Remote Access Mathematics of Computation
Green Open Access

Mathematics of Computation

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

Request Permissions   Purchase Content 
 
 

 

Constructively well-posed approximation methods with unity inf-sup and continuity constants for partial differential equations


Authors: Tan Bui-Thanh, Leszek Demkowicz and Omar Ghattas
Journal: Math. Comp. 82 (2013), 1923-1952
MSC (2010): Primary 65N30; Secondary 65N12, 65N15, 65N22
DOI: https://doi.org/10.1090/S0025-5718-2013-02697-X
Published electronically: April 23, 2013
MathSciNet review: 3073186
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: Starting from the generalized Lax-Milgram theorem and from the fact that the approximation error is minimized when the continuity and inf-sup constants are unity, we develop a theory that provably delivers well-posed approximation methods with unity continuity and inf-sup constants for numerical solution of linear partial differential equations. We demonstrate our single-framework theory on scalar hyperbolic equations to constructively derive two different $ hp$ finite element methods. The first one coincides with a least squares discontinuous Galerkin method, and the other appears to be new. Both methods are proven to be trivially well-posed, with optimal $ hp$-convergence rates. The numerical results show that our new discontinuous finite element method, namely a discontinuous Petrov-Galerkin method, is more accurate, has optimal convergence rate, and does not seem to have nonphysical diffusion compared to the upwind discontinuous Galerkin method.


