Abstract
Finite element method is probably the most popular numerical method used in different fields of applications nowadays. While approximation properties of the classical finite element method, as well as its various modifications, are well understood, stability of the method is still a crucial problem in practice. Therefore, alternative approaches based not on an approximation of continuous differential equations, but working directly with discrete structures associated with these equations, have gained an increasing interest in recent years. Finite element exterior calculus is one of such approaches. The finite element exterior calculus utilises tools of algebraic topology, such as de Rham cohomology and Hodge theory, to address the stability of the continuous problem. By its construction, the finite element exterior calculus is limited to triangulation based on simplicial complexes. However, practical applications often require triangulations containing elements of more general shapes. Therefore, it is necessary to extend the finite element exterior calculus to overcome the restriction to simplicial complexes. In this paper, the script geometry, a recently introduced new kind of discrete geometry and calculus, is used as a basis for the further extension of the finite element exterior calculus.
REFERENCES
- 1. P.G. Ciarlet, The finite element method for elliptic problems, North-Holland Publishing Company, 1978. Google Scholar
- 2. V.S. Ryaben’kii, Method of difference potentials and its applications, Springer-Verlag, 2002. Google ScholarCrossref
- 3. K. Gürlebeck, A. Hommel, On finite difference potentials and their applications in a discrete function theory, Mathematical Methods in the Applied Sciences, 25, 2002, 1563–1576. Google Scholar
- 4. L. Lovász, Discrete analytic functions: an exposition, Surveys in Differential Geometry, Volume 9, pp. 241–273, 2004. https://doi.org/10.4310/SDG.2004.v9.n1.a7, Google ScholarCrossref
- 5. F. Brackx, H. De Schepper, F. Sommen, L. Van de Voorde, Discrete Clifford analysis: a germ of function theory, Hypercomplex Analysis, Birkhäuser Basel, 2009, 37–53. Google Scholar
- 6. D.N. Arnold, R.S. Falk, R. Winther, Finite element exterior calculus, homological techniques, and applications, Acta Numerica, pp. 1–155, 2006. https://doi.org/10.1017/S0962492906210018, Google ScholarCrossref
- 7. D.N. Arnold, R.S. Falk, R. Winther, Finite element exterior calculus: from Hodge theory to numerical stability, Bulletin (New Series) of the American Mathematical Society, 47(2), pp. 281–354, 2010. https://doi.org/10.1090/S0273-0979-10-01278-4, Google ScholarCrossref
- 8. Y. Bazilevs, L. Beirão da Veiga, J.A. Cottrell, T.J.R. Hughes, G. Sangalli, Isogeometric analysis: approximation, stability and error estimates for h-refined meshes, Mathematical Models and Methods in Applied Sciences, 16(7), 2006. https://doi.org/10.1142/S0218202506001455, Google ScholarCrossref
- 9. K. Gürlebeck, U. Kähler, D. Legatiuk, Error estimates for the coupling of analytical and numerical solutions. Complex Analysis and Operator Theory, 11(5), pp. 1221–1240, 2017. https://doi.org/10.1007/s11785-016-0583-y, Google ScholarCrossref
- 10. K. Gürlebeck, D. 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, 2014. Google Scholar
- 11. P. Cerejeiras, U. Kähler, F. Sommen, A. Vajiac, Script geometry, Modern Trends in Hypercomplex Analysis, pp. 79–110, 2016. https://doi.org/10.1007/978-3-319-42529-0_5, Google ScholarCrossref
Published by AIP Publishing.
Please Note: The number of views represents the full text views from December 2016 to date. Article views prior to December 2016 are not included.
