User:BeaRiviere/DGTestbox

Scalar elliptic equation
A scalar elliptic equation is of the form



\begin{align} -\partial_{xx} u &= f(x) \quad \text{for} \quad x\in (a,b) \\ u(x) &= g(x)\,\quad\text{for}\,\quad x=a,b \end{align} $$

This equation is the steady-state heat equation, where $$ u $$ is the temperature. Space discretization is the same as above. We recall that the interval $$(a,b)$$ is partitioned into $$N+1$$ intervals of length $$h$$.

We introduce jump $$[\cdot]$$ and average $$\{\cdot\}$$ of functions at the node $$ x_k$$:



[v]|_{x_k} = v(x_k^+)-v(x_k^-), \quad \{v\}|_{x_k} = 0.5 (v(x_k^+)+v(x_k^-)) $$

The interior penalty discontinuous Galerkin (IPDG) method is: find $$u_h$$ satisfying



A(u_h,v_h) + A_{\partial}(u_h,v_h) = \ell(v_h) + \ell_\partial(v_h) $$ where the bilinear forms $$A$$ and $$A_\partial$$ are

A(u_h,v_h) = \sum_{k=1}^{N+1} \int_{x_{k-1}}^{x_k}\partial_x u_h \partial_x v_h -\sum_{k=1}^N \{ \partial_x u_h\}_{x_k} [v_h]_{x_k} +\epsilon\sum_{k=1}^N \{ \partial_x v_h\}_{x_k} [u_h]_{x_k} +\frac{\sigma}{h} \sum_{k=1}^N [u_h]_{x_k} [v_h]_{x_k} $$ and

A_\partial(u_h,v_h) = \partial_x u_h(a) v_h(a) -\partial_x u_h(b) v_h(b) -\epsilon \partial_x v_h(a) u_h(a) + \epsilon\partial_x v_h(b) u_h(b) +\frac{\sigma}{h} \big(u_h(a) v_h(a) + u_h(b) v_h(b)\big) $$ The linear forms $\ell$ and $\ell_\partial$ are

\ell(v_h) = \int_a^b f v_h $$ and

\ell_\partial(v_h) = -\epsilon \partial_x v_h(a) g(a) + \epsilon\partial_x v_h(b) g(b) +\frac{\sigma}{h} \big( g(a) v_h(a) + g(b) v_h(b) \big) $$