# 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

\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

\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

\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}

03060d68fce4e60e37dcbeb58434483f6257d5aa
ipynb
Jupyter Notebook
slides/2021-11-03-discussion.ipynb
cu-numpde/numpde
e5e1a465a622eba56900004f9a503412407cdccf
MIT
2022-01-01T20:54:51.000Z
2022-01-01T20:54:51.000Z
slides/2021-11-03-discussion.ipynb
amta3208/fall21
e5e1a465a622eba56900004f9a503412407cdccf
MIT
slides/2021-11-03-discussion.ipynb
amta3208/fall21
e5e1a465a622eba56900004f9a503412407cdccf
MIT
2022-01-01T20:54:46.000Z
2022-01-01T20:54:46.000Z
Qwen/Qwen-72B
1. YES
2. YES