Basics

Okay, say we have a volume $\Omega$ with boundary $\partial\Omega$, on which we are solving $\nabla^2u=0$, with $u=\bar{u}$ on $\partial\Omega_1$ and $\nabla u\cdot\hat{n}=\bar{q}$ on $\partial\Omega_2$ ($\hat{n}$ is the outward-pointing unit normal vector). We can approximately solve it using the following weighted residual equation in the Galerkin Finite Element formulation:

$$\int_\Omega\phi\nabla^2udV = \int_{\partial\Omega_2}\phi(\nab... ...A} -\int_{\partial\Omega_1}(u-\bar{u})\nabla\phi\cdot\vec{dA}.$$ (6.1)

Integrating the left side by parts and subtracting $\int_{\partial\Omega}\phi \nabla u\cdot\vec{dA}$ gives

$$-\int_\Omega\nabla\phi\cdot\nabla udV = -\int_{\partial\Omega... ...ec{dA} +\int_{\partial\Omega_1}\bar{u}\nabla\phi\cdot\vec{dA},$$ (6.2)

and another integration plus $\int_{\partial\Omega}u\nabla\phi\cdot\vec{dA}$gives

$$\int_\Omega u\nabla^2\phi dV = -\int_{\partial\Omega_2}\phi\... ...ec{dA} +\int_{\partial\Omega_1}\bar{u}\nabla\phi\cdot\vec{dA},$$ (6.3)

which this book calls the ``starting point for the boundary element method''.