Some Examples of the Poisson Equation  . (ku) = f

[Taken from J. N. Reddy's Book "Introduction to the Finite Element Method", J.N. Reddy, McGraw Hill Publishers, 2nd Edition]

Field of Application

Primary Variable u

Material Constant k

Source Variable f

Secondary Variables d, du/dx, du/dy 

Heat Transfer

Temperature T

Conductivity k

Heat Source Q

Heat Flow q [comes from conduction k T/n and convection h(T-T)


Irrotational Flow of an Ideal Fluid


Stream Function y


Velocity Potential f


Density r


Density r


Mass Production s (normally zero)

Mass Production s (normally zero)



y /x = -v;

y /y = u

f /x = -v;

f /y = u


Groundwater Flow


Piezometric Head f


Permeability K


Recharge Q


Seepage: q = k f/dn


u = -k f/dx ,

v = -k f/dy


Torsion of Members with Constant Cross-Section


Stress Function Y


k = 1

G = Shear Modulus


f = 2

q = angle of twist per unit length


Gqdf/dx = -syz

Gqdf/dx = -sxz




Scalar Potential f


Dielectric Constant e


Charge Density r


Displacement Flux density Dn




Magnetic Potential f


Permeability m


Charge density r


Magnetic Flux density Bn


Transverse Deflection of Elastic Membranes


Transverse deflection u


Tension T in membrane


Transversely distributed Load


Normal force q

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