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Piecewise polynomial interpolation in Muckenhoupt weighted Sobolev spaces and applications

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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.

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References

  1. Acosta, G.: Lagrange and average interpolation over 3D anisotropic elements. J. Comput. Appl. Math. 135(1), 91–109 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  2. Agler, J., McCarthy, J.E.: Pick interpolation and Hilbert function spaces. In: Graduate Studies in Mathematics, vol. 44. American Mathematical Society, Providence (2002)

  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)

  4. Apel, T.: Interpolation of non-smooth functions on anisotropic finite element meshes. M2AN. Math. Model. Numer. Anal. 33(6), 1149–1185 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  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)

    Article  MATH  Google Scholar 

  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)

    Article  MATH  MathSciNet  Google Scholar 

  7. Babuška, I.: Error-bounds for finite element method. Numer. Math. 16, 322–333 (1970/1971)

  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)

  9. Bartels, S., Nochetto, R.H., Salgado, A.J.: A total variation diminishing interpolation operator and applications. Math. Comput. (2014, accepted)

  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)

    Article  MATH  MathSciNet  Google Scholar 

  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)

    Article  MATH  MathSciNet  Google Scholar 

  12. Besov, O.V., Il’in, V.P., Nikol’skiĭ, S.M.: Integralnye predstavleniya funktsii i teoremy vlozheniya. 2nd ed. Fizmatlit “Nauka”, Moscow (1996)

  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)

  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)

  15. Caffarelli, L., Silvestre, L.: An extension problem related to the fractional Laplacian. Commun. Partial Differ. Equ. 32(7–9), 1245–1260 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  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)

    Article  MATH  Google Scholar 

  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)

    Article  MATH  MathSciNet  Google Scholar 

  18. Cavalheiro, A.C.: A theorem on global regularity for solutions of degenerate elliptic equations. Commun. Math. Anal. 11(2), 112–123 (2011)

    MATH  MathSciNet  Google Scholar 

  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)

    Article  MATH  MathSciNet  Google Scholar 

  20. Chen, L., Nochetto, R.H., Otárola, E., Salgado, A.J.: Multilevel methods for nonuniformly elliptic operators. arXiv:1403.4278. (2014)

  21. Chen, Y.: Regularity of solutions to the Dirichlet problem for degenerate elliptic equation. Chin. Ann. Math. Ser. B 24(4), 529–540 (2003)

    Article  MATH  Google Scholar 

  22. Chua, S.-K.: Extension theorems on weighted Sobolev spaces. Indiana Univ. Math. J. 41(4), 1027–1076 (1992)

    Article  MATH  MathSciNet  Google Scholar 

  23. Ciarlet, P.G.: The finite element method for elliptic problems. In: Classics in Applied Mathematics, vol. 40. SIAM, Philadelphia (2002)

  24. Clément, P.: Approximation by finite element functions using local regularization. RAIRO Analyse Numérique 9(R-2), 77–84 (1975)

  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)

    Article  MATH  MathSciNet  Google Scholar 

  26. Diening, L., Ružička, M.: Interpolation operators in Orlicz–Sobolev spaces. Numer. Math. 107(1), 107–129 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  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-Uribe

  28. Dupont, T., Scott, L.R.: Polynomial approximation of functions in Sobolev spaces. Math. Comp. 34(150), 441–463 (1980)

    Article  MATH  MathSciNet  Google Scholar 

  29. Durán, R.G.: On polynomial approximation in Sobolev spaces. SIAM J. Numer. Anal. 20(5), 985–988 (1983)

    Article  MATH  MathSciNet  Google Scholar 

  30. Durán, R.G.: Quasi-optimal estimates for finite element approximations using Orlicz norms. Math. Comp. 49(179), 17–23 (1987)

    Article  MATH  MathSciNet  Google Scholar 

  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)

  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)

    Article  MATH  MathSciNet  Google Scholar 

  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)

    Article  MATH  MathSciNet  Google Scholar 

  34. Eriksson, K.: Improved accuracy by adapted mesh-refinements in the finite element method. Math. Comput. 44(170), 321–343 (1985)

    Article  MATH  MathSciNet  Google Scholar 

  35. Ern, A., Guermond, J.-L.: Theory and practice of finite elements. In: Applied Mathematical Sciences, vol. 159. Springer, New York (2004)

  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)

    Article  MATH  MathSciNet  Google Scholar 

  37. Figueroa, L.E., Süli, E.: Greedy approximation of high-dimensional Ornstein–Uhlenbeck operators. Found. Comput. Math. 12(5), 573–623 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  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)

  39. French, D.A.: The finite element method for a degenerate elliptic equation. SIAM J. Numer. Anal. 24(4), 788–815 (1987)

    Article  MATH  MathSciNet  Google Scholar 

  40. Gilbarg, D., Trudinger, N.S.: Elliptic partial differential equations of second order. In: Classics in Mathematics. Springer, Berlin (2001). Reprint of the 1998 edition

