Boundary elements

Beginning with constant elements, if one assumes that u and $\nabla u\cdot\hat{n}$ are constant over each element j of which the surface is comprised, then equation 6.11 becomes

$$\frac{u^i}{2}+u_j\int_{\partial\Omega_j}\nabla u^*_i\cdot\vec{dA} = (\nabla u\cdot\hat{n})_j\int_{\partial\Omega_j}u^*_i dA,$$ (6.12)

where summation is implied for like indices (Einstein notation). Defining the matrices $\hat{H}_{ij}$ and G_ij according to the left- and right-side integrals gives the matrix equation

$$\frac{u^i}{2}+\hat{H}_{ij}u_j=G_{ij}(\nabla u\cdot\hat{n})_j,$$ (6.13)

which can be simplified further by defining $H_{ij}=\hat{H}_{ij}+\frac{1}{2}\delta_{ij}$ to give

$$H_{ij}u_j=G_{ij}(\nabla u\cdot\hat{n})_j.$$ (6.14)

Rearranging the matrix columns and vector rows so that unknowns go on the left and knowns on the right (i.e. for nodes j in $\partial\Omega_1$, u_j goes on the right and $(\nabla u\cdot\hat{n})_j$ on the left; vice versa for $\partial\Omega_2$) gives the matrix equation which solves the problem.

Note that for constant and linear elements, $\hat{H}_{ii}$ is zero because $\nabla u^*_i$ is orthogonal to the surface normal. Also, G_ii for constant elements can be calculated analytically, and in 2-D is equal to

$$G_{ii}=\int_{\partial\Omega_i}u^*_idA = \frac{1}{2\pi}\int_{\... ...i}\vert r_1\vert\left\{\ln\left(\frac{1}{r_1}\right)+1\right\}$$ (6.15)

where |r₁| is half the element width.