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1

The solution $u\in H^1(\Omega)$ for the Poisson problem is continuous-differentiable and therefore we can apply the div-theorem such that we end up with the weak form \begin{align} -\int_\Omega \Delta u v dx= \int_\Omega \nabla u \nabla v \ dx - \underbrace{\int_{\Gamma} \nabla u \cdot n v \ ds}_{=0\text{ since } u=0 \text{ on } \Gamma} =\int_\Omega f v \ ...

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The face integrals don't show up when using continuous basis functions, and the weak form that you wrote down indeed assumes that the basis functions are continuous. The finite volume method is a special case of the finite element method when you allow discontinuous basis functions instead of only continuous ones. For a discontinuous Galerkin (or DG) ...

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Consider a FV method as an approximation of the integral conservation law. Starting from the one-dimensional, scalar conservation equation \begin{equation} u_t + f(u,\nabla u)_x =0, \end{equation} with the definitions of the step sizes $\Delta x$ and $\Delta t$ \begin{equation} t_n = n \Delta t,~~~x_i = i \Delta x,~~~x_{i+1/2}=\frac{1}{2}\left(x_i + x_{i+...

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