Interpolation estimates for $H^1$ into $P_1$

Question

As far as I can tell, nodal interpolation estimates proven with Bramble / Hilbert require higher regularity of the functions being interpolated. E.g. with linear elements in 2 dimensions, one needs $v \in W^{m,p}$ with $m p > 2, p \geq 1$ in order to have

$$ |v - I(v)|_{W^{r,p}} \leq C h^{m-r} |v|_{W^{m,p}}, $$

where $I$ is the nodal interpolant, $| \cdot |$ are seminorms in the indicated spaces and $0 \leq r \leq m$. In particular note that the bound is using the $m$-th derivative of $v$.

For my application I only have $m = 1$ and $p \leq 2$. More specifically this is what I want:

Let $V_h$ be a continuous $P_1$ space over a polygonal domain $U \subset \mathbb{R}^2$ with a quasi-uniform mesh, and $(v_h) \subset V_h$ a sequence bounded in $W^{1,2}(U)$. For each $h$ let $I_h:C^0 \rightarrow V_h$ be some interpolant (I'd use the nodal interpolant but I'm not working with $W^{2,2}$) and $q$ some quadratic function. To simplify maybe take $q(x)=x^2$. I want to show

$$ \|q(v_h) - I_h(q(v_h))\|_{L^2(U)} \leq C h^\alpha \|D q(v_h) \|_{L^2(U)}$$

for any $\alpha > 0$ which makes this work. Reading some papers, I see people change $V_h$ to be discontinuous and claim to have bounds using just the first derivative of $q \circ v_h$, but I don't know where to find precise statements and proofs of the properties of the interpolant in this case. I'm happy to make $V_h$ discontinuous, I just want the damn thing to converge :)

Any help, anyone?

Answer 1

What I was looking for is Clément interpolation, which is defined by averaging a target function $v \in L^1$ over cells sharing a node and assigning this value to the interpolant at that node (or equivalently by local $L^2$ projections). It is explained in detail in §6.3 of these lecture notes of Volker John at the WIAS.

Specifically, the estimate obtained for $v \in W^{1,q}$ is

$$ \| D^k(v - I_h^{\operatorname{cle}}(v)) \|_{L^q} \leq C h^{l-k} \| D^l v\|_{L^q}$$

for all $0 \leq k \leq l \leq 2 $ and $1 \leq q \leq \infty$, assuming a nice enough mesh.

Alternatively, I guess I could just use global $L^2$ projection instead of interpolation, which guarantees the convergence I want, but I need to use the same operator in other places and I don't want my code to be performing costly projections.