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The \(p\) and \(hp\) versions of the finite element method for problems with boundary layers. (English) Zbl 0853.65115
Summary: We study the uniform approximation of boundary layer functions \(\exp (-x/d)\) for \(x\in (0,1)\), \(d\in (0,1]\), 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 \({\mathcal O}(p^{-1}\sqrt {\ln p})\) in the energy norm error which is uniform in \(d\), and show that this rate is sharp (up to the \(\sqrt {\ln p}\) term) when robust estimates uniform in \(d\in (0,1]\) are considered. For the \(p\) version with variable mesh (i.e., the \(hp\) version), we show that exponential convergence, uniform in \(d\in (0,1]\), is achieved by taking the first element at the boundary layer to be of size \({\mathcal O}(pd)\). 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.

MSC:
65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
35B30 Dependence of solutions to PDEs on initial and/or boundary data and/or on parameters of PDEs
65N15 Error bounds for boundary value problems involving PDEs
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