$H^1$-conforming approximation for elliptic PDE with discontinuous coefficient? everyone
Suppose I want to solve the diffusion equation $$-\nabla\cdot a \nabla u=f, \\u=0 \text{ on } \partial \Omega,$$ $f \in L^2(\Omega)$, $\partial \Omega$ is smooth.
I use the standard node-based linear elements on a tetrahedral mesh.
I know that to find some $u \in H^2(\Omega)$, I need to make assumption on $a$: $a(x) \in C^1(\Omega)$ (Evans) or $a(x) \in C^0(\bar{\Omega})$ (another textbook). In this case I can safely build approximation in $H^1_0(\Omega)$.
But I also know that I can define $a$ to be merely piece-wise constant in my code without any harm, and I have seen a lot of such computations.
The question: to which space $u$ belong to in the case of the piece-wise constant coefficient? I would be grateful for a reference.
 A: It is quite standard to have $a$ piecewise constant, or more generally a piecewise constant diffusion matrix. Here we consider $a:\Omega\to\mathbb{R}$, but shall require additional regularity.
In the stated problem, you have already assumed that the gradient exists. So your question actually boils down to whether or not a solution to the variational problem exists: find $u\in H^1_0(\Omega)$ such that
$$\langle a\nabla u,\nabla v\rangle_{\Omega} = \langle f,v \rangle_{\Omega}\quad\forall\,v\in H^1_0(\Omega).$$
If a solution to the above exists, then it follows that it is in $H^1_0(\Omega)$. We require to make the assumption that there exist real numbers $\underline{a},\overline{a}>0$ such that
$$ \underline{a} \le a(x) \le \overline{a}\quad\forall\,x\in\Omega.$$
Under this assumption the norm $|v|_{a,H^1_0(\Omega)}^2:=\langle a\nabla v,\nabla v\rangle_{\Omega}$ is well defined and equivalent to the $H^1_0$ norm, thus existence and uniqueness follows from the Riesz Representation Theorem.
Note that the condition above covers the piecewise non-negative constant case, but does not allow for $a$ to be negative. If $a<0$ for all $x$, then the above could be rewritten, and a solution would also be guaranteed. The problems arise when $a$ is allowed to switch between positive and negative over $\Omega$.
