Advertisement

Numerische Mathematik

, Volume 132, Issue 1, pp 85–130 | Cite as

Piecewise polynomial interpolation in Muckenhoupt weighted Sobolev spaces and applications

  • Ricardo H. Nochetto
  • Enrique Otárola
  • Abner J. SalgadoEmail author
Article
First Online:

Abstract

We develop a constructive piecewise polynomial approximation theory in weighted Sobolev spaces with Muckenhoupt weights for any polynomial degree. The main ingredients to derive optimal error estimates for an averaged Taylor polynomial are a suitable weighted Poincaré inequality, a cancellation property and a simple induction argument. We also construct a quasi-interpolation operator, built on local averages over stars, which is well defined for functions in \(L^1\). We derive optimal error estimates for any polynomial degree on simplicial shape regular meshes. On rectangular meshes, these estimates are valid under the condition that neighboring elements have comparable size, which yields optimal anisotropic error estimates over \(n\)-rectangular domains. The interpolation theory extends to cases when the error and function regularity require different weights. We conclude with three applications: nonuniform elliptic boundary value problems, elliptic problems with singular sources, and fractional powers of elliptic operators.

Mathematics Subject Classification

35J70 35J75 65D05 65N30 65N12 
This is a preview of subscription content, log in to check access.

Notes

Acknowledgments

We dedicate this paper to R.G. Durán, whose work at the intersection of real and numerical analysis has been inspirational to us.

