Numerische Mathematik

, Volume 135, Issue 4, pp 1073–1119 | Cite as

Convergence and optimality of \({\mathbf {hp}}\)-AFEM

  • Claudio Canuto
  • Ricardo H. Nochetto
  • Rob Stevenson
  • Marco Verani
Article

Abstract

We design and analyze an adaptive hp-finite element method (\({\mathbf {hp}}\)-AFEM) in dimensions \(n=1,2\). The algorithm consists of iterating two routines: \({\mathbf {hp}}\)-NEARBEST finds a near-best hp-approximation of the current discrete solution and data to a desired accuracy, and REDUCE improves the discrete solution to a finer but comparable accuracy. The former hinges on a recent algorithm by Binev for adaptive hp-approximation, and acts as a coarsening step. We prove convergence and instance optimality.

Mathematics Subject Classification

65N30 65N12 65N50 

References

  1. 1.
    Ainsworth, M., Senior, B.: An adaptive refinement strategy for \(hp\)-finite element computations. Appl. Numer. Math. 26(1–2), 165–178 (1998)MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    Babuška, I., Craig, A., Mandel, J., Pitkäranta, J.: Efficient preconditioning for the \(p\)-version finite element method in two dimensions. SIAM J. Numer. Anal. 28(3), 624–661 (1991)MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    Bank, R., Parsania, A., Sauter, S.: Saturation estimates for hp-finite element methods. Comput. Vis. Sci. 16(5), 195–217 (2013)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Binev, P.: Instance optimality for hp-type approximation. Oberwolfach Rep. 39, 14–16 (2013)Google Scholar
  5. 5.
    Binev, P.: Tree approximation for \(hp\)-adaptivity (in preparation)Google Scholar
  6. 6.
    Binev, P., Dahmen, W., DeVore, R.: Adaptive finite element methods with convergence rates. Numer. Math. 97(2), 219–268 (2004)MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    Binev, P., DeVore, R.: Fast computation in adaptive tree approximation. Numer. Math. 97(2), 193–217 (2004)MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Braess, D., Pillwein, V., Schöberl, J.: Equilibrated residual error estimates are \(p\)-robust. Comput. Methods Appl. Mech. Eng. 198(13–14), 1189–1197 (2009)MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    Brix, K., Campos Pinto, M., Canuto, C., Dahmen, W.: Multilevel preconditioning of discontinuous Galerkin spectral element methods. Part I: geometrically conforming meshes. IMA J. Numer. Anal. 35(4), 1487–1532 (2015)Google Scholar
  10. 10.
    Bürg, M., Dörfler, W.: Convergence of an adaptive \(hp\) finite element strategy in higher space-dimensions. Appl. Numer. Math. 61(11), 1132–1146 (2011)MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    Canuto, C., Nochetto, R.H., Verani, M.: Adaptive Fourier–Galerkin methods. Math. Comput. 83, 1645–1687 (2014)MathSciNetCrossRefMATHGoogle Scholar
  12. 12.
    Canuto, C., Nochetto, R.H., Verani, M.: Contraction and optimality properties of adaptive Legendre–Galerkin methods: the 1-dimensional case. Comput. Math. Appl. 67(4), 752–770 (2014)MathSciNetCrossRefMATHGoogle Scholar
  13. 13.
    Canuto, C., Simoncini, V., Verani, M.: Contraction and optimality properties of an adaptive Legendre–Galerkin method: the multi-dimensional case. J. Sci. Comput. 63(3), 769–798 (2015)MathSciNetCrossRefMATHGoogle Scholar
  14. 14.
    Canuto, C., Hussaini, M.Y., Quarteroni, A., Zang, T.A.: Spectral Methods. Fundamentals in Single Domains. Scientific Computation. Springer, Berlin (2006)Google Scholar
  15. 15.
