Hostname: page-component-78c5997874-mlc7c Total loading time: 0 Render date: 2024-11-10T16:13:34.639Z Has data issue: false hasContentIssue false
  • English
  • Français

Interpolation, Projection and Hierarchical Bases in Discontinuous Galerkin Methods

Part of: Numerical approximation and computational geometry Partial differential equations, initial value and time-dependent initial-boundary value problems

Published online by Cambridge University Press:  05 August 2015

Lutz Angermann  and
Christian Henke
Lutz Angermann*
Affiliation:
Institut für Mathematik, Technische Universität Clausthal, Erzstraße 1, D-38678 Clausthal-Zellerfeld, Germany
Christian Henke
Affiliation:
Institut für Mathematik, Technische Universität Clausthal, Erzstraße 1, D-38678 Clausthal-Zellerfeld, Germany
*
*Email addresses: lutz.angermann@tu-clausthal.de (L. Angermann), henke@math.tu-clausthal.de (C. Henke)
Get access

Abstract

The paper presents results on piecewise polynomial approximations of tensor product type in Sobolev-Slobodecki spaces by various interpolation and projection techniques, on error estimates for quadrature rules and projection operators based on hierarchical bases, and on inverse inequalities.

Keywords

Interpolationquadraturehierarchical basesinverse inequalities

MSC classification

Secondary: 65M60: Finite elements, Rayleigh-Ritz and Galerkin methods, finite methods 65D32: Quadrature and cubature formulas
Type
Research Article
Information
Numerical Mathematics: Theory, Methods and Applications , Volume 8 , Issue 3 , August 2015 , pp. 425 - 450
Copyright
Copyright © Global-Science Press 2015 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[1]Adams, R.A. and Fournier, J.J.F.. Sobolev spaces, volume 140 of Pure and Applied Mathematics (Amsterdam). Elsevier/Academic Press, Amsterdam, 2nd edition, 2003.Google Scholar
[2]Baran, M.. New approach to Markov inequality in Lp norms. In Approximation theory, volume 212 of Monogr. Textbooks Pure Appl. Math., pages 75–85. Dekker, New York, 1998.Google Scholar
[3]Bergh, J. and Löfström, J.. Interpolation spaces. An introduction. Springer-Verlag, Berlin, 1976. Grundlehren der Mathematischen Wissenschaften, No. 223.CrossRefGoogle Scholar
[4]Bernardi, C. and Maday, Y.. Spectral methods. In Handbook of numerical analysis, Vol. V, pages 209–485. North-Holland, Amsterdam, 1997.Google Scholar
[5]Chen, Q. and Babuška, I.. Polynomial interpolation of real functions I: Interpolation in an interval. Technical Note BN 1153, University of Maryland, College Park, 1993.Google Scholar
[6]Calvetti, D., Golub, G.H., Gragg, W.B., and Reichel, L.. Computation of Gauss-Kronrod quadrature rules. Math. Comp., 69(231):1035–1052, 2000.CrossRefGoogle Scholar
[7]Canuto, C., Hussaini, M.Y., Quarteroni, A., and Zang, T.A.. Spectral methods. Scientific Computation. Springer, Berlin, 2007. Evolution to complex geometries and applications to fluid dynamics.Google Scholar
[8]Ciarlet, P.G.. Basic error estimates for elliptic problems. In Ciarlet, P.G. and Lions, J.L., editors, Handbook of numerical analysis. II: Finite element methods (Part 1), pages 17–351. North-Holland, Amsterdam-London-New York-Tokyo, 1991.CrossRefGoogle Scholar
[9]Canuto, C. and Quarteroni, A.. Approximation results for orthogonal polynomials in Sobolev spaces. Math. Comp., 38(157):67–86, 1982.CrossRefGoogle Scholar
[10]DeVore, R.A. and Lorentz, G.G.. Constructive approximation. Springer-Verlag, Berlin-Heidelberg-New York, 1993.CrossRefGoogle Scholar
[11]Ern, A. and Guermond, J.-L.. Theory and practice of finite elements, volume 159 of Applied Mathematical Sciences. Springer-Verlag, New York, 2004.CrossRefGoogle Scholar
