Abstract
In this chapter our goal is to develop a unified and general framework for enriching finite element approximations via the use of additional enrichment functions. A crucial point in such an approach is to determine conditions on enrichment functions which guarantee that they generate a well-defined finite element. We start by giving under some conditions an abstract general theorem characterizing the existence of any enriched finite element approximation. After proving four key lemmas, we then establish under a unisolvence condition a more practical characterization result. We show that this proposed method easily allows us to establish a new class of enriched non-conforming finite elements in any dimension. This new family is inspired by the Han rectangular element and the nonconforming rotated element of Rannacher and Turek. They are all obtained as applications of a new family of multivariate trapezoidal, midpoint, and Simpson type cubature formulas, which employ integrals over facets. In addition, we provide analogously a general class of perturbed trapezoidal and midpoint cubature formulas, and use them to build a new enriched nonconforming finite element of Wilson-type.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
Achchab, B., Agouzal, A., Bouihat, K.: A simple nonconforming quadrilateral finite element. C. R. Acad. Sci. Paris Ser. I 352, 529–533 (2014)
Achchab, B., Bouihat, K., Guessab, A., Schmeisser, G.: A general approach to the construction of nonconforming finite elements on convex polytope. Appl. Math. Comput. 268, 916–923 (2015)
Achchab, B., Guessab, A., Zaim, Y.: A new class of nonconforming finite elements for the enrichment of Q 1 element on convex polytope. Appl. Math. Comput. 271, 657–668 (2015)
Atkinson, K.E.: An Introduction to Numerical Analysis, 2nd edn. Addison-Wesley, Reading, MA (1975)
Babu\(\check{\mathrm{s}}\) ka, I., Banerjee, U., Osborn, J.E.: Survey of meshless and generalized finite element methods: a unified approach. Acta Numer. 12 (1), 1–125 (2003)
Babu\(\check{\mathrm{s}}\) ka, I., Banerjee, U., Osborn, J.: Generalized finite element methods: main ideas, results and perspective. Int. J. Comput. Methods 1 (1), 67–103 (2004)
Belytschko, T., Krongauz, Y., Organ, D., Fleming, M., Krysl, P.: Meshless methods: an overview and recent developments. Comput. Meth. Appl. Mech. Eng. 139 (1–4), 3–47 (1996)
Benzley, S.E.: Representation of singularities with isoparametric finite elements. Int. J. Numer. Methods Eng. 8, 537–545 (1974)
Byskov, E.: The calculation of stress intensity factors using the finite element method with cracked elements. Int. J. Fract. Mech. 6, 159–167 (1970)
Cai, Z., Douglas, J. Jr., Ye, X.: A stable nonconforming quadrilateral finite element method for the stationary Stokes and Navier-Stokes equations. Calcolo 36, 215–232 (1999)
Cai, Z., Douglas, J. Jr., Santos, J.E., Sheen, D., Ye, X.: Nonconforming quadrilateral finite elements: a correction. Calcolo 37, 253–254 (2000)
Ciarlet, P.G.: The Finite Element Method for Elliptic Problems. North Holland, Amsterdam (1980)
De, S., Bathe, K.J.: The method of finite spheres. Comput. Mech. 25 (4), 329–345 (2000)
Douglas, J. Jr., Santos, J.E., Sheen, D., Ye, X.: Non conforming Galerkin methods based on quadrilateral elements for second order elliptic problems. RAIRO Math. Model. Anal. Numer. 33, 747–770 (1999)
Fix, G., Gulati, S., Wakoff, G.I.: On the use of singular functions with finite element approximations. J. Comput. Phys. 13, 209–228 (1973)
Fries, T.P., Belytschko, T.: The intrinsic XFEM: a method for arbitrary discontinuities without additional unknowns. Int. J. Numer. Methods Eng. 68 (13), 1358–385 (2006)
Han, H.: Nonconforming elements in the mixed finite element method. J. Comput. Math. 2 (3), 223–233 (1984)
Khoei, A.R.: Extended Finite Element Method, Theory and Applications. Wiley, New York (2015)
Li, S., Liu, W.K.: Meshfree Particle Methods. Springer, Berlin (2004)
Lin, Q., Tobiska, L., Zhou, A.: Superconvergence and extrapolation of nonconforming low order finite elements applied to the Poisson equation. IMA J. Numer. Anal. 25, 160–181 (2005)
Liu, G.: Meshfree Methods: Moving Beyond the Finite Element Method. CRC, Boca Raton, FL (2002)
Mao, S.P., Chen, S.C., Shi, D.: Convergence and superconvergence of a nonconforming finite element on anisotropic meshes. Int. J. Numer. Anal. Model. 4, 16–38 (2007)
Rannacher, R., Turek, S.: Simple nonconforming quadrilateral Stokes element. Numer. Methods Partial Differ. Equ. 8 (2), 97–111 (1992)
Silvester, J.R.: Determinants of block matrices. Math. Gaz. 84, 460–467 (2000)
Zienkiewicz, O.C.: The Finite Element Method, 3rd edn. McGraw-Hill, New York (1977)
Acknowledgements
Allal Guessab and Yassine Zaim would like to thank the Volubilis Hubert Curien Program (Grant No. MA/13/286) for financial support.
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2017 Springer International Publishing AG
About this chapter
Cite this chapter
Guessab, A., Zaim, Y. (2017). A Unified and General Framework for Enriching Finite Element Approximations. In: Govil, N., Mohapatra, R., Qazi, M., Schmeisser, G. (eds) Progress in Approximation Theory and Applicable Complex Analysis. Springer Optimization and Its Applications, vol 117. Springer, Cham. https://doi.org/10.1007/978-3-319-49242-1_22
Download citation
DOI: https://doi.org/10.1007/978-3-319-49242-1_22
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-49240-7
Online ISBN: 978-3-319-49242-1
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)