A More General Version of the Hybrid-Trefftz Finite Element Model by Application of TH-Domain Decomposition

  • Ismael Herrera
  • Martin Diaz
  • Robert Yates
Conference paper
Part of the Lecture Notes in Computational Science and Engineering book series (LNCSE, volume 40)


In recent years the hybrid-Trefftz finite element (hT-FE) model, which originated in the work by Jirousek and his collaborators and makes use of an independently defined auxiliary inter-element frame, has been considerably improved. It has indeed become a highly efficient computational tool for the solution of difficult boundary value problems In parallel and to a large extent independently, a general and elegant theory of Domain Decomposition Methods (DDM) has been developed by Herrera and his coworkers, which has already produced very significant numerical results. Theirs is a general formulation of DDM, which subsumes and generalizes other standard approaches. In particular, it supplies a natural theoretical framework for Trefftz methods. To clarify further this point, it is important to spell out in greater detail than has been done so far, the relation between Herrera's theory and the procedures studied by researchers working in standard approaches to Trefftz method (Trefftz-Jirousek approach). As a contribution to this end, in this paper the hybrid-Trefftz finite element model is derived in considerable detail, from Herrera's theory of DDM. By so doing, the hT-FE model is generalized to non-symmetric systems (actually, to any linear differential equation, or system of such equations, independently of its type) and to boundary value problems with prescribed jumps. This process also yields some numerical simplifications.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. I. Herrera. Unified Approach to Numerical Methods. Part 1. Green's Formulas for Operators in Discontinuous Fields. Numerical Methods for Partial Differential Equations, 1(1):12–37, 1985.MathSciNetCrossRefGoogle Scholar
  2. I. Herrera. On Jirousek Method and its Generalizations. Computer Assisted. Mech. Eng. Sci., 8:325–342, 2001.zbMATHGoogle Scholar
  3. I. Herrera. A Unified Theory of Domain Decomposition Methods. In I. Herrera, D. E. Keyes, O. B. Widlund, and R. Yates, editors, 14th International Conference on Domain Decomposition Methods, pages 243–248, 2003.Google Scholar
  4. I. Herrera, R. Yates, and M. Diaz. General Theory of Domain Decomposition: Indirect Methods. Numerical Methods for Partial Differential Equations, 18(3):296–322, 2002.MathSciNetCrossRefGoogle Scholar
  5. J. Jirousek. Basis for development of large finite elements locally satisfying all field equations. Comp. Meth. Appl. Mech. Eng., 14:65–92, 1978.zbMATHCrossRefGoogle Scholar
  6. J. Jirousek and N. Leon. A powerful finite element for plate bending. Comp. Meth. Appl. Mech. Eng., 12:77–96, 1977.MathSciNetCrossRefGoogle Scholar
  7. J. Jirousek and Wroblewski. T-elements: State of the Art and Future Trends. Archives of Computational Methods in Engineering, 3–4:323–434, 1996.MathSciNetGoogle Scholar
  8. J. Jirousek and A. P. Zielinski. Survey of Trefftz-Type Element Formulations. Compu. and Struct., 63:225–242, 1997.CrossRefGoogle Scholar
  9. Q. H. Qin. The Trefftz Finite and Boundary Element Method. WIT Press. Southampton, 2000.Google Scholar
  10. E. Trefftz. Ein Gegenstck zum Ritzschen Verfahren. In Proceedings 2nd International Congress of Applied mechanics, pages 131–137, Zurich, 1926.Google Scholar
  11. A. P. Zielinski. On trial functions applied in the generalized Trefftz method. Advances in Engineering Software, 24:147–155, 1995.zbMATHCrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2005

Authors and Affiliations

  • Ismael Herrera
    • 1
  • Martin Diaz
    • 2
  • Robert Yates
    • 1
  1. 1.Instituto de GeofísicaUniversidad Nacional Autónoma de México (UNAM)México
  2. 2.Instituto Mexicano del PetróleoPetróleo

Personalised recommendations