References

  1. 1.
    Acosta, G.: Lagrange and average interpolation over 3D anisotropic elements. J. Comput. Appl. Math. 135(1), 91–109 (2001)zbMATHMathSciNetCrossRefGoogle Scholar
  2. 2.
    Agler, J., McCarthy, J.E.: Pick interpolation and Hilbert function spaces. In: Graduate Studies in Mathematics, vol. 44. American Mathematical Society, Providence (2002)Google Scholar
  3. 3.
    Agnelli, J.P., Garau, E.M., Morin, P.: A posteriori error estimates for elliptic problems with Dirac measure terms in weighted spaces. ESAIM: Math. Modell. Numer. Anal. 48(11), 1557–1581 (2014)Google Scholar
  4. 4.
    Apel, T.: Interpolation of non-smooth functions on anisotropic finite element meshes. M2AN. Math. Model. Numer. Anal. 33(6), 1149–1185 (1999)zbMATHMathSciNetCrossRefGoogle Scholar
  5. 5.
    Araya, R., Behrens, E., Rodríguez, R.: An adaptive stabilized finite element scheme for a water quality model. Comput. Methods Appl. Mech. Eng. 196(29–30), 2800–2812 (2007)zbMATHCrossRefGoogle Scholar
  6. 6.
    Arroyo, D., Bespalov, A., Heuer, N.: On the finite element method for elliptic problems with degenerate and singular coefficients. Math. Comp. 76(258), 509–537 (2007)zbMATHMathSciNetCrossRefGoogle Scholar
  7. 7.
    Babuška, I.: Error-bounds for finite element method. Numer. Math. 16, 322–333 (1970/1971)Google Scholar
  8. 8.
    Babuška, I., Aziz, A.K.: Survey lectures on the mathematical foundations of the finite element method. In: The Mathematical Foundations of the Finite Element Method with Applications to Partial Differential Equations (Proc. Sympos., Univ. Maryland, Baltimore, Md., 1972), pp. 1–359. Academic Press, New York. With the collaboration of G. Fix and R. B. Kellogg (1972)Google Scholar
  9. 9.
    Bartels, S., Nochetto, R.H., Salgado, A.J.: A total variation diminishing interpolation operator and applications. Math. Comput. (2014, accepted)Google Scholar
  10. 10.
    Belhachmi, Z., Bernardi, Ch., Deparis, S.: Weighted Clément operator and application to the finite element discretization of the axisymmetric Stokes problem. Numer. Math. 105(2), 217–247 (2006)zbMATHMathSciNetCrossRefGoogle Scholar
  11. 11.
    Bernardi, Ch., Canuto, C., Maday, Y.: Generalized inf-sup conditions for Chebyshev spectral approximation of the Stokes problem. SIAM J. Numer. Anal. 25(6), 1237–1271 (1988)zbMATHMathSciNetCrossRefGoogle Scholar
  12. 12.
    Besov, O.V., Il’in, V.P., Nikol’skiĭ, S.M.: Integralnye predstavleniya funktsii i teoremy vlozheniya. 2nd ed. Fizmatlit “Nauka”, Moscow (1996)Google Scholar
  13. 13.
    Brenner, S.C., Scott, L.R.: The mathematical theory of finite element methods. In: Texts in Applied Mathematics, vol. 15. 3rd ed. Springer, New York (2008)Google Scholar
  14. 14.
    Cabré, X., Sire, Y.: Nonlinear equations for fractional Laplacians ii: existence, uniqueness and qualitative properties of solutions. Trans. Amer. Math. Soc. (2014, To appear)Google Scholar
  15. 15.
    Caffarelli, L., Silvestre, L.: An extension problem related to the fractional Laplacian. Commun. Partial Differ. Equ. 32(7–9), 1245–1260 (2007)zbMATHMathSciNetCrossRefGoogle Scholar
  16. 16.
    Capella, A., Dávila, J., Dupaigne, L., Sire, Y.: Regularity of radial extremal solutions for some non-local semilinear equations. Commun. Partial Differ. Equ. 36(8), 1353–1384 (2011)zbMATHCrossRefGoogle Scholar
  17. 17.
    Casas, E.: \(L^2\) estimates for the finite element method for the Dirichlet problem with singular data. Numer. Math. 47(4), 627–632 (1985)zbMATHMathSciNetCrossRefGoogle Scholar
  18. 18.
    Cavalheiro, A.C.: A theorem on global regularity for solutions of degenerate elliptic equations. Commun. Math. Anal. 11(2), 112–123 (2011)zbMATHMathSciNetGoogle Scholar
  19. 19.
    Chanillo, S., Wheeden, R.L.: Weighted Poincaré and Sobolev inequalities and estimates for weighted Peano maximal functions. Am. J. Math. 107(5), 1191–1226 (1985)zbMATHMathSciNetCrossRefGoogle Scholar
  20. 20.
    Chen, L., Nochetto, R.H., Otárola, E., Salgado, A.J.: Multilevel methods for nonuniformly elliptic operators. arXiv:1403.4278. (2014)
  21. 21.
    Chen, Y.: Regularity of solutions to the Dirichlet problem for degenerate elliptic equation. Chin. Ann. Math. Ser. B 24(4), 529–540 (2003)zbMATHCrossRefGoogle Scholar
  22. 22.
    Chua, S.-K.: Extension theorems on weighted Sobolev spaces. Indiana Univ. Math. J. 41(4), 1027–1076 (1992)zbMATHMathSciNetCrossRefGoogle Scholar
  23. 23.
    Ciarlet, P.G.: The finite element method for elliptic problems. In: Classics in Applied Mathematics, vol. 40. SIAM, Philadelphia (2002)Google Scholar
  24. 24.
    Clément, P.: Approximation by finite element functions using local regularization. RAIRO Analyse Numérique 9(R-2), 77–84 (1975)Google Scholar
