The \(p\) and \(hp\) versions of the finite element method for problems with boundary layers.

*(English)*Zbl 0853.65115Summary: 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 |

##### Keywords:

boundary layer; singularly perturbed problem; \(p\) version; \(hp\) version; spectral element method
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\textit{C. Schwab} and \textit{M. Suri}, Math. Comput. 65, No. 216, 1403--1429 (1996; Zbl 0853.65115)

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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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