Numerical Solution of Irrotational Fluid Flow Problem Using Finite Element Method

International Journal of Mathematics Trends and Technology (IJMTT)
© 2018 by IJMTT Journal
Volume-62 Number-1
Year of Publication : 2018
Authors : Rajesh Kumar Pal


MLA Style: Rajesh Kumar Pal "Numerical Solution of Irrotational Fluid Flow Problem Using Finite Element Method" International Journal of Mathematics Trends and Technology 62.1 (2018): 36-45.

APA Style: Rajesh Kumar Pal (2018). Numerical Solution of Irrotational Fluid Flow Problem Using Finite Element Method. International Journal of Mathematics Trends and Technology, 62(1), 36-45.

The Finite Element Method has its advancement over other finite difference methods due to taken into consideration the irregular shape domain of problem. The domain of interest on which problem is defined has to be subdivided into sub domains. Herein, two irregular-shaped domain are considered and elliptic equations are solved in each domain with the boundary conditions having varies nature i.e. Dirichlet, Neumann and mixed conditions are applied. Detail discretisation of Finite Element Method and Weighted Residual Techniques are also discussed with their prerequisite conditions. So this Method is a novel fast elliptic solver that can serves as a feasible alternative for numerical solutions and the results compare well with those of [18].

[1] Barrett, K.E. and Diminish, G., Finite Element solutions of convective diffusion problems, Int. J. Num. Meth. Eng., 14, 1511-1524, 1979.
[2] Bennour, H. and Said, M. J., Finite Element solutions of convective diffusion problems, Int. J. Num. Meth. Eng., 14, 1511-1524, 1979.
[3] Bercovier, M. and Engelman,M., Numerical Solution of Poisson‟s equation with Dirichlet Boundary conditions, Int. J. open problems compt. Math.,Vol. 5, No. 4, 2012.
[4] Bramble, J.H., Pasciak, J.E. and Schatz, A.H., „The construction of preconditioners for elliptic problems‟, Mathematics of Computation, Vol. 53, No.187, pp.1-24, 1989.
[5] Burggraf, O.R., Analytical and Numerical studies of the structure of steady separated flows, J.Fluid Mech., 24, 113-151,1966.
[6] Cai, Z., Kim, S. and Kong, S., A finite element method using singular function for Poisson equation: mixed boundary conditions, Comput. Methods Appl. Mech. Engrg., 195, pp 2635-2648, 2006.
[7] Chang , S.C., Solution of Elliptic Partial Differential Equations by fast Poisson solvers using a Local Relaxation Factor. I. One step method‟ NSA technical paper 2529, 1986.
[8] Chaudhary, T.U. and Patel, D.M., Finite Element Solution of Poisson‟s equation in a Homogeneous Medium, International Research Journal of Engineering and Tech. (IRJET), Vol. 02, Issue 09, 2015.
[9] Girault,V. and Raviart, P.A ., Finite element approximation of the Navier-Stokes equations, Lecture notes in Mathematics 749, Springer, Berlin, 1979.
[10] Hockney, R.W., A fast direct solution of Poisson‟s equation using Fourier analysis, J. assoc. Comput. Mach., 12,95-113,1965.
[11] Hyman, J.M. and Manteuffel, T.A., High order sparse factorization methods for elliptic boundary value problems in R. Vichneretsky and R.S.Stepleman. Eds. Advances in computer methods for partial differential equations. V(IMACS, New Brunswick . NS. 1984) 551-555.
[12] Kikuchi, F. and Saito, H., Remarks on a posteriori error estimation for finite element solutions, J. Comput. Appl. Meth., 199(2), pp 329-336, 2007.
[13] Liniger, W, Odeh,F. And Hara, V., A second order sparse factorization method for Poisson‟s equation with mixed boundary conditions, J. Comp. And App. Math., 44,201-218,1992.
[14] Macormic,S.F., „Multigrid methods for variational problems: Further results‟ SIAM J. Numer. Anal., Vol.21 ,1984,PP.255-263.
[15] Mills, R.D., On the closed motion of a fluid in a square cavity, J. Roy. Aero. Soc., 69,116-120,1965.
[16] Sharma,P.K. and Agarwal , M.K., Fast finite difference direct solver for Poisson‟s equation, Indian J. Phy. Nat. Sci. Vol.10, Sec. B, 1989.
[17] Taylor, C and Hughes, T.G., Finite Element Programming of N-S equations, Pineridge Press Limited, Swansea, U.K., 1981.
[18] Tuann, S.Y. and Olson, M.D., Review of computing methods for recirculating flows, J. Comp. Phys. 29,1-17,1978.
[19] Zienkiewicz, O.C., The Finite Element Method, Tata McGraw-Hill, New Delhi.

Poisson’s equation, Weighted Residual Techniques, Finite Element Methods, Galerkin method.