5.1. Finite Element Methods for Two-Dimensional Problems
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where [J(e,11)]-l is the inverse matrix of [J] in Eq. (5.1.48), and orfoJoe and
OrfoJ011 can be calculated by Eqs. (5.1.45) and (5.1.46). As a result, Eq. (5.1.49)
expresses orfoJox and orfoJoy as functions of e and 11. Using the same method,
the expressions of orfoj/ox and orfoj/oy in terms of e and 11 can be obtained.
Now, the calculation of Aij associated with element (e) in Eq. (5.1.28) becomes
the calculation of the following integral:
Al]> = f~l f~l !(e, 11)de d11·
(5.1.50)
The Gauss quadratic formula is often used to complete the numerical integration. The calculations of B!j> and F?) are similar. With these being completed, matrices [A], [B] and vector F can be formed .
. .
5.1.5 Treatment of Boundary Conditions
In the formulas derived above, we have considered the first- and second-types
of boundary conditions. However, to deal with the third-type of boundary
condition, the derivation procedure in the Galerkin method should be
modified slightly. The boundary condition with known solute flux can be
given as folIo ws:
(D grad C - VC)·n = -g3, (X,Y)E(r 3 )·
Consider the general advection-dispersion equation
~~ - div(D grad C - VC) = O.
Its Galerkin equations are
f IR) {~~ - div(DgradC - VC)}rfoidXdY = 0,
(i = 1,2, ... ,N)
where Cis the trial function defined by Eq. (5.1.5).
(5.1.51)
(5.1.52)
(5.1.53)
Applying Green's formula to the second term (including both advection
and dispersion) of the integrand in Eq. (5.1.53), we obtain
f IR) {~: rfoi + (DgradC - vC).gradrfoi}dxdY
- r rfoi(D grad C - VC)·ndr = 0,
Jr3)
(i = 1,2, ... ,N).
(5.1.54)
Substituting Eq. (5.1.5) and boundary condition (5.1.51) into the above line