Abstract
Natural superconvergence of the least-squares finite element method is surveyed for the one-and two-dimensional Poisson equation. For two-dimensional problems, both the families of Lagrange elements and Raviart-Thomas elements have been considered on uniform triangular and rectangular meshes. Numerical experiments reveal that many superconvergence properties of the standard Galerkin method are preserved by the least-squares finite element method.
Similar content being viewed by others
References
M. Ainsworth, J. T. Oden: A Posteriori Error Estimation in Finite Element Analysis. Pure and Applied Mathematics. Wiley Interscience, John Wiley & Sons, New York, 2000.
I. Babuška: Error bounds for finite element method. Numer. Math. 16 (1971), 322–333.
I. Babuška, T. Strouboulis: The Finite Element Method and its Reliability. Clarendon Press, Oxford, 2001.
I. Babuška, T. Strouboulis, C. S. Upadhyay, S. K. Gangaraj: Computer-based proof of the existence of superconvergence points in the finite element method; superconvergence of the derivatives in finite element solutions of Laplace, Poisson, and the elasticity equations. Numer. Methods Partial Differ. Equations 12 (1996), 347–392.
D. M. Bedivan: Error estimates for least squares finite element methods. Comput. Math. Appl. 43 (2002), 1003–1020.
P. B. Bochev, M. D. Gunzburger: Finite element methods of least-squares type. SIAM Rev. 40 (1998), 789–837.
J. H. Brandts: Superconvergence and a posteriori error estimation for triangular mixed finite elements. Numer. Math. 68 (1994), 311–324.
J. H. Brandts: Superconvergence for triangular order k = 1 Raviart-Thomas mixed finite elements and for triangular standard quadratic finite element methods. Appl. Numer. Math. 34 (2000), 39–58.
J. H. Brandts, Y. P. Chen: Superconvergence of least-squares mixed finite element methods. Int. J. Numer. Anal. Model. 3 (2006), 303–311.
J. H. Brandts, Y. P. Chen, J. Yang: A note on least-squares mixed finite elements in relation to standard and mixed finite elements. IMA J. Numer. Anal. 26 (2006), 779–789.
F. Brezzi: On the existence, uniqueness and approximation of saddle-point problems arising from Lagrangian multipliers. Rev. Franc. Automat. Inform. Rech. Operat. 8 (1974), 129–151.
F. Brezzi, J. Douglas, Jr., M. Fortin, L. D. Marini: Efficient rectangular mixed finite elements in two and three space variables. Mathematical Modelling and Numerical Analysis 21 (1987), 581–604.
F. Brezzi, J. Douglas, Jr., L. D. Marini: Two families of mixed finite elements for second order elliptic problems. Numer. Math. 47 (1985), 217–235.
Z. Cai, J. Ku: The L 2 norm error estimates for the div least-squares method. SIAM J. Numer. Anal. 44 (2006), 1721–1734.
Z. Cai, R. D. Lazarov, T. A. Manteuffel, S. F. McCormick: First-order system least squares for second-order partial differential equations. I. SIAM J. Numer. Anal. 31 (1994), 1785–1799.
G. F. Carey, Y. Shen: Convergence studies of least-squares finite elements for first-order systems. Commun. Appl. Numer. Methods 5 (1989), 427–434.
C. M. Chen: Structure Theory of Superconvergence of Finite Elements. Hunan Science Press, Hunan, 2001. (In Chinese.)
C. M. Chen, Y. Q. Huang: High Accuracy Theory of Finite Element Methods. Hunan Science and Technology Press, Hunan, 1995. (In Chinese.)
Y. Chen: Superconvergence of mixed finite element methods for optimal control problems. Math. Comput. 77 (2008), 1269–1291.
Z. Chen: Finite Element Methods and Their Applications. Scientific Computation. Springer, Berlin, 2005.
J. Douglas, T. Dupont: Galerkin approximations for the two point boundary problem using continuous, piecewise polynomial spaces. Numer. Math. 22 (1974), 99–109.
J. Douglas, T. Dupont, L. Wahlbin: Optimal L ∞ error estimates for Galerkin approximations to solutions of two-point boundary value problems. Math. Comput. 29 (1975), 475–483.
J. Douglas, J. Wang: Superconvergence of mixed finite element methods on rectangular domains. Calcolo 26 (1989), 121–133.
R. Durán: Superconvergence for rectangular mixed finite elements. Numer. Math. 58 (1990), 287–298.
R. E. Ewing, R. D. Lazarov, J. Wang: Superconvergence of the velocity along the Gauss lines in mixed finite element methods. SIAM J. Numer. Anal. 28 (1991), 1015–1029.
R. E. Ewing, M. M. Liu, J. Wang: Superconvergence of mixed finite element approximations over quadrilaterals. SIAM J. Numer. Anal. 36 (1998), 772–787.
