Discontinuous Galerkin method

Remarks

The equation is the weak form on one element, obtained by multiplying by $v$ and integrating by parts once. The boundary flux $F(u) \cdot n_K$ is replaced by $\hat H$, because $u_h$ has two values on $\partial K$. Since the test functions live on one element, the mass matrix is block diagonal and explicit time stepping is cheap. Taking $v = 1$ on $K$ shows that the integral of $u_h$ over $K$ changes only through the fluxes on $\partial K$.

Example: for $k = 0$ and $d = 1$, with cells $K_i$ of width $\Delta x$, $u_h = u_i$ on $K_i$. Here $\nabla v = 0$, and the conservation property of $\hat H$ turns the flux on the left face into $-\hat H(u_{i-1}, u_i)$, with $\hat H(a, b)$ short for $\hat H(a, b, +1)$. The equation reads $\Delta x \, u_i' + \hat H(u_i, u_{i+1}) - \hat H(u_{i-1}, u_i) = 0$, a finite volume scheme.

Non-example: the continuous Galerkin method with continuous piecewise polynomials uses a space with continuity across elements and no numerical flux. It does not have the element-by-element form above.