2021-11-03 Software discussion¶
Last time¶
Running multigrid with PETSc
Projects
GitHub codesearch, JOSS search
Decide what’s next: Finite Element or Finite Volume
Today¶
Project discussion
Weak forms
Integration by parts¶
One dimension¶
it’s the product rule backwards
(34)¶\[\begin{align} \int_a^b d(uv) &= \int_a^b u dv + \int_a^b v du \\
(uv)_a^b &= \int_a^b u dv + \int_a^b v du \\
\int_a^b u dv &= (uv)_a^b - \int_a^b v du
\end{align}\]
you can move the derivative to the other term; it’ll cost you a minus sign and a boundary term
Multiple dimensions¶
(35)¶\[\begin{align}
\int_\Omega v \nabla\cdot \mathbf f = -\int_\Omega \nabla v \cdot \mathbf f + \int_{\partial \Omega} v \mathbf f \cdot \mathbf n
\end{align}\]
Strong form¶
\[ -\nabla\cdot(\kappa \nabla u) = 0 \]
Weak form¶
multiply by a test function and integrate by parts
(36)¶\[\begin{align} -\int_\Omega v \nabla\cdot(\kappa \nabla u) = 0, \forall v \\
\int_\Omega \nabla v \cdot \kappa \nabla u - \int_{\partial\Omega} v \underbrace{\kappa \nabla u \cdot \mathbf n}_{\text{boundary condition}} = 0
\end{align}\]