  25. 25.
    D’Angelo, C.: Finite element approximation of elliptic problems with Dirac measure terms in weighted spaces: applications to one- and three-dimensional coupled problems. SIAM J. Numer. Anal. 50(1), 194–215 (2012)zbMATHMathSciNetCrossRefGoogle Scholar
  26. 26.
    Diening, L., Ružička, M.: Interpolation operators in Orlicz–Sobolev spaces. Numer. Math. 107(1), 107–129 (2007)zbMATHMathSciNetCrossRefGoogle Scholar
  27. 27.
    Duoandikoetxea, J.: Fourier analysis. In: Graduate Studies in Mathematics, vol. 29. American Mathematical Society, Providence (2001). Translated and revised from the 1995 Spanish original by David Cruz-UribeGoogle Scholar
  28. 28.
    Dupont, T., Scott, L.R.: Polynomial approximation of functions in Sobolev spaces. Math. Comp. 34(150), 441–463 (1980)zbMATHMathSciNetCrossRefGoogle Scholar
  29. 29.
    Durán, R.G.: On polynomial approximation in Sobolev spaces. SIAM J. Numer. Anal. 20(5), 985–988 (1983)zbMATHMathSciNetCrossRefGoogle Scholar
  30. 30.
    Durán, R.G.: Quasi-optimal estimates for finite element approximations using Orlicz norms. Math. Comp. 49(179), 17–23 (1987)zbMATHMathSciNetCrossRefGoogle Scholar
  31. 31.
    Durán, R.G., Lombardi, A.L.: Error estimates on anisotropic \(Q_1\) elements for functions in weighted Sobolev spaces. Math. Comput. 74(252), 1679–1706 (2005, electronic)Google Scholar
  32. 32.
    Durán, R.G., Lombardi, A.L., Prieto, M.I.: Superconvergence for finite element approximation of a convection–diffusion equation using graded meshes. IMA J. Numer. Anal. 32(2), 511–533 (2012)zbMATHMathSciNetCrossRefGoogle Scholar
  33. 33.
    Durán, R.G., López, F.: García. Solutions of the divergence and Korn inequalities on domains with an external cusp. Ann. Acad. Sci. Fenn. Math. 35(2), 421–438 (2010)zbMATHMathSciNetCrossRefGoogle Scholar
  34. 34.
    Eriksson, K.: Improved accuracy by adapted mesh-refinements in the finite element method. Math. Comput. 44(170), 321–343 (1985)zbMATHMathSciNetCrossRefGoogle Scholar
  35. 35.
    Ern, A., Guermond, J.-L.: Theory and practice of finite elements. In: Applied Mathematical Sciences, vol. 159. Springer, New York (2004)Google Scholar
  36. 36.
    Fabes, E.B., Kenig, C.E., Serapioni, R.P.: The local regularity of solutions of degenerate elliptic equations. Commun. Partial Differ. Equ. 7(1), 77–116 (1982)zbMATHMathSciNetCrossRefGoogle Scholar
  37. 37.
    Figueroa, L.E., Süli, E.: Greedy approximation of high-dimensional Ornstein–Uhlenbeck operators. Found. Comput. Math. 12(5), 573–623 (2012)zbMATHMathSciNetCrossRefGoogle Scholar
  38. 38.
    Franchi, B., Gutiérrez, C.E., Wheeden, R.L.: Two-weight Sobolev–Poincaré inequalities and Harnack inequality for a class of degenerate elliptic operators. Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. Rend. Lincei (9) Mat. Appl. 5(2), 167–175 (1994)Google Scholar
  39. 39.
    French, D.A.: The finite element method for a degenerate elliptic equation. SIAM J. Numer. Anal. 24(4), 788–815 (1987)zbMATHMathSciNetCrossRefGoogle Scholar
  40. 40.
    Gilbarg, D., Trudinger, N.S.: Elliptic partial differential equations of second order. In: Classics in Mathematics. Springer, Berlin (2001). Reprint of the 1998 editionGoogle Scholar
  41. 41.
    Gol’dshtein, V., Ukhlov, A.: Weighted Sobolev spaces and embedding theorems. Trans. Am. Math. Soc. 361(7), 3829–3850 (2009)zbMATHMathSciNetCrossRefGoogle Scholar
  42. 42.
    Gopalakrishnan, J., Pasciak, J.E.: The convergence of V-cycle multigrid algorithms for axisymmetric Laplace and Maxwell equations. Math. Comput. 75(256), 1697–1719 (2006)zbMATHMathSciNetCrossRefGoogle Scholar
  43. 43.
    Griebel, M., Scherer, K., Schweitzer, A.: Robust norm equivalencies for diffusion problems. Math. Comput. 76(259), 1141–1161 (2007, electronic)Google Scholar
  44. 44.
    Grisvard, P.: Elliptic problems in nonsmooth domains. In: Monographs and Studies in Mathematics, vol. 24. Pitman (Advanced Publishing Program), Boston (1985)Google Scholar
  45. 45.
    Gurka, P., Opic, B.: Continuous and compact imbeddings of weighted Sobolev spaces. I. Czechoslovak Math. J. 38(113)(4), 730–744 (1988)Google Scholar
  46. 46.
    Hajłasz, P.: Sobolev spaces on an arbitrary metric space. Potential Anal. 5(4), 403–415 (1996)zbMATHMathSciNetGoogle Scholar
  47. 47.
    Hajłasz, P., Koskela, P.: Sobolev met Poincaré. Mem. Am. Math. Soc. 145(688), x+101 (2000)Google Scholar
  48. 48.
    Haroske, D.D., Skrzypczak, L.: Entropy and approximation numbers of embeddings of function spaces with Muckenhoupt weights. I. Rev. Mat. Complut. 21(1), 135–177 (2008)zbMATHMathSciNetGoogle Scholar
  49. 49.