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

  • 1. L. Demkowicz, J. Gopalakrishnan, A class of discontinuous Petrov-Galerkin methods. Part I: The transport equation, Computer Methods in Applied Mechanics and Engineering 199 (23-24) (2010) 1558-1572. MR 2630162 (2011e:65263)
  • 2. L. Demkowicz, J. Gopalakrishnan, A class of discontinuous Petrov-Galerkin methods. Part II: Optimal test functions, Numerical methods for Partial Differential Equations 27 (1) (2011) 70-105. MR 2743600 (2011k:65155)
  • 3. L. Demkowicz, J. Gopalakrishnan, A class of discontinuous Petrov-Galerkin methods. Part IV: The optimal test norm and time-harmonic wave propagation in 1D, Journal Computational Physics 230 (7) (2011) 2406-2432. MR 2772923 (2012d:65294)
  • 4. I. Babuška, A. Aziz, Survey lectures on the mathematical foundations of the finite element method, in: A. Aziz (Ed.), The Mathematical Foundations of the Finite Element Method with Applications to Partial Differential Equations, Academic Press, New York, 1972, pp. 3-359. MR 0421106 (54:9111)
  • 5. J. T. Oden, L. F. Demkowicz, Applied functional analysis, CRC Press, 2010. MR 2599487 (2011d:46001)
  • 6. A. Ern, J.-L. Guermond, Theory and Practice of Finite Elements, Vol. 159 of Applied Mathematical Sciences, Spinger-Verlag, 2004. MR 2050138 (2005d:65002)
  • 7. I. Babuška, Error bounds for finite element method, Numerische Mathematik 16 (1971) 322-333. MR 0288971 (44:6166)
  • 8. D. B. Szyld, The many proofs of an identity on the norm of oblique projections, Numer. Algorithms 42 (2006) 309-323. MR 2279449 (2007k:46040)
  • 9. L. Demkowicz, ``Babuška $ \leftrightarrow $ Brezzi?'', Tech. Rep. 06-08, Institute for Computational Engineering and Sciences, the University of Texas at Austin (April 2006).
  • 10. J. Xu, L. Zikatanov, Some observations on Babuška and Brezzi theories, Tech. Rep. AM222, Penn State University, http://www.math.psu.edu/ccma/reports.html (September 2000).
  • 11. P. Azerad, Analyse des équations de Navier-Stokes en bassin peu profond et de l'équation de transport, Ph.D. thesis, Neuchatel (1996).
  • 12. C. Johnson, J. Pitkäranta, An analysis of the discontinuous Galerkin method for a scalar hyperbolic equation, Mathematics of Computation 46 (173) (1986) 1-26. MR 815828 (88b:65109)
  • 13. I. Babuška, M. Suri, The $ hp$-version of the finite element method with quasiuniform meshes, Mathematical Modeling and Numerical Analysis 21 (1987) 199-238. MR 896241 (88d:65154)
  • 14. I. Babuška, M. Suri, The optimal convergence rate of the $ p$-version of the finite element method, SIAM J. Numer. Anal. 24 (4) (1987) 750-776. MR 899702 (88k:65102)
  • 15. I. Babuška, M. Suri, The $ p$ and $ h-p$ version of the finite element method, basic principles and properties, SIAM Review 36 (4) (1994) 578-632. MR 1306924 (96d:65184)
  • 16. C. Schwab, $ p$- and $ hp$-finite element methods: Theory and applications in solid and fluid mechanics, Oxford University Press, Oxford, 1998. MR 1695813 (2000d:65003)
  • 17. P. Houston, M. Jensen, E. Süli, $ hp$-Discontinuous Galerkin finite element methods with least-squares stabilization, Journal of Scientific Computing 17 (1-4) (2002) 3-25. MR 1910549
  • 18. N. Nguyen, G. Rozza, D. Huynh, A. Patera, Reduced basis approximation and a posteriori error estimation for parametrized parabolic PDEs; Application to real-time Bayesian parameter estimation, in: L. Biegler, G. Biros, O. Ghattas, M. Heinkenschloss, D. Keyes, B. Mallick, L. Tenorio, B. van Bloemen Waanders, K. Willcox (Eds.), Large Scale Inverse Problems and Quantification of Uncertainty, John Wiley & Sons, 2011.
  • 19. T. Bui-Thanh, K. Willcox, O. Ghattas, Model reduction for large-scale systems with high-dimensional parametric input space, SIAM Journal on Scientific Computing 30 (2008) 3270-3288. MR 2452388 (2009g:90084)
  • 20. T. Bui-Thanh, Model-constrained optimization methods for reduction of parameterized large-scale systems, Ph.D. thesis, MIT (2007).
  • 21. P. G. Ciarlet, The finite element method for elliptic problems, Vol. 40 of Classics in Applied Mathematics, SIAM, Philadelphia, PA, 2002, reprint of the 1978 original [North-Holland, Amsterdam]. MR 0520174 (58:25001)
  • 22. B. Cockburn, G. E. Karniadakis, C.-W. Shu, Discontinuous Galerkin Methods: Theory, Computation and Applications, Lecture Notes in Computational Science and Engineering, Vol. 11, Springer-Verlag, Berlin, Heidelberg, New York, 2000. MR 1842160 (2002b:65004)
  • 23. B. Cockburn, J. Gopalakrishnan, R. Lazarov, Unified hybridization of discontinuous Galerkin, mixed, and continuous Galerkin methods for second order elliptic problems, SIAM J. Numer. Anal. 47 (2009) 1319-1365. MR 2485455 (2010b:65251)
  • 24. W. H. Reed, T. R. Hill, Triangular mesh methods for the neutron transport equation, Tech. Rep. LA-UR-73-479, Los Alamos Scientific Laboratory (1973).
  • 25. P. LeSaint, P. A. Raviart, On a finite element method for solving the neutron transport equation, in: C. de Boor (Ed.), Mathematical Aspects of Finite Element Methods in Partial Differential Equations, Academic Press, 1974, pp. 89-145. MR 0658142 (58:31918)
  • 26. T. Bui-Thanh, L. Demkowicz, O. Ghattas, A fast algorithm for inverse transport equation using a discontinuous Petrov-Galerkin method, In preparation.
  • 27. T. E. Peterson, A note on the convergence of the discontinuous Galerkin method for a scalar hyperbolic equation, SIAM J. Numer. Anal. 28 (1) (1991) 133-140. MR 1083327 (91m:65250)
  • 28. J. Gopalakrishnan, W. Qiu, An analysis of the practical DPG method, to appear in Math. Comp.
  • 29. M. Drela, Two-dimensional transonic aerodyanmic design and analysis using the Euler equations, Ph.D. thesis, MIT (1986).

Similar Articles

Retrieve articles in Mathematics of Computation with MSC (2010): 65N30, 65N12, 65N15, 65N22

Retrieve articles in all journals with MSC (2010): 65N30, 65N12, 65N15, 65N22


Additional Information

Tan Bui-Thanh
Affiliation: Institute for Computational Engineering & Sciences, The University of Texas at Austin, Austin, Texas 78712
Email: tanbui@ices.utexas.edu

Leszek Demkowicz
Affiliation: Institute for Computational Engineering & Sciences, The University of Texas at Austin, Austin, Texas 78712
Email: leszek@ices.utexas.edu

Omar Ghattas
Affiliation: Institute for Computational Engineering & Sciences, The University of Texas at Austin, Austin, Texas 78712
Email: omar@ices.utexas.edu

DOI: https://doi.org/10.1090/S0025-5718-2013-02697-X
Keywords: Inf-sup condition, inf-sup constant, generalized Lax-Milgram theorem, discontinuous Galerkin method, discontinuous Petrov-Galerkin method, consistency, finite element method, stability, convergence, well-posedness hyperbolic partial differential equations
Received by editor(s): April 18, 2011
Received by editor(s) in revised form: October 31, 2011
Published electronically: April 23, 2013
Article copyright: © Copyright 2013 American Mathematical Society

American Mathematical Society