Numerical Experiments Using MATLAB: Superconvergence of Nonconforming Finite Element Approximation for Second-Order Elliptic Problems
- 1 Department of Mathematics and Computer Science, University of Arkansas at Pine Bluff, Pine Bluff, Arkansas, USA
- 2 US Food and Drug Administration, National Center for Toxicology Research, Jefferson, Arkansas, USA
- 3 Department of Mathematics and Computer Science, University of Arkansas at Pine Bluff, Pine Bluff, Arkansas, USA
Abstract
The superconvergence in the finite element method is a phenomenon in which the fi-nite 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 L2-projections. However, since the conforming finite element method (CFEM) requires a strong continuity, it is not easy to construct such finite elements for the complex partial differential equations. Thus, the nonconforming finite element method (NCFEM) is more appealing computationally due to better stability and flexibility properties compared to CFEM. The objective of this paper is to establish a general superconvergence result for the nonconforming finite element approximations for second-order elliptic problems by L2-projection methods by applying the idea presented in Wang. MATLAB codes are published at https://github.com/annaleeharris/Superconvergence-NCFEM for anyone to use and to study. The results of numerical experiments show great promise for the robustness, reliability, flexibility and accuracy of superconvergence in NCFEM by L2- projections.
- Croouzeix, M. and Raviart, P.A. (1973) Conforming and Nonconforming Finite Element Methods for Solving the Stationary Stokes Equations. R.A.I.R.O. R, 3, 33-76.
- Douglas Jr, J., Santos, J.E., Sheen, D. and Ye, X. (1999) Nonconforming Galerkin Methods Based on Quadrilateral Elements for Second Order Elliptic Problems. Mathematical Modelling and Numerical Analysis, 33, 747-770. https://doi.org/10.1051/m2an:1999161
- Girault, V. and Raviart, P.A. (1986) Finite Element Methods for the Navier-Stokes Equations: Theory and Algorithms. Springer, Berlin. https://doi.org/10.1007/978-3-642-61623-5
- Wang, J. (2000) A Superconvergence Analysis for Finite Element Solutions by the Least-Square Surface Fitting on Irregular Meshes for Smooth Problems. Journal of Mathematical Study, 33, 229-243.
- Ciarlet, P.G. (1978) The Finite Element Method for Elliptic Problems. North-Holland, New York.
- Ye, X. (2002) Superconvergence of Nonconforming Finite Element Method for the Stokes Equation. Numer. Method for PDE, 18, 143-154. https://doi.org/10.1002/num.1036
- Brenner, S.C. and Scott, L.R. (1994) The Mathematical Theory of Finite Element Methods. Springer-Verlag, Berlin. https://doi.org/10.1007/978-1-4757-4338-8