    Haroske, D.D., Skrzypczak, L.: Entropy and approximation numbers of embeddings of function spaces with Muckenhoupt weights, II. General weights. Ann. Acad. Sci. Fenn. Math. 36(1), 111–138 (2011)zbMATHMathSciNetCrossRefGoogle Scholar
  50. 50.
    Heinonen, J., Kilpeläinen, T., Martio, O.: Nonlinear potential theory of degenerate elliptic equations. In: Oxford Mathematical Monographs. The Clarendon Press, Oxford University Press, Oxford Science Publications, New York (1993)Google Scholar
  51. 51.
    Kufner, A.: Weighted Sobolev Spaces. A Wiley-Interscience Publication, Wiley, New York (1985)zbMATHGoogle Scholar
  52. 52.
    Kufner, A., Opic, B.: How to define reasonably weighted Sobolev spaces. Comment. Math. Univ. Carolin. 25(3), 537–554 (1984)zbMATHMathSciNetGoogle Scholar
  53. 53.
    Kufner, A., Sändig, A.-M.: Some applications of weighted Sobolev spaces. In: Teubner Texts in Mathematics, vol. 100. BSB B. G. Teubner Verlagsgesellschaft, Leipzig (1987)Google Scholar
  54. 54.
    Lai, M.-J., Schumaker, L.L.: Spline functions on triangulations. In: Encyclopedia of Mathematics and its Applications, vol. 110. Cambridge University Press, Cambridge (2007)Google Scholar
  55. 55.
    Landkof, N.S.: Foundations of Modern Potential Theory. Springer, New York (1972). Translated from the Russian by A. P. Doohovskoy, Die Grundlehren der mathematischen Wissenschaften, Band, 180Google Scholar
  56. 56.
    Li, H.: A-priori analysis and the finite element method for a class of degenerate elliptic equations. Math. Comput. 78(266), 713–737 (2009)zbMATHCrossRefGoogle Scholar
  57. 57.
    Mamedov, F.I., Amanov, R.A.: On some nonuniform cases of weighted Sobolev and Poincaré inequalities. Algebra i Analiz 20(3), 163–186 (2008). (in Russian)MathSciNetGoogle Scholar
  58. 58.
    Muckenhoupt, B.: Weighted norm inequalities for the Hardy maximal function. Trans. Am. Math. Soc. 165, 207–226 (1972)zbMATHMathSciNetCrossRefGoogle Scholar
  59. 59.
    Nikol’skiĭ, S.M.: Approximation of functions of several variables and imbedding theorems. Springer, New York (1975)CrossRefGoogle Scholar
  60. 60.
    Nochetto, R.H., Otárola, E., Salgado, A.J.: A PDE approach to fractional diffusion in general domains: a priori error analysis. Found. Comput. Math. 1–59. doi: 10.1007/s10208-014-9208-x (2014)
  61. 61.
    Nochetto, R.H., Siebert, K.G., Veeser, A.: Theory of adaptive finite element methods: an introduction. In: DeVore, R., Kunoth, A. (eds.) Multiscale. Nonlinear and Adaptive Approximation, pp. 409–542. Springer, Berlin (2009)Google Scholar
  62. 62.
    Pérez, C.: Two weighted norm inequalities for Riesz potentials and uniform \(L^p\)-weighted Sobolev inequalities. Indiana Univ. Math. J. 39(1), 31–44 (1990)zbMATHMathSciNetCrossRefGoogle Scholar
  63. 63.
    Schatz, A.H., Wahlbin, L.B.: Interior maximum norm estimates for finite element methods. Math. Comput. 31(138), 414–442 (1977)zbMATHMathSciNetCrossRefGoogle Scholar
  64. 64.
    Scott, L.R.: Finite element convergence for singular data. Numer. Math. 21, 317–327 (1973/74)Google Scholar
  65. 65.
    Scott, L.R., Zhang, S.: Finite element interpolation of nonsmooth functions satisfying boundary conditions. Math. Comput. 54(190), 483–493 (1990)zbMATHMathSciNetCrossRefGoogle Scholar
  66. 66.
    Seidman, T.I., Gobbert, M.K., Trott, D.W., Kružík, M.: Finite element approximation for time-dependent diffusion with measure-valued source. Numer. Math. 122(4), 709–723 (2012)zbMATHMathSciNetCrossRefGoogle Scholar
  67. 67.
    Sobolev, S.L.: On a theorem of functional analysis. Mat. Sb 4(46), 471–497 (1938)zbMATHGoogle Scholar
  68. 68.
    Sobolev, S.L.: Nekotorye primeneniya funkcional’ nogo analiza v matematičeskoĭ fizike. Izdat. Leningrad. Gos. Univ, Leningrad (1950)Google Scholar
  69. 69.
    Stinga, P.R., Torrea, J.L.: Extension problem and Harnack’s inequality for some fractional operators. Commun. Partial Differ. Equ. 35(11), 2092–2122 (2010)zbMATHMathSciNetCrossRefGoogle Scholar
  70. 70.
    Turesson, B.O.: Nonlinear potential theory and weighted Sobolev spaces. In: Lecture Notes in Mathematics, vol. 1736. Springer, Berlin (2000)Google Scholar
  71. 71.
    Ziemer, W.P.: Weakly differentiable functions. In: Graduate Texts in Mathematics, vol. 120. Springer, New York (1989)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2015

Authors and Affiliations

  • Ricardo H. Nochetto
    • 1
  • Enrique Otárola
    • 2
    • 3
  • Abner J. Salgado
    • 4
    Email author
  1. 1.Department of Mathematics, Institute for Physical Science and TechnologyUniversity of MarylandCollege ParkUSA
  2. 2.Department of MathematicsUniversity of MarylandCollege ParkUSA
  3. 3.Department of Mathematical SciencesGeorge Mason UniversityFairfaxUSA
  4. 4.Department of MathematicsUniversity of TennesseeKnoxvilleUSA

Personalised recommendations