Abstract
The purpose of this paper is to present extended results on convergence of a method for coupling of an analytical and a finite element solution for a boundary value problem with a singularity. As an example of a singular problem we consider a solution to the Lamé-Navier equation with a singularity caused by a crack. The main idea of such an approach is to construct a continuous coupling between an analytical solution near a singularity and a finite element solution through the whole interaction interface. In this paper we present a basic theory of convergence for a method of coupling, particularly we discuss some points which show a difference to the standard theory of the finite element method.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
R.A. Adams, J.J.F. Fournier, Sobolev spaces, Elsevier B.V., 2003.
T. Belytschko, R. Gracie, G. Ventura, A review of extended/generalized finite element methods for material modelling. Modelling and Simulation in Materials Science and Engineering. Vol. 17, 2009.
S. Bock, K. Gürlebeck, D. Legatiuk, Convergence of the finite element method with holomorphic functions. AIP Conference proceedings 1558 (2013), 513.
S. Bock, K. Gürlebeck, D. Legatiuk, On a special finite element based on holomorphic functions, AIP Conference proceedings 1479 (2012), 308.
S. Bock, K. Gürlebeck, D. Legatiuk, On the continuous coupling between analytical and finite element solutions, Le Hung Son & Wolfgang Tutschke eds. Interactions between real and complex analysis, pp. 3 - 19. ISBN 978-604-67- 0032-6, Science and Technics Publishing House, Hanoi, 2012.
Philippe G. Ciarlet, The finite element method for elliptic problems, North- Holland Publishing Company, 1978.
Philip J. Davis, Interpolation and Approximation, Dover Publications, Inc., 1975.
Dieter Gaier, Lecture on complex approximation, Birkhäuser Boston Inc., 1987.
Klaus Gürlebeck and Dmitrii Legatiuk, On the continuous coupling of finite elements with holomorphic basis functions. Hypercomplex Analysis: New perspectives and applications, ISBN 978-3-319-08770-2, Birkhäuser, Basel.
D. Legatiuk, K. Gürlebeck, G. Morgenthal, Modelling of concrete hinges through coupling of analytical and finite element solutions. Bautechnik Sonderdruck, ISSN 0932-8351, A 1556, April 2013.
H. Liebowitz, Fracture, an advanced treatise. Volume II: Mathematical fundamentals, Academic Press, 1968.
A.I. Lurie, Theory of elasticity, Foundations of engineering mechanics, Springer- Verlag Berlin Heidelberg, 2005. (translated from Russian)
N. I. Mußchelischwili, Einige Grundaufgaben der mathematischen Elastizitätstheorie, VEB Fachbuchverlag Leipzig, 1971.
R. Piltner, Some remarks on finite elements with an elliptic hole, Finite elements in analysis and design, Volume 44, Issues 12-13, (2008).
R. Piltner, Special finite elements with holes and internal cracks, International journal for numerical methods in engineering, Volume 21, 1985.
Larry Schumaker, Spline Functions on Triangulation, Cambridge University Press, 2007.
Author information
Authors and Affiliations
Corresponding author
Additional information
To Klaus Gürlebeck
The research of the first author is supported by the German Research Foundation (DFG).
The second author acknowledges the financial support of MOET-Vietnam & DAAD.
Rights and permissions
About this article
Cite this article
Legatiuk, D., Nguyen, H.M. Improved Convergence Results for the Finite Element Method with Holomorphic Functions. Adv. Appl. Clifford Algebras 24, 1077–1092 (2014). https://doi.org/10.1007/s00006-014-0501-1
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00006-014-0501-1