[12]Fejér, L.. Bestimmung derjenigen Abszissen eines Intervalles, für welche die Quadrat-summe der Grundfunktionen der Lagrangeschen Interpolation im Intervalle ein möglichst kleines Maximum besitzt. Ann. Scuola Norm. Sup. Pisa Cl. Sci. (2), 1(3):263–276, 1932.Google Scholar
[13]Gagliardo, E.. Caratterizzazione delle Tracce sulla Frontiera Relative ad Alcune Classi di Funzioni in n Variabli. Rend. Sem. Mat. Padova, 27:284–305, 1957.Google Scholar
[14]Georgoulis, E.H.. Discontinuous Galerkin methods on shape-regular and anisotropic meshes. PhD thesis, Oxford University Computing Laboratory, 2003.Google Scholar
[15]Gudi, T., Nataraj, N., and Pani, A.K.. hp-discontinuous Galerkin methods for strongly nonlinear elliptic boundary value problems. Numer. Math., 109(2):233–268, 2008.CrossRefGoogle Scholar
[16]Houston, P., Schwab, C., and Süli, E.. Discontinuous hp-finite element methods for advection-diffusion-reaction problems. SIAM J. Numer. Anal., 39(6):2133–2163, 2002.CrossRefGoogle Scholar
[17]Hille, E., Szegö, G., and Tamarkin, J.D.. On some generalizations of a theorem of A. Markoff. Duke Math. J., 3(4):729–739, 1937.CrossRefGoogle Scholar
[18]Jerison, D. and Kenig, C.E.. The inhomogeneous Dirichlet problem in Lipschitz domains. J. Funct. Anal., 130(1):161–219, 1995.CrossRefGoogle Scholar
[19]Johnson, W.P.The curious history of Faà di Bruno’s formula. Amer. Math. Monthly, 109(3):217–234, 2002.Google Scholar
[20]Kalyabin, G.A.Theorems on extension, multipliers and diffeomorphisms for generalized Sobolev-Liouville classes in domains with Lipschitz boundary. Trudy Mat. Inst. Steklov., 172:173–186, 353, 1985. Studies in the theory of functions of several real variables and the approximation of functions.Google Scholar
[21]Karniadakis, G.E. and Sherwin, S.J.Spectral/hp element methods for computational fluid dynamics. Numerical Mathematics and Scientific Computation. Oxford University Press, New York, 2nd edition, 2005.CrossRefGoogle Scholar
[22]Milovanović, G.V., Mitrinović, D.S., and Rassias, Th.M.Topics in polynomials: extremal problems, inequalities, zeros. World Scientific Publishing Co. Inc., River Edge, NJ, 1994.CrossRefGoogle Scholar
[23]Nečas, J.Les méthodes directes en théorie des équations elliptiques. Academia, Prague, 1967.Google Scholar
[24]Nikolski, S.M.Inequalities for entire functions of exponential type and their application to the theory of differentiable functions of several variables. In Trudy Mat. Inst. Steklova, vol. 38, pages 244–278. Izdat. Akad. Nauk SSSR, Moscow, 1951. English transl.: Amer. Math. Soc. Transl. 80(1969), (2), 1–38.Google Scholar
[25]Quarteroni, A. and Valli, A.Numerical approximation of partial differential equations. Springer-Verlag, Berlin-Heidelberg-New York, 1994.CrossRefGoogle Scholar
[26]Stein, E.M.Singular integrals and differentiability properties of functions. Princeton Mathematical Series, No. 30. Princeton University Press, Princeton, N.J., 1970.Google Scholar
[27]Sudirham, J.J., van der Vegt, J.J.W., and van Damme, R.M.J.Space-time discontinuous Galerkin method for advection-diffusion problems on time-dependent domains. Appl. Numer. Math., 56(12):1491–1518, 2006.CrossRefGoogle Scholar
[28]Timan, A.F.Theory of approximation of functions of a real variable. Translated by Berry, J.International Series of Monographs on Pure and Applied Mathematics, vol. 34. Pergamon Press, Oxford etc., 1963.Google Scholar
[29]Triebel, H.Interpolation theory, function spaces, differential operators, volume 18 of North-Holland Mathematical Library. North-Holland Publishing Co., Amsterdam, 1978.Google Scholar