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3.3.1 Galerkin's Method for the Scalar Differential Operator

Equation (3.9) is written as

$\displaystyle \int_{\mathcal{V}}N_i \left[\vec{\nabla}\cdot(\utilde{a}\cdot\ve...
...}\right] \mathrm{d}V - \int_{\mathcal{V}}N_if \mathrm{d}V = 0,   i\in[1;n],$ (3.14)

where Galerkin's method ($ W_i = N_i$ ) and the scalar differential operator (3.2) are used. After applying the first scalar Green's theorem

$\displaystyle \int_{\mathcal{V}}W\left[\vec{\nabla}\cdot(\utilde{a}\cdot\vec{u}...
...m{d}A - \int_{\mathcal{V}}\vec{\nabla}W\cdot\utilde{a}\cdot\vec{u} \mathrm{d}V$ (3.15)

(3.14) is modified to read

$\displaystyle \int_{\partial\mathcal{V}}N_i \vec{n}\cdot\utilde{a}\cdot\vec{\n...
...\tilde{u} \mathrm{d}V - \int_{\mathcal{V}}N_if \mathrm{d}V = 0,  i\in[1;n].$ (3.16)

Since $ N_i$ vanishes on $ \mathcal{A}_D$ for $ i\in[1;n]$ the boundary integral from (3.16) reads

$\displaystyle \int_{\partial\mathcal{V}}N_i \vec{n}\cdot\utilde{a}\cdot\vec{\n...
...\nabla}\tilde{u} \mathrm{d}A \simeq \int_{\mathcal{A}_N}N_i u_n \mathrm{d}A.$ (3.17)

In (3.17) the conormal derivative $ \vec{n}\cdot\utilde{a}\cdot\vec{\nabla}\tilde{u}$ corresponds with the Neumann boundary condition on $ \mathcal{A}_N$ (3.4) [35,37]

$\displaystyle \vec{n}\cdot\utilde{a}\cdot\vec{\nabla}\tilde{u} \simeq u_n = \vec{n}\cdot\utilde{a}\cdot\vec{\nabla}u  \mathrm{on} \mathcal{A}_N.$ (3.18)

Equation (3.16) leads with (3.5) and (3.17) to the linear equation system (3.12), where $ [K]$ and $ \{d\}$ are given by

\begin{displaymath}\begin{split}K_{ij} & = - \int_{\mathcal{V}}\vec{\nabla}N_i\c...
..._N}N_i u_n \mathrm{d}A,  i\in[1;n], j\in[1;n], \end{split}\end{displaymath} (3.19)

with $ \mathcal{L}[v]$ from (3.2).


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Next: 3.3.2 Galerkin's Method for Up: 3.3 Galerkin's Method Previous: 3.3 Galerkin's Method   Contents

A. Nentchev: Numerical Analysis and Simulation in Microelectronics by Vector Finite Elements