Remote Access Mathematics of Computation
Green Open Access

Mathematics of Computation

ISSN 1088-6842(online) ISSN 0025-5718(print)



A posteriori error estimation for $ hp$-adaptivity for fourth-order equations

Authors: Peter K. Moore and Marina Rangelova
Journal: Math. Comp. 79 (2010), 677-705
MSC (2000): Primary 65M60; Secondary 65M15
Published electronically: July 22, 2009
MathSciNet review: 2600539
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: A posteriori error estimates developed to drive hp-adaptivity for second-order reaction-diffusion equations are extended to fourth-order equations. A $ C^1$ hierarchical finite element basis is constructed from Hermite-Lobatto polynomials. A priori estimates of the error in several norms for both the interpolant and finite element solution are derived. In the latter case this requires a generalization of the well-known Aubin-Nitsche technique to time-dependent fourth-order equations. We show that the finite element solution and corresponding Hermite-Lobatto interpolant are asymptotically equivalent. A posteriori error estimators based on this equivalence for solutions at two orders are presented. Both are shown to be asymptotically exact on grids of uniform order. These estimators can be used to control various adaptive strategies. Computational results for linear steady-state and time-dependent equations corroborate the theory and demonstrate the effectiveness of the estimators in adaptive settings.

References [Enhancements On Off] (What's this?)

