# Introduction to Finite Element Techniques in Computational Fluid Dynamics

## Abstract

The finite element method (FEM) is a technique for solving partial differential equations (PDEs). Its first essential characteristic is that the continuum field, or domain, is subdivided into cells, called elements, which form a grid. The elements have either a triangular or a quadrilateral form. The elements can be rectilinear or can be curved. The grid itself need not be structured. Due to this unstructured form, very complex geometries can be handled with ease. This is clearly the most important advantage of the method and is not shared by the finite difference method (FDM) which needs a structured grid. Therefore in the FDM the domain is to be regularized by mapping a complex domain into a series of rectangular regions. The finite volume method (FVM), on the other hand, has the same geometric characteristic as the FEM.

## Keywords

Finite Element Method Weighting Function Computational Fluid Dynamics Shape Function Finite Difference Method
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

## References

1. 1.
Zienkiewicz, O.C. and Morgan, K. Finite Elements and Approximation, John Wiley, 1983.Google Scholar
2. 2.
Reddy, J.N. An Introduction to the Finite Element Method, McGraw-Hill, 1984.Google Scholar
3. 3.
Cuvelier, C., Segal, A. and Van Steenhoven, A.A. Finite Element Methods and Navier—Stokes Equations, Reidel, 1986.Google Scholar
4. 4.
Zienkiewicz, O.C. The Finite Element Method, McGraw-Hill, 1977.Google Scholar
5. 5.
Oden, J.T. and Reddy J.N. An Introduction to the Mathematical Theory of Finite Elements, John Wiley, 1976.Google Scholar
6. 6.
VKI LS 1987–05: Finite element calculation methods and their application to turbomachinery flows.Google Scholar
7. 7.
Christie, I., Griffiths, D.F., Mitchell, A.R. and Zienckiewicz O.C. ‘Finite element methods for second order differential equations with significant first derivatives’, Int. J. Num. Meth. Engng., Vol. 10, pp. 1389–1396, 1976.
8. 8.
Dick, E. ‘Steady laminar flow over a downwind-facing step as a critical test case for the upwind Petrov—Galerkin finite element method’, Applied Scientific Research, Vol. 39, pp. 321–328, 1982.
9. 9.
VKI LS 1986–04: Computational fluid dynamics.Google Scholar
10. 10.
Hughes, J.R. ‘Recent progress in the development and understanding of SUPG methods with special reference to the compressible Euler and Navier—Stokes equations’, Int. J. Num. Meth. in Fluids,Vol. 7, pp. 1261–1275, 1987.Google Scholar
11. 11.
Sani, R.L., Gresho, P.M. Lee, R.L., Griffiths, D.F. and Engelman M. ‘The cause and cure of the spurious pressures generated by certain FEM solutions of the incompressible Navier—Stokes equations’, Int. J. Num. Meth. in Fluids,Vol. 1, pp. 17–43 and 171–204, 1981.Google Scholar
12. 12.
Fortin, M. ‘Old and new finite elements for incompressible flows’, Int. J. Num. Meth. in Fluids,Vol. 1, pp. 347–364, 1981.Google Scholar
13. 13.
Gresho, Ph.M. ‘The finite element method in viscous incompressible flows’, in ‘Recent advances in computational fluid dynamics’, Lecture Notes in Engineering, Vol. 43, pp. 148–190 Springer, 1989.Google Scholar
14. 14.
Löhner R., Morgan K., Peraire J. and Vandati, M. ‘Finite element flux-corrected transport (FEM-FCT) for the Euler and Navier—Stokes equations’, Int. J. Num. Meth. in Fluids,Vol. 7, pp. 1093–1109, 1987.Google Scholar