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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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##### References:
 [1] D. N. Arnold and R. S. Falk. Asymptotic analysis of the boundary layer for the Reissner-Mindlin plate model. SIAM J. Math. Anal. 27: 486–514, 1996. · Zbl 0846.73027 [2] Ivo Babuška and Manil Suri, On locking and robustness in the finite element method, SIAM J. Numer. Anal. 29 (1992), no. 5, 1261 – 1293. · Zbl 0763.65085 · doi:10.1137/0729075 · doi.org [3] I. Babuška and B. A. Szabo. Lecture notes on finite element analysis, (to appear). [4] I. A. Blatov and V. V. Strygin. On estimates best possible in order in the Galerkin finite element method for singularly perturbed boundary value problems. Russian Acad. Sci. Dokl. Math., 47:93–96, 1993. · Zbl 0812.65078 [5] Claudio Canuto, Spectral methods and a maximum principle, Math. Comp. 51 (1988), no. 184, 615 – 629. · Zbl 0699.65080 [6] Eugene C. Gartland Jr., Uniform high-order difference schemes for a singularly perturbed two-point boundary value problem, Math. Comp. 48 (1987), no. 178, 551 – 564, S5 – S9. · Zbl 0621.65088 [7] W. B. Liu and J. Shen. A new efficient spectral Galerkin method for singular perturbation problems, Preprint, Department of Mathematics, Penn State University, State College Pa (1994). · Zbl 0891.76066 [8] W. B. Liu and T. Tang. Boundary layer resolving methods for singularly perturbed problems, submitted to I.M.A. J. Numer. Anal. [9] F. W. J. Olver, Error bounds for the Liouville-Green (or \?\?\?) approximation, Proc. Cambridge Philos. Soc. 57 (1961), 790 – 810. · Zbl 0168.14003 [10] I. S. Gradshteyn and I. M. Ryzhik, Table of integrals, series, and products, Academic Press [Harcourt Brace Jovanovich, Publishers], New York-London-Toronto, Ont., 1980. Corrected and enlarged edition edited by Alan Jeffrey; Incorporating the fourth edition edited by Yu. V. Geronimus [Yu. V. Geronimus] and M. Yu. Tseytlin [M. Yu. Tseĭtlin]; Translated from the Russian. · Zbl 0521.33001 [11] H. Hakula, Y. Leino, and J. Pitkäranta. Scale resolution, layers and high-order numerical modeling of shells, to appear in Comp. Meth. Appl. Mech. Eng., 1996. · Zbl 0918.73111 [12] H. Kraus. Thin elastic shells: an introduction to the theoretical foundations and the analysis of their static and dynamic behavior. New York, Wiley 1967. · Zbl 0266.73052 [13] A. H. Schatz and L. B. Wahlbin, On the finite element method for singularly perturbed reaction-diffusion problems in two and one dimensions, Math. Comp. 40 (1983), no. 161, 47 – 89. · Zbl 0518.65080 [14] Karl Scherer, On optimal global error bounds obtained by scaled local error estimates, Numer. Math. 36 (1980/81), no. 2, 151 – 176. · Zbl 0495.65006 · doi:10.1007/BF01396756 · doi.org [15] Christoph Schwab and Manil Suri, Locking and boundary layer effects in the finite element approximation of the Reissner-Mindlin plate model, Mathematics of Computation 1943 – 1993: a half-century of computational mathematics (Vancouver, BC, 1993) Proc. Sympos. Appl. Math., vol. 48, Amer. Math. Soc., Providence, RI, 1994, pp. 367 – 371. · Zbl 0815.73061 · doi:10.1090/psapm/048/1314872 · doi.org [16] C. Schwab, M. Suri, and C. Xenophontos. The $$hp$$ finite element method for problems in mechanics with boundary layers (to appear). · Zbl 0959.74073 [17] C. Schwab and S. Wright, Boundary layers of hierarchical beam and plate models, J. Elasticity 38 (1995), no. 1, 1 – 40. · Zbl 0834.73040 · doi:10.1007/BF00121462 · doi.org [18] G. I. Shishkin, Grid approximation of singularly perturbed parabolic equations with internal layers, Soviet J. Numer. Anal. Math. Modelling 3 (1988), no. 5, 393 – 407. Translated from the Russian. · Zbl 0825.65062 [19] R. Vulanović, D. Herceg, and N. Petrović, On the extrapolation for a singularly perturbed boundary value problem, Computing 36 (1986), no. 1-2, 69 – 79 (English, with German summary). · Zbl 0576.34019 · doi:10.1007/BF02238193 · doi.org [20] C. A. Xenophontos. The $$hp$$ version of the finite element method for singularly perturbed problems in unsmooth domains. Ph.D. Dissertation, UMBC, 1996.
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