## Abstract

We may not be properly poised to talk about the Green element method **(GEM)** without making reference to the boundary element method **(BEM)** since both methods are founded on the same singular integral theory. The theoretical developments in boundary element **(BE)** circles can be traced to the eighteenth century when related theories of ideal fluid flow (potential flow) and of integral transforms were established, but it was not until the 1960’s that flurries of research activities on **BEM** intensified because of its acclaimed advantage of being a *boundary-only* method. Because the earlier boundary element applications were directed at elliptic boundary-value problems, *the free-space Green’s function* of the differential operator, which is *Inr* in 2-D. spatial dimensions and 1/*r* in 3-D. spatial dimensions, is amenable to Green’s identity which transforms the differential equation into an integral one that can essentially be implemented on the boundary (along a line for 2-D. and on a surface for 3-D.) of the computational domain. In the classical approach, a system of discrete equations is obtained from the integral equation by subdividing the boundary into segments over which distributions are prescribed for the primary variable and the flux (Jaswon [1], Jaswon and Ponter [2], Symm [3], Liggett [4], Liu and Liggett [5], Fairweather *et al.* [6], Rizzo [7], and others). These research thrusts were considered successful when viewed against the background that contemporary methods, like the finite element method **(FEM)** solved similar problems by discretizing the entire computational domain. This led to claims by most investigators that **BEM** was more superior than existing computational methods in terms of accuracy and computational efficiency. It is not in doubt that the integral replication of the differential equation provided by the boundary element theory evolves quite naturally, making use of the response function (free-space Green’s function) to a unit instantaneous input. It is only at the computational stage that the distribution of the dependent variable is approximated by some interpolating polynomial function. It is, thus, expected that for problems where the unit response function can be obtained, **BEM** achieves secondorder accuracy. The claim that **BEM** is superior to **FEM** in computational efficiency is not, in most cases, arrived at after actual CPU comparisons are carried, but it is alluded to on the basis of the boundary-only character of the method.

## Keywords

Source Node Boundary Element Boundary Element Method Domain Integration Green Element## Preview

