Next, they can be solved numerically using the Finite Element Method^{[1]} where the problem is represented as
∫ | v^{T} A(u) dΩ = | ∫ | [v_{1}A_{1}(u) + v_{2}A_{2}(u) + v_{3}A_{3}(u)] dΩ = 0 |
Ω | Ω | ||
∫ | r^{T} B(u) dΓ = | ∫ | [r_{1}B_{1}(u) + r_{2}B_{2}(u) + r_{3}B_{3}(u)] dΓ = 0 |
Γ | Γ |
where
u = | [ |
n_{+} n_{-} φ |
] |
and v and r are sets of arbitrary functions equal in number to the number of equations (or components of u) involved. A_{1}(u), A_{2}(u) and A_{3}(u) are given by the following formulas, respectively
A_{1}(u) = ∇^{T} | ( | -k_{+} u^{T} | [ |
1 0 0 |
] | ∇ u^{T} | [ |
0 0 1 |
] | - D_{+} ∇ u^{T} | [ |
1 0 0 |
] | ) | + [1, 0, 0] ∂/∂t u |
A_{2}(u) = ∇^{T} | ( | -k_{-} u^{T} | [ |
0 1 0 |
] | ∇ u^{T} | [ |
0 0 1 |
] | - D_{-} ∇ u^{T} | [ |
0 1 0 |
] | ) | + [0, 1, 0] ∂/∂t u |
A_{3}(u) = ε_{0}ε ∇^{T}∇ u^{T} | [ |
0 0 1 |
] | + [1, -1, 0] ze u. |
where k_{i} = D_{i}z_{i}e / (k_{B}T), i = {+, -}.
An expression B(u) gives the boundary conditions on Γ, however, we choose a forced type of boundary conditions on Γ i. e.
φ - φ_{boun} = 0 |
n_{i} - n_{i, boun} = 0. |
Let us approximate unknown u functions by the expansion
u ≈ | ∑ | N_{a} Ι | [ |
n_{+}^{a} n_{-}^{a} φ^{a} |
] | = N u^{bar} |
a |
where Ι is the unit matrix. In place of any function v we put a set of approximate functions
v ≈ | ∑ | w_{b} Ι δu^{bar}_{b} |
b |
where δu^{bar}_{b} are arbitrary parameters. Putting w_{b} = N_{b} we end up with the Galerkin's formulation of the problem. The original basis (shape) functions are used as weighting ones.