FEM Approach

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.


References

  1. ^ O. C. Zienkiewicz, R. L. Taylor and J. Z. Zhu, The Finite Element Method: Its Basis and Fundamentals, Sixth edition., Elsevier 2005