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Next, they can be solved numerically using the Finite Element Method[1] where the problem is represented as
| ∫ | vT A(u) dΩ = | ∫ | [v1A1(u) + v2A2(u) + v3A3(u)] dΩ = 0 |
| Ω | Ω | ||
| ∫ | rT B(u) dΓ = | ∫ | [r1B1(u) + r2B2(u) + r3B3(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. A1(u), A2(u) and A3(u) are given by the following formulas, respectively
| A1(u) = ∇T | ( | -k+ uT | [ |
1 0 0 |
] | ∇ uT | [ |
0 0 1 |
] | - D+ ∇ uT | [ |
1 0 0 |
] | ) | + [1, 0, 0] ∂/∂t u |
| A2(u) = ∇T | ( | -k- uT | [ |
0 1 0 |
] | ∇ uT | [ |
0 0 1 |
] | - D- ∇ uT | [ |
0 1 0 |
] | ) | + [0, 1, 0] ∂/∂t u |
| A3(u) = ε0ε ∇T∇ uT | [ |
0 0 1 |
] | + [1, -1, 0] ze u. |
where ki = Dizie / (kBT), 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 - φboun = 0 |
| ni - ni, boun = 0. |
Let us approximate unknown u functions by the expansion
| u ≈ | ∑ | Na Ι | [ |
n+a n-a φa |
] | = N ubar |
| a | ||||||
where Ι is the unit matrix. In place of any function v we put a set of approximate functions
| v ≈ | ∑ | wb Ι δubarb |
| b | ||
where δubarb are arbitrary parameters. Putting wb = Nb we end up with the Galerkin formulation of the problem. The original basis (shape) functions are used as weighting ones.