Research ArticleOpen AccessGoogle Scholar indexed
Numerical Experiments Using MATLAB: Superconvergence of Conforming Finite, Element Approximation for Second Order, Elliptic Problems
Department of Mathematics and Computer Science, University of Arkansas at Pine Bluff, Pine Bluff, USA
US Food and Drug Administration, National Center for Toxicology Research, Jefferson, USA
Department of Mathematics and Computer Science, University of Arkansas at Pine Bluff, Pine Bluff, USA
Department of Mathematics and Computer Science, University of Arkansas at Pine Bluff, Pine Bluff, USA
- 1 Department of Mathematics and Computer Science, University of Arkansas at Pine Bluff, Pine Bluff, USA
- 2 US Food and Drug Administration, National Center for Toxicology Research, Jefferson, USA
- 3 Department of Mathematics and Computer Science, University of Arkansas at Pine Bluff, Pine Bluff, USA
- 4 Department of Mathematics and Computer Science, University of Arkansas at Pine Bluff, Pine Bluff, USA
Copy link · social · email
Abstract
The superconvergence in the finite element method is a phenomenon in which the finite element approximation converges to the exact solution at a rate higher than the optimal order error estimate. Wang proposed and analyzed superconvergence of the conforming finite element method by L 2 -projections. The goal of this paper is to perform numerical experiments using MATLAB to support and to verify the theoretical results in Wang for the superconvergence of the conforming finite element method (CFEM) for the second order elliptic problems by L 2 -projection methods. MATLAB codes are published at https://github.com/annaleeharris/Superconvergence-CFEM for anyone to use and to study.
KeywordsConforming Finite Element MethodsSuperconvergence<em>L<sup>2</sup></em>-ProjectionSecond Order Elliptic EquationmMATLAB
- Bramble, J.H. and Schatz, A.H. (1977) Higher Order Local Accuracy by Averaging in the Finite Element Method. Mathematics of Computation, 31, 94-111. https://doi.org/10.1090/S0025-5718-1977-0431744-9
- Douglas Jr., J. and Wang, J. (1989) A Superconvergence for Mixed Finite Element Methods on Rectangular Domains. Calcolo, 26, 121-134. https://doi.org/10.1007/BF02575724
- Ewing, R.E., Lazarov, R. and Wang, J. (1991) Superconvergence of the Velocity along the Gauss Lines in Mixed Finite Element Methods. SIAM Journal on Numerical Analysis, 28, 1015-1029. https://doi.org/10.1137/0728054
- Krizek, M. and Neittaanmaki, P. (1984) Superconvergence Phenomenon in the Finite Element Method Arising from Averaging Gradient. Numerische Mathematik, 45, 105-116. https://doi.org/10.1007/BF01379664
- Lazarov, R., Andreev, A.B. and Hatri, M. (1984) Superconvergence of the Gradients in the Finite Element Method for Some Elliptic and Parabolic Problems. In: Variational-Difference Methods in Mathematical Physics, Part II, Moscow, 13-25.
- Lin, Q. (1992) Global Error Expansion and Superconvergence for Higher Order Interpolation of Finite Element. Journal of Computational Mathematics, 10, 286-289.
- Oganesyan, I.A. and Rukhovetz, L.A. (1969) Study of the Rate of Convergence of Variational Difference Scheme for Second-Order Elliptic Equations in Two-Dimensional Field with a Smooth Boundary. USSR Computational Mathematics and Mathematical Physics, 9, 158-183. https://doi.org/10.1016/0041-5553(69)90159-1
- Wahlbin, L.B. (1995) Superconvergence in Galerkin Finite Element Methods. Springer, New York. https://doi.org/10.1007/BFb0096835
- Zlamal, M. (1978) Superconvergence and Reduced Integration in the Finite Element Method. Mathematics of Computation, 32, 663-685. https://doi.org/10.2307/2006479
- Schatz, A.H., Sloan, I.H. and Wahlbin, L.B. (1996) Superconvergence in Finite Element Methods and Meshes That Are Symmetric with Respect to a Point. SIAM Journal on Numerical Analysis, 33, 505-521. https://doi.org/10.1137/0733027
- Douglas, J., Dupont, T. and Wheeler, M.F. (1974) An Estimate and a Superconvergence Result for a Galerkin Method for Elliptic Equations Based on Tensor Products of Piecewise Polynomials. RAIRO: Analyse Numérique, 8, 61-66.
- Douglas, J. and Dupont, T. (1973) Some Superconvergence Results for Galerkin Methods for the Approximation Solution of Two-Point Boundary Value Problems. Topics in Numerical Analysis, 89-92.