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## References

- 1.Jaswon, M.A., “Integral Equation Methods in Potential Theory, I”,
*Proc. Roy. Soc. Ser.*, A 275, pp. 23–32, 1963.Google Scholar - 2.Jaswon, M.A. and A.R. Ponter, “An Integral Equation Solution of the Torsion Problem”,
*Proc. Roy. Soc. Ser.*, A 275, pp. 237–246. 1963.Google Scholar - 3.Symm, G.T., “Integral Equation Methods in Potential Theory, II”,
*Proc. Roy. Soc. Ser.*, A 275, pp. 33–46, 1963.Google Scholar - 4.Liggett, J.A., “Location of Free Surface in Porous Media”,
*J. Hyd Div.*,*ASCE*, 103 (HYA), pp. 353–365, 1977.Google Scholar - 5.Liu, P. L-F. and J.A. Liggett, “An Efficient Numerical Method of Two-Dimensional Steady Groundwater Problems”,
*Water Resources Res.*, 14 (3), pp. 385–390, 1978.CrossRefGoogle Scholar - 6.Fairweather, G., F.J. Rizzo, D.J. Shippy and Y.S. Wu, “On the Numerical Solution of Two-Dimensional Potential Problems by an Improved Boundary Integral Equation Method”,
*J. Comp. Phy.*, 31, pp. 96–112, 1979.CrossRefGoogle Scholar - 7.Rizzo, F.J., “An Integral Equation approach to Boundary Value Problems of Classical Elastostatics”, Quart. Appl. Math (25), 1967.Google Scholar
- 8.Clements, D.L., “A Boundary Integral Equation Method for the Numerical Solution of a Second-order Elliptic Equation with Variable Coefficients”,
*J Austrl. Math. Soc.*, 22B, pp. 218–228, 1980.CrossRefGoogle Scholar - 9.Cheng, A. H-D., “Darcy’s flow with variable permeability–A Boundary Integral Solution”,
*Water Resources Res.*, 20 (7), pp. 980–984, 1984.CrossRefGoogle Scholar - 10.Lafe, O.E., “On Boundary Integral Formulation for the Direct and Inverse Problems in Heterogeneous and Deformable Aquifers”, Paper presented at
*Proceedings of the International Atomic Agency/UNESCO Seminar on Applications of Isotope and Nuclear Techniques in Hydrology of Arid and Semi-Arid Lands*, Adana, Turkey, 1985a.Google Scholar - 11.Lafe, O.E., “A Boundary Integral Formulation for the Direct and Inverse Problem in Heterogeneous and Compressible Porous Media”, Paper presented at
*Proceedings*,*NATO Advanced Study Institute on the Fundamentals of Transport Phenomena in Porous Media*, Newark, Del., 1985b.Google Scholar - 12.Lafe, O.E. and A. H-D. Cheng, “A Perturbation Boundary Element Code for Steady State groundwater Flow in Heterogeneous Aquifers”,
*Water Resources Res.*, 23 (6), pp. 1079–1084, 1987.CrossRefGoogle Scholar - 13.Rizzo, F.J. and D.J. Shippy, “A Method of Solution for Certain Problems of Transient Heat Conduction”,
*AIAA*, 8 (11), pp. 2004–2009, 1970.CrossRefGoogle Scholar - 14.Liggett, J.A. and P.L-F. Liu, “Unsteady Flow in Confined Aquifers: A comparison of Two Boundary Integral Methods”,
*Water Resources Res.*, 15 (4), pp. 861–9886, 1979.CrossRefGoogle Scholar - 15.Taigbenu, A.E., AH-D. Cheng and J.A. Liggett, “Boundary Integral Solution to Seawater Intrusion into Coastal Aquifers”,
*Water Resources Res.*, 20 (8), pp. 1150–1158, 1984.CrossRefGoogle Scholar - 16.Cheng, A. H-D and J.A. Liggett, “Boundary Integral Equation Methods for Linear Porous-Elasticity with Applications to Soil Consolidation”,
*Int. J Numer. Meth. Engrg.*, 20, pp. 255–278, 1984.CrossRefGoogle Scholar - 17.Cheng, A. H-D and P. L-F Liu, “Seepage Forces on a Pipeline buried in a Poroelastic seabed under wave Loading”,
*Appl. Ocean Res.*, 8, pp. 22–32, 1986.CrossRefGoogle Scholar - 18.Chang, Y.P., C.S. Kang and D.J. Chen, “The Use of Fundamental Green’s Functions for the Solution of Problems of Heat Conduction in Anisotropic Media”,
*Int. J. Heat Mass Transfer*, 16, pp. 1905–1918, 1973.CrossRefGoogle Scholar - 19.Shaw, R.P., “An Integral Equation Approach to Diffusion”,
*Int. J Heat Mass Transfer*$117, pp. 693–699, 1974.Google Scholar - 20.Brebbia, C.A. and L.C. Wrobel, “The Boundary Element Method for steady State and Transient Heat Conduction”,
*Numerical Methods in Thermal Problems*, ed. by R.W. Lewis and K. Morgan, Prineridge Press, Swansea, Wales, 1979.Google Scholar - 21.Banerjee, P.K. and R. Butterfield,
*Boundary Element Methods in Engineering Science*, McGraw-Hill, London, U.K., 1981.Google Scholar - 22.Wu, J.C., “Fundamental Solutions and Numerical Methods for Flow Problems”,
*Int. J Numer. Meth. Fluids*, 4, pp. 185–201, 1984.CrossRefGoogle Scholar - 23.Taigbenu, A.E. and J.A. Liggett, “Boundary Element Calculations for Diffusion Equation”,
*J. Engrg. Mechanics*,*ASCE*, 111 (3), pp. 311–328, 1985.Google Scholar - 24.Taigbenu, A.E., “A new Boundary Element Formulation applied to unsteady aquifer problems”,
*Ph.D Thesis*, Cornell University, 1985.Google Scholar - 25.Taigbenu, A.E. and J.A. Liggett, “An Integral Formulation Applied to the Diffusion and Boussinesq Equations”,
*Int. J. Numer. Meth. Engrg.*, 23, pp. 1057–1079, 1986a.CrossRefGoogle Scholar - 26.Taigbenu, A.E. and J.A. Liggett, “An Integral Solution for the Diffusion-Advection Equation”,
*Water Resources Res.*, 22 (8) pp. 1237–1246, 1986b.CrossRefGoogle Scholar - 27.Liggett J.A. and A.E. Taigbenu, “Calculation of Diffusion, Advection-Diffusion and Boussinesq Flow by Integral Methods”,
*Proc. VI Int. Conf on the Finite Element Method in Water Resources*, Lisbon, Portugal; pp. 723–733, 1986.Google Scholar - 28.Brebbia, C.A. and D. Nardini, “Dynamic analysis in Solid Mechanics by an alternative Boundary Element Procedure”,
*Int. J. Soil Dynam. Earthquate Engrg.*, 2, pp. 228–233, 1983.CrossRefGoogle Scholar - 29.Wrobel, L.C., and C.A. Brebbia, and D. Nardini, “The Dual Reciprocity Boundary Element Formulation for Transient Heat Conduction”,
*Proc. VI Int. Conf. on Finite Elements in Water Resources*, pp. 801–812, 1986.Google Scholar - 30.Aral, M.M. and Y Tang, “A New Boundary Element Formulation for Time-Dependent Confined and Unconfined Aquifer Problems”,
*Water Resources Res.*, 24 (6), pp. 831–842, 1988.CrossRefGoogle Scholar - 31.Brebbia, C.A. and P. Skerget, “Diffusion-Advection Problems using Boundary Elements”,
*Proc. 5th Int. Conf. on Finite Elements in Water Resources*, Burlington Vermont, USA, pp. 747–768, 1984.Google Scholar - 32.Taigbenu, A.E., “The Green Element Method,”
*Int. J for Numerical Methods in Engineering*, 38, pp. 2241–2263, 1995.CrossRefGoogle Scholar - 33.Taigbenu, A.E., and O.O.Onyejekwe, “Green Element simulations of the Transient Unsaturated Flow Equation,”
*Applied Mathematical Modelling*, 19, pp. 675–684, 1995.CrossRefGoogle Scholar - 34.Taigbenu, A.E., “Green Element Solutions of the 1-D. Steady Groundwater flow in Heterogeneous Aquifers”,
*Ife Journal of Technology*, 5 (2), pp. 29–43, 1996.Google Scholar - 35.Taigbenu„ A.E. and O.O. Onyejekwe, “A Mixed Green element formulation for transient Burgers’ equation”,
*Mt. J Num. Methods in Fluids*, 24, pp. 563–578, 1997.CrossRefGoogle Scholar