R. E. Ewing, J. Wang: Analysis of mixed finite element methods on locally refined grids. Numer. Math. 63 (1992), 183–194.
L. Gastaldi, R. H. Nochetto: Optimal L ∞-error estimates for nonconforming and mixed finite element methods of lowest order. Numer. Math. 50 (1987), 587–611.
L. Gastaldi, R. H. Nochetto: Sharp maximum norm error estimates for general mixed finite element approximations to second order elliptic equations. RAIRO, Modélisation Math. Anal. Numér. 23 (1989), 103–128.
B.-N. Jiang: The Least-Squares Finite Element Method. Theory and Applications in Computational Fluid Dynamics and Electromagnetics. Springer, Berlin, 1998.
M. Křížek, P. Neittaanmäki: Bibliography on superconvergence. Finite element methods. Superconvergence, post-processing, and a posteriori estimates (M. Křížek, P. Neittaanmäki, R. Stenberg, eds.). Marcel Dekker, New York, 1998, pp. 315–348.
M. Křížek, P. Neittaanmäki: Finite Element Approximation of Variational Problems and Applications. Pitman Monographs and Surveys in Pure and Applied Mathematics, 50. Longman Scientific & Technical, Harlow; copublished in the United States with John Wiley & Sons, Inc., New York, 1990.
J. Li, M. F. Wheeler: Uniform convergence and superconvergence of mixed finite element methods on anisotropically refined grids. SIAM J. Numer. Anal. 38 (2000), 770–798.
Q. Lin, J. Lin: Finite Element Methods: Accuracy and Improvement. Science Press, Beijing, 2006.
Q. Lin, J. H. Pan: High accuracy for mixed finite element methods in Raviart-Thomas element. J. Comput. Math. 14 (1996), 175–182.
Q. Lin, N. Yan: Construction and Analysis of High Efficient Finite Elements. Hebei University Press, Hebei, 1996. (In Chinese.)
R. Lin, Z. Zhang: Natural superconvergent points of triangular finite elements. Numer. Methods Partial Differ. Equations 20 (2004), 864–906.
R. Lin, Z. Zhang: Convergence analysis for least-squares approximations to solutions of second-order two-point boundary value problems. Submitted.
A. I. Pehlivanov, G. F. Carey: Error estimates for least-squares mixed finite elements. RAIRO, Modélisation Math. Anal. Numér. 28 (1994), 499–516.
A. I. Pehlivanov, G. F. Carey, R. D. Lazarov: Least-squares mixed finite elements for second-order elliptic problems. SIAM J. Numer. Anal. 31 (1994), 1368–1377.
A. I. Pehlivanov, G. F. Carey, R. D. Lazarov, Y. Shen: Convergence analysis of least-squares mixed finite elements. Computing 51 (1993), 111–123.
P. A. Raviart, J. M. Thomas: A mixed finite element method for second order elliptic problems. In: Mathematical Aspects of the Finite Element Method. Lecture Notes Math. 606 (I. Galligani, E. Magenes, eds.). Springer, Berlin, 1977, pp. 292–315.
R. Verfürth: A Review of Posteriori Error Estimation and Adaptive Mesh-Refinement Techniques. Wiley-Teubner, Chichester-Stuttgart, 1996.
L. B. Wahlbin: Superconvergence in Galerkin Finite Flement Methods. Lecture Notes Math. 1605. Springer, Berlin, 1995.
M. F. Wheeler: An optimal L ∞ error estimate for Galerkin approximations to solutions of two-point boundary value problems. SIAM J. Numer. Anal. 10 (1973), 914–917.
N. Yan: Superconvergence Analysis and a Posteriori Error Estimation in Finite Element Methods. Science Press, Beijing, 2008.
Z. Zhang: Derivative superconvergence points in finite element solutions of Poisson equation for the serendipity and intermediate families. A theoretical justification. Math. Comput. 67 (1998), 541–552.
Z. Zhang: Recovery techniques in finite element methods. In: Adaptive Computations: Theory and Algorithms (T. Tang, J. Xu, eds.). Science Publisher, 2007, pp. 297–365.
Q. Zhu: High Accuracy and Post-Processing Theory of the Finite Element Method. Science Press, Beijing, 2008. (In Chinese.)
O. C. Zienkiewicz, R. L. Taylor, J. Z. Zhu: The Finite Element Method, 6th ed. Mc-Graw-Hill, London, 2005.
Author information
Authors and Affiliations
Corresponding author
Additional information
Dedicated to Ivan Hlaváček on the occasion of his 75th birthday
The second author was supported in part by the US National Science Foundation under Grant DMS-0612908.
Rights and permissions
About this article
Cite this article
Lin, R., Zhang, Z. Numerical study of natural superconvergence in least-squares finite element methods for elliptic problems. Appl Math 54, 251–266 (2009). https://doi.org/10.1007/s10492-009-0016-6
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10492-009-0016-6