I am learning about the finite element method in an abstract Banach and Hilbert space setting, especially for an application to differential forms, and I am a bit lost on the big picture about the purpose and the necessity of quasi-interpolants.
My understanding of the abstract Galerkin setting, so far, is that we start with two infinite dimensional normed spaces $U$ and $V$ that need to be complete (to allow formulating what convergence means). Then we regard the continuous problem: given a possibly unbounded operator $L:U\to V^*$ and a piece of data $f\in V^*$ which is the continuous dual space of $V$ it displays as
$$\text{seek }u\in U\text{ such that }L\,u=f$$
This is approximated with a Galerkin method by picking finite dimensional normed subspaces $U_h\subset U$ and $V_h\subset V$. To make this suitable for real world methods, one allows $L_h$ to differ from the restriction $L|_{U_h}$ and $f_h$ to differ from the restriction $f|_{V_h^*}$. Therefore the discrete problem is
$$\text{seek }u_h\in U_h\text{ such that }L_h\,u_h=f_h$$
Now, the error which we are interested in to control is $\|u-u_h\|_{U}$ where "control" means "bounding towards zero as $h$ tends to zero".
When $U$ is an inner product space, then we can define a best approximation $u_h^{\text{best}}$ of $u$ in $U_h$ as the orthogonal projection of $u$ onto $U_h$ using this inner product. This gives the best approximation error
$$\|u-u_h^{\text{best}}\|_U=\inf_{v_h\,\in\,U_h}\|u-v_h\|_U$$
By the triangle inequality we can bound $\|u-u_h\|_{U}$ with the the error from $u$ to $u_h^{\text{best}}$ plus the error from $u_h^{\text{best}}$ to $u_h$.
$$\|u-u_h\|_{U}\le \|u-u_h^{\text{best}}\|_U+\|u_h^{\text{best}}-u_h\|_{U_h}$$
When $U$ is not an inner product space, then we can still define the best approximation error with the infimum, but (how) do we still obtain a best approximation (the element)? Something like
$$u_h^{\text{best}}=\operatorname*{arginf}_{v_h\,\in\,U_h}\|u-v_h\|_U$$
If $U_h$ takes the infimum (does it?), isn't it a minimium then? I could think that the ill-posedness of such a definition could be one reason to introduce an interpolator that actually obtains a value in $U_h$.
Update: Yes, I think so. Since $U_h$ is finite dimensional and
But then, commonly $U$ is chosen to be the space of square integrable funtions $L^2$ over some domain and this comes with an inner product. Still people introduce quasi-interpolants $\mathcal I_h$ on complete inner product spaces. So maybe for concrete finite element theories (a combination of finite element spaces and a boundary value problem) it turns out to be helpful