  • 1. S. Adjerid, A posteriori error estimates for fourth-order elliptic problems, Comput. Meth. Appl. Mech. Engrg. 191, (2002), 2539-2559. MR 1902705 (2003j:65105)
  • 2. S. Adjerid, P.K. Moore and J. Teresco, eds., ADAPT'03: Adaptive methods for partial differential equations and large-scale computation, Appl. Numer. Math. 52, (2005). MR 2116907
  • 3. S. Adjerid, I. Babuska and J.E. Flaherty, A posteriori error estimation for the finite element method-of-lines solution of parabolic problems, Math. Model. Meth. Appl. Sci. 9, (1999), 261-286. MR 1674560 (2000a:65117)
  • 4. M. Ainsworth and J.T. Oden, A Posteriori Error Estimation in Finite Element Analysis, Wiley/Interscience, New York, 2000. MR 1885308 (2003b:65001)
  • 5. I. Babuska, J.E. Flaherty, W.D. Henshaw, J.E. Hopcroft, J.E. Oliger and T. Tezduyar, eds., Modeling, Mesh Generation, and Adaptive Numerical Methods for Partial Differential Equations, Springer-Verlag, New York, 1995. MR 1370242 (96g:65002)
  • 6. J.B. van den Berg and R.C. Vandervorst, Stable patterns for fourth-order parabolic equations, Duke Math. J. 115, (2002), 513-558. MR 1940411 (2003j:35136)
  • 7. K.E. Brenan, S.L. Campbell and L.R. Petzold, Numerical Solution of Initial Value Problems in Differential-Algebraic Equations, North-Holland, New York, 1989. MR 1101809 (92e:65001)
  • 8. C.J. Budd and R. Kuske, Localized periodic patterns for the nonsymmetric generalized Swift-Hohenberg equation, Phys. D 208, (2005), 73-95. MR 2167908 (2007d:35141)
  • 9. J. Cahn and J. Hilliard, Free energy of a nonuniform system. I. Interfacial free energy, J. Chem. Phys. 28, (1958), 258-267.
  • 10. Y. Cheng and C.-W. Shu, A discontinuous Galerkin finite element method for time dependent partial differential equations with higher order derivatives, Math. Comput. 77, (2008), 699-730. MR 2373176 (2008m:65252)
  • 11. P. Cvitanovic, R.L. Davidchack and E. Siminos, State space geometry of a spatio-temporally chaotic Kuramoto-Sivashinsky flow, arXiv:0709.2944v1, (2007), 1-27.
  • 12. G.T. Dee and W. van Saarloos, Bistable systems with propagating fronts leading to pattern formation, Phys. Rev. Lett. 60, (1988), 2641-2644.
  • 13. L. Demkowicz, Computing with hp-Adaptive Finite Elements. Volume 1. One and Two Dimensional Elliptic and Maxwell Problems, Chapman & Hall/CRC, Boca Raton, 2007. MR 2267112 (2007k:65003)
  • 14. D.J. Eyre, Systems of Cahn-Hilliard equations, SIAM J. Appl. Math. 53, (1993), 1686-1712. MR 1247174 (94j:73010)
  • 15. A. Ern and J.-L. Guermond, Theory and Practice of Finite Elements, Springer, New York, 2004. MR 2050138 (2005d:65002)
  • 16. M. Gameiro, K. Mischaiskow and T. Wanner, Evolution of pattern complexity in the Cahn-Hilliard theory of phase separation, Acta Materialia 53, (2005), 693-704.
  • 17. M'F. Hilali, S. Métens, P. Borckmans and G. Dewel, Pattern selection in the generalized Swift-Hohenberg model, Phys. Rev. E 31, (1995), 2046-2055.
  • 18. Y. Kuramoto and T. Tsuzuki, Persistent propagation of concentration waves in dissipative media far from thermal equilibrium, Progress Theor. Phys. 55, (1976), 356-368.
  • 19. J. Lawson, M. Berzins, and P.M. Dew, Balancing space and time errors in the method of lines for parabolic equations, SIAM J. Sci. Stat. Comput. 12, (1991), 573-594. MR 1093207 (92a:65252)
  • 20. P.K. Moore, Comparison of adaptive methods for one-dimensional parabolic systems, Appl. Numer. Math. 16, (1995), 471-488. MR 1325260 (95m:65170)
  • 21. P.K. Moore, Interpolation error-based a posteriori error estimation for two-point boundary value problems and parabolic equations in one space dimension, Numer. Math. 90, (2001), 149-177. MR 1868766 (2002h:65111)
  • 22. P.K. Moore, Applications of Lobatto polynomials to an adaptive finite element method: a posteriori error estimates for hp-adaptivity and grid-to-grid interpolation, Numer. Math. 94, (2003), 367-401. MR 1974560 (2004e:65105)
  • 23. P.K. Moore, Implicit interpolation error-based error estimation for reaction-diffusion equations in two space dimensions, Comput. Meth. Appl. Mech. Engrg. 192, (2003), 4379-4401. MR 2008075 (2004i:65107)
  • 24. P.K. Moore, Interpolation error-based a posteriori error estimation for hp-refinement using first and second derivative jumps, Appl. Numer. Math. 48, (2004), 63-82. MR 2027822 (2004k:65122)
  • 25. P.K. Moore, An implicit interpolation error-based error estimation strategy for hp-adaptivity in one space dimension, J. Numer. Math. 12, (2004), 143-167. MR 2062583 (2005b:65087)
  • 26. P.K. Moore, HP4: a hp-adaptive finite element code for solving fourth-order equations in one space dimension, preprint.
  • 27. J.T. Oden and G.F. Carey, Finite Elements. Mathematical Aspects. Volume IV, Prentice Hall, Inc., Englewood Cliffs, NJ, 1983. MR 767806 (86m:65001d)
  • 28. E.D. Rainville, Special Functions, Chelsea Publishing Co., New York, 1960. MR 0107725 (21:6447)
  • 29. M. Rangelova, Error estimation for fourth order partial differential equations, Ph.D. Thesis, Southern Methodist University, Dallas, TX, 2007.
  • 30. G.I. Sivashinsky, Nonlinear analysis of hydrodynamic instability in laminar flames, I. Acta Astronautica 4, (1977), 1177-1206. MR 0502829 (58:19741)
  • 31. G. Strang and G.J. Fix, An Analysis of the Finite Element Method, Prentice Hall, Inc., Englewood Cliffs, NJ, 1973. MR 0443377 (56:1747)
  • 32. Y.S. Smyrlis and D.T. Papageorgiou, Computational study of chaotic and ordered solutions of the Kuramoto-Sivashinsky equation, ICASE Report 96-12, (1996), 1-32.
  • 33. E. Süli and I. Mozolevski, hp-version interior penalty DGFEMs for the biharmonic equation, Comput. Meth. Appl. Mech. Engrg. 196, (2007), 1851-1863. MR 2298696 (2008c:65350)
  • 34. R. Verfürth, A Review of a Posteriori Error Estimation and Adaptive Mesh-Refinement Techniques, Wiley-Teubner, Chichester, 1996.
  • 35. R. Wait and A.R. Mitchell, Finite Element Analysis and Applications, John Wiley & Sons, Chichester, 1985. MR 817440 (87i:65192)
  • 36. R. Wang, P. Keast, and P. Muir, A high-order global spatially adaptive collocation method for 1-D parabolic PDEs, Appl. Numer. Math. 50, (2004), 239.250. MR 2066739
  • 37. A. Vande Wouwer, Ph. Saucez and W.E. Schiesser, eds., Adaptive Method of Lines, Chapman & Hall/CRC, Boca Raton, 2001. MR 1851429 (2002c:65004)
  • 38. D. Yu, Asymptotically exact a-posteriori error estimator for elements of bi-even degree, Math. Numer. Sinica 19, (1991), 89-101.
  • 39. D. Yu, Asymptotically exact a-posteriori error estimator for elements of bi-odd degree, Math. Numer. Sinica 19, (1991), 307-314.
  • 40. Z. Zhang, Superconvergence of spectral collocation and p-version methods in one dimensional problems, Math. Comput. 74, (2005), 1621-1636. MR 2164089 (2006h:65199)

Similar Articles

Retrieve articles in Mathematics of Computation with MSC (2000): 65M60, 65M15

Retrieve articles in all journals with MSC (2000): 65M60, 65M15

Additional Information

Peter K. Moore
Affiliation: Department of Mathematics, Southern Methodist University, Dallas, Texas 75275

Marina Rangelova
Affiliation: eVerge Group, Plano, Texas 75093

Keywords: A posteriori error estimation, fourth-order equations, adaptivity
Received by editor(s): December 11, 2007
Received by editor(s) in revised form: April 14, 2009
Published electronically: July 22, 2009
Article copyright: © Copyright 2009 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

American Mathematical Society