    Canuto, C., Verani, M.: On the numerical analysis of adaptive spectral/\(hp\) methods for elliptic problems. In: Analysis and Numerics of Partial Differential Equations. Springer INdAM Series, vol. 4, pp. 165–192. Springer, Milan (2013)Google Scholar
  16. 16.
    Cascon, J.M., Kreuzer, Ch., Nochetto, R.H., Siebert, K.G.: Quasi-optimal convergence rate for an adaptive finite element method. SIAM J. Numer. Anal. 46(5), 2524–2550 (2008)MathSciNetCrossRefMATHGoogle Scholar
  17. 17.
    Carstensen, C., Feischl, M., Page, M., Praetorius, D.: Axioms of adaptivity. Comput. Math. Appl. 67(6), 1195–1253 (2014)MathSciNetCrossRefMATHGoogle Scholar
  18. 18.
    Dahmen, W., Scherer, K.: Best approximation by piecewise polynomials with variable knots and degrees. J. Approx. Theory 26(1), 1–13 (1979)MathSciNetCrossRefMATHGoogle Scholar
  19. 19.
    Demkowicz, L., Oden, J.T., Rachowicz, W., Hardy, O.: Toward a universal \(h\)-\(p\) adaptive finite element strategy. I. Constrained approximation and data structure. Comput. Methods Appl. Mech. Eng. 77(1–2), 79–112 (1989)MathSciNetCrossRefMATHGoogle Scholar
  20. 20.
    Demkowicz, L., Rachowicz, W., Devloo, Ph.: A fully automatic \(hp\)-adaptivity. J. Sci. Comput. 17(1–4), 127–155 (2002)Google Scholar
  21. 21.
    DeVore, R., Scherer, K.: Variable knot, variable degree spline approximation to \(x^\beta \). In: Quantitative Approximation (Proceedings of the International Symposium. Bonn, 1979), pp. 121–131. Academic Press, New York (1980)Google Scholar
  22. 22.
    Dörfler, W.: A convergent adaptive algorithm for Poisson’s equation. SIAM J. Numer. Anal. 33(3), 1106–1124 (1996)MathSciNetCrossRefMATHGoogle Scholar
  23. 23.
    Dörfler, W., Heuveline, V.: Convergence of an adaptive \(hp\) finite element strategy in one space dimension. Appl. Numer. Math. 57(10), 1108–1124 (2007)MathSciNetCrossRefMATHGoogle Scholar
  24. 24.
    Eibner, T., Melenk, J.M.: An adaptive strategy for \(hp\)-FEM based on testing for analyticity. Comput. Mech. 39(5), 575–595 (2007)MathSciNetCrossRefMATHGoogle Scholar
  25. 25.
    Ern, A., Vohralík, M.: Polynomial-degree-robust a posteriori estimates in a unified setting for conforming, nonconforming, discontinuous Galerkin, and mixed discretizations. SIAM J. Numer. Anal. 53(2), 1058–1081 (2015)MathSciNetCrossRefMATHGoogle Scholar
  26. 26.
    Gui, W., Babuška, I.: The \(h,\;p\) and \(h\)-\(p\) versions of the finite element method in \(1\) dimension. II. The error analysis of the \(h\)- and \(h\)-\(p\) versions. Numer. Math. 49(6), 613–657 (1986)MathSciNetCrossRefMATHGoogle Scholar
  27. 27.
    Gui, W., Babuška, I.: The \(h,\;p\) and \(h\)-\(p\) versions of the finite element method in \(1\) dimension. III. The adaptive \(h\)-\(p\) version. Numer. Math. 49(6), 659–683 (1986)MathSciNetCrossRefMATHGoogle Scholar
  28. 28.
    Guo, B., Babuška, I.: The \(hp\)-version of the finite element method i: the basic approximation results. Comput. Mech. 1, 21–41 (1986)CrossRefMATHGoogle Scholar
  29. 29.
    Guo, B., Babuška, I.: The \(hp\)-version of the finite element method ii: general results and applications. Comput. Mech. 1, 203–226 (1986)CrossRefMATHGoogle Scholar
  30. 30.
    Guo, B., Babuška, I.: Regularity of the solutions for elliptic problems on nonsmooth domains in \( R^3\). II. Regularity in neighbourhoods of edges. Proc. R. Soc. Edinb. Sect. A 127(3), 517–545 (1997)Google Scholar
  31. 31.
    Houston, P., Senior, B., Süli, E.: \(hp\)-discontinuous Galerkin finite element methods for hyperbolic problems: error analysis and adaptivity. Int. J. Numer. Methods Fluids 40(1–2), 153–169 (2002) [ICFD Conference on Numerical Methods for Fluid Dynamics (Oxford, 2001)]Google Scholar