  41. Gol’dshtein, V., Ukhlov, A.: Weighted Sobolev spaces and embedding theorems. Trans. Am. Math. Soc. 361(7), 3829–3850 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  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)

    Article  MATH  MathSciNet  Google Scholar 

  43. Griebel, M., Scherer, K., Schweitzer, A.: Robust norm equivalencies for diffusion problems. Math. Comput. 76(259), 1141–1161 (2007, electronic)

  44. Grisvard, P.: Elliptic problems in nonsmooth domains. In: Monographs and Studies in Mathematics, vol. 24. Pitman (Advanced Publishing Program), Boston (1985)

  45. Gurka, P., Opic, B.: Continuous and compact imbeddings of weighted Sobolev spaces. I. Czechoslovak Math. J. 38(113)(4), 730–744 (1988)

  46. Hajłasz, P.: Sobolev spaces on an arbitrary metric space. Potential Anal. 5(4), 403–415 (1996)

    MATH  MathSciNet  Google Scholar 

  47. Hajłasz, P., Koskela, P.: Sobolev met Poincaré. Mem. Am. Math. Soc. 145(688), x+101 (2000)

  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)

    MATH  MathSciNet  Google Scholar 

  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)

    Article  MATH  MathSciNet  Google Scholar 

  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)

  51. Kufner, A.: Weighted Sobolev Spaces. A Wiley-Interscience Publication, Wiley, New York (1985)

    MATH  Google Scholar 

  52. Kufner, A., Opic, B.: How to define reasonably weighted Sobolev spaces. Comment. Math. Univ. Carolin. 25(3), 537–554 (1984)

    MATH  MathSciNet  Google Scholar 

  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)

  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)

  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, 180

  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)

    Article  MATH  Google Scholar 

  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)

    MathSciNet  Google Scholar 

  58. Muckenhoupt, B.: Weighted norm inequalities for the Hardy maximal function. Trans. Am. Math. Soc. 165, 207–226 (1972)

    Article  MATH  MathSciNet  Google Scholar 

  59. Nikol’skiĭ, S.M.: Approximation of functions of several variables and imbedding theorems. Springer, New York (1975)

    Book  Google Scholar 

  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. 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)

  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)

    Article  MATH  MathSciNet  Google Scholar 

  63. Schatz, A.H., Wahlbin, L.B.: Interior maximum norm estimates for finite element methods. Math. Comput. 31(138), 414–442 (1977)

    Article  MATH  MathSciNet  Google Scholar 

  64. Scott, L.R.: Finite element convergence for singular data. Numer. Math. 21, 317–327 (1973/74)

  65. Scott, L.R., Zhang, S.: Finite element interpolation of nonsmooth functions satisfying boundary conditions. Math. Comput. 54(190), 483–493 (1990)

    Article  MATH  MathSciNet  Google Scholar 

  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)

    Article  MATH  MathSciNet  Google Scholar 

  67. Sobolev, S.L.: On a theorem of functional analysis. Mat. Sb 4(46), 471–497 (1938)

    MATH  Google Scholar 

  68. Sobolev, S.L.: Nekotorye primeneniya funkcional’ nogo analiza v matematičeskoĭ fizike. Izdat. Leningrad. Gos. Univ, Leningrad (1950)

    Google Scholar 

  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)

    Article  MATH  MathSciNet  Google Scholar 

  70. Turesson, B.O.: Nonlinear potential theory and weighted Sobolev spaces. In: Lecture Notes in Mathematics, vol. 1736. Springer, Berlin (2000)

  71. Ziemer, W.P.: Weakly differentiable functions. In: Graduate Texts in Mathematics, vol. 120. Springer, New York (1989)

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Acknowledgments

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

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Authors and Affiliations

  1. Department of Mathematics, Institute for Physical Science and Technology, University of Maryland, College Park, MD, 20742, USA

    Ricardo H. Nochetto

  2. Department of Mathematics, University of Maryland, College Park, MD, 20742, USA

    Enrique Otárola

  3. Department of Mathematical Sciences, George Mason University, Fairfax, VA, 22030, USA

    Enrique Otárola

  4. Department of Mathematics, University of Tennessee, Knoxville, TN, 37996, USA

    Abner J. Salgado

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  1. Ricardo H. Nochetto
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  2. Enrique Otárola
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  3. Abner J. Salgado
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Correspondence to Abner J. Salgado.

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R. H. Nochetto has been partially supported by NSF Grants DMS-1109325 and DMS-1411808.

E. Otárola has been partially supported by the Conicyt-Fulbright Fellowship Beca Igualdad de Oportunidades and NSF Grants DMS-1109325 and DMS-1411808.

A. J. Salgado has been partially supported by NSF Grant DMS-1418784.

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Nochetto, R.H., Otárola, E. & Salgado, A.J. Piecewise polynomial interpolation in Muckenhoupt weighted Sobolev spaces and applications. Numer. Math. 132, 85–130 (2016). https://doi.org/10.1007/s00211-015-0709-6

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