What is the difference between $u_h$ and $I_h(u)$ in finite element literature?

Question

In finite element books, we have estimates for $$||u-u_h||$$ and also estimates for $$||u - I_h(u)||$$ where $I_h(u): V \mapsto V_h$ projects a function from the infinite dimensional space to the finite dimensional one. For instance, $V = H^1$ and $V_h \subset V$ is the set of piecewise poly of degree $r$.

What is the difference between $u_h$ and $I_h(u)$ ?

Here's my attempt:

Usually the interpolant is defined as (whenever point evaluation makes sense) $$I_h(u)(x) = \sum_{i=1}^N u(x_i)\phi_i(x) $$

Also, we know that $u_h \in V_h$, so there must exists a set of coefficients $\{U_i\}_i$ (called DoFs) such that $$u_h(x)= \sum_{i=1}^N U_i \phi_i(x)$$

So it seems to me that $u_h$ and $I_h(u)$ are really similar, but the only difference is in the coefficients: if $U_i$ coincides with $u(x_i)$ for every $i$, then the interpolant coincides with the discrete solution $u_h$. In practice, we obtain $\{ U_i\}_i$ by solving linear systems, so they will not be equal to $u(x_i)$

Answer 1

The other answer has everything you already need, but it's also worth pointing out that $u_h$ is computable whereas $I_hu$ is not: The latter requires you to know the exact solution, which in general we of course don't know (and if we did, we didn't need to compute a Galerkin approximation $u_h$).

Answer 2

Let's assume $u$ is the solution to the variational problem and $u_h$ is the Galerkin approximation of the solution on the subspace $V_h \subset V$.

By Cea's lemma you have: \begin{equation} ||u-u_h||_V \leq C\inf\limits_{v_h \in V_h}||u-v_h||_V \end{equation} for some positive constant $C$. Now you define the projection $I_h:V \rightarrow V_h$. As you already mentioned if $I_h$ involves point evaluations you need that $u$ is smooth enough such that $I_h(u)$ is well defined. If we have enough smoothness, we can state the inequality: \begin{equation} \inf\limits_{v_h \in V_h}||u-v_h||_V \leq ||u-I_h(u)||_V \end{equation} because $I(u)\in V_h$ by definition. Therefore overall: \begin{equation} ||u-u_h||_V \leq C||u-I_h(u)||_V \end{equation} So you see that it might be possible from the estimate that the two norms are equal, but this is generally not always the case and depends on how exactly you define $I_h$.