  32. 32.
    Houston, P., Süli, E.: A note on the design of \(hp\)-adaptive finite element methods for elliptic partial differential equations. Comput. Methods Appl. Mech. Eng. 194(2–5), 229–243 (2005)MathSciNetCrossRefMATHGoogle Scholar
  33. 33.
    Mavriplis, C.: Adaptive mesh strategies for the spectral element method. Comput. Methods. Appl. Mech. Eng. 116, 77–86 (1994)MathSciNetCrossRefMATHGoogle Scholar
  34. 34.
    Melenk, J.M., Wohlmuth, B.I.: On residual-based a posteriori error estimation in \(hp\)-FEM. Adv. Comput. Math. 15(1–4), 311–331 (2002)MathSciNetMATHGoogle Scholar
  35. 35.
    Morin, P., Nochetto, R.H., Siebert, K.G.: Data oscillation and convergence of adaptive FEM. SIAM J. Numer. Anal. 38(2), 466–488 (2000) (electronic)Google Scholar
  36. 36.
    Nochetto, R.H., Siebert, K.G., Veeser, A.: Theory of adaptive finite element methods: an introduction. In: Multiscale, Nonlinear and Adaptive Approximation, pp. 409–542. Springer, Berlin (2009)Google Scholar
  37. 37.
    Oden, J.T., Demkowicz, L., Rachowicz, W., Westermann, T.A.: Toward a universal \(h\)-\(p\) adaptive finite element strategy. II. A posteriori error estimation. Comput. Methods Appl. Mech. Eng. 77(1–2), 113–180 (1989)MathSciNetCrossRefMATHGoogle Scholar
  38. 38.
    Oden, J.T., Patra, A., Feng, Y.: An \(hp\) Adaptive Strategy, vol. 157, pp. 23–46. ASME Publication, New York (1992)Google Scholar
  39. 39.
    Rachowicz, W., Oden, J.T., Demkowicz, L.: Toward a universal \(h\)-\(p\) adaptive finite element strategy. III. Design of \(h\)-\(p\) meshes. Comput. Methods Appl. Mech. Eng. 77(1–2), 181–212 (1989)MathSciNetCrossRefMATHGoogle Scholar
  40. 40.
    Schwab, Ch.: \(p\)- and \(hp\)-Finite Element Methods. Oxford University Press, Oxford (1998)MATHGoogle Scholar
  41. 41.
    Scott, L.R., Zhang, S.: Finite element interpolation of nonsmooth functions satisfying boundary conditions. Math. Comput. 54(190), 483–493 (1990)MathSciNetCrossRefMATHGoogle Scholar
  42. 42.
    Schmidt, A., Siebert, K.G.: A posteriori estimators for the \(h\)-\(p\) version of the finite element method in 1D. Appl. Numer. Math. 35(1), 43–66 (2000)MathSciNetCrossRefMATHGoogle Scholar
  43. 43.
    Schötzau, D., Schwab, C., Wihler, T.: \(hp\)-dgfem for elliptic problems in polyhedra i: stability and quasi-optimality on geometric meshes. SIAM J. Numer. Anal. 51(3), 1610–1633 (2013)MathSciNetCrossRefMATHGoogle Scholar
  44. 44.
    Schötzau, D., Schwab, C., Wihler, T.: \(hp\)-dgfem for elliptic problems in polyhedra ii: exponential convergence. SIAM J. Numer. Anal. 51(4), 2005–2035 (2013)MathSciNetCrossRefMATHGoogle Scholar
  45. 45.
    Stevenson, R.: Optimality of a standard adaptive finite element method. Found. Comput. Math. 7(2), 245–269 (2007)MathSciNetCrossRefMATHGoogle Scholar
  46. 46.
    Veeser, A.: Approximating gradients with continuous piecewise polynomial functions. Found. Comput. Math. (2015). doi:10.1007/s10208-015-9262-z MATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2016

Authors and Affiliations

  • Claudio Canuto
    • 1
  • Ricardo H. Nochetto
    • 2
  • Rob Stevenson
    • 3
  • Marco Verani
    • 4
  1. 1.Dipartimento di Scienze MatematichePolitecnico di TorinoTurinItaly
  2. 2.Department of Mathematics and Institute for Physical Science and TechnologyUniversity of MarylandCollege ParkUSA
  3. 3.Korteweg-de Vries Institute for MathematicsUniversity of AmsterdamAmsterdamThe Netherlands
  4. 4.MOX, Dipartimento di MatematicaPolitecnico di MilanoMilanItaly

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