instead of M 1 .…”

A numerical study on Neumann-Neumann methods forhpapproximations on geometrically refined boundary layer meshes II. Three-dimensional problems

“…For singularity resolution, n is required to be proportional to the polynomial degree k; see [3,5]. For boundary layers, the width of the thinnest layer mesh needs to be comparable to that of the boundary layer; see [36,51,52].…”

“…We note that this is a genuine hp approximation. As shown in [3,5,36,51], in order to obtain exponential convergence in presence of singularities in polyhedral domains, the number of layers n must be at least equal to the spectral degree k, thus better accuracy is achieved by simultaneously increasing the polynomial degree and the number of layers. In our experiments we have chosen n = k.…”