Nonlinear Laplace/Poisson Equation

  • Akpofure E. Taigbenu


In the previous chapter, we presented the boundary element and Green element formulations for the linear version of the steady 1-D. second-order differential equation given by eq. (2.1) in which the parameter K assumed a constant value. We used the example of flow in a confined aquifer of uniform thickness to derive the linear form of the differential equation. Here we relax that condition and consider cases where K is a function of the spatial variable x (heterogeneous case), and a function of the dependent variable h (nonlinear case). Nonlinear heterogeneous problems are frequently encountered in many engineering applications — heat flux through a material whose thermal properties are nonlinear, infiltration into unsaturated soils, elastic deformation of a bar of varying section subject to axial loads, etc. We elect to use a heat transfer example to derive the nonlinear equation of the form of eq. (2.1).


Boundary Element Green Element Element Equation Picard Scheme Dual Reciprocity Boundary Element Method 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Cheng, A. H-D. and D. Ouaza, “Groundwater flow,” in chapter 8 of Boundary Element Techniques in Geomechanics, eds. G.D. Manolis and T.G. Davies, CMP/Elsevier, 1993.Google Scholar
  2. 2.
    Lafe, O.E. and A.H-D. Cheng, “A Perturbation Boundary Element Code for Groundwater flow in Heterogeneous Aquifer,” Water Resources Research, 23, pp. 1079–1084, 1987.CrossRefGoogle Scholar
  3. 3.
    Partridge, P.W., C.A. Brebbia and L.C. Wrobel, The Dual Reciprocity Boundary Element method, CMP/Elsevier, 1991.Google Scholar
  4. 4.
    Cheng, A.H-D., “Darcy’s flow with variable Permeability–a Boundary Integral Solution,” Water Resources Research, 20, pp. 980–984, 1984.CrossRefGoogle Scholar
  5. 5.
    Taigbenu, A.E., “The Green Element Method,” Int. J. for Numerical Methods in Engineering, 38, pp 2241–2263, 1995.CrossRefGoogle Scholar
  6. 6.
    Taigbenu, A.E., and 0.0.Onyejekwe, “Green Element simulations of the Transient Unsaturated Flow Equation,” Applied Mathematical Modelling, 19, pp. 675–684, 1995.CrossRefGoogle Scholar
  7. 7.
    Strang, G. and G.J. Fix, An Analysis of the Finite Element Method, Prentice Hall, Englewood Cliffs, NJ., 1973.Google Scholar
  8. 8.
    Zienkiewicz, O.C. and R.L. Taylor, The Finite Element Method, 4th edition, Vol. I, McGraw-Hill, NY., 1989.Google Scholar
  9. 9.
    Zienkiewicz, O.C., The Finite Element Method, 3ed., McGraw-Hill, London, 1977.Google Scholar
  10. 10.
    Reddy, J.N., An Introduction to the Finite Element Method, McGraw-Hill, Inc. NY., 1984.Google Scholar
  11. 11.
    Rao, S.S., The Finite Element Method in Engineering, 2ed. Pergamon Press plc, Oxford, 1989.Google Scholar
  12. 12.
    Ottosen, N.S. and H. Petersson, Introduction to the Finite Element Method, Prentice Hall Int. Ltd. UK., 1992.Google Scholar
  13. 13.
    Hornbeck, R.W., Numerical Methods, Quantum Pub. Inc., U.S.A., 1975.Google Scholar

Copyright information

© Springer Science+Business Media New York 1999

Authors and Affiliations

  • Akpofure E. Taigbenu
    • 1
  1. 1.Department of Civil and Water EngineeringNational University of Science and TechnologyBulawayoZimbabwe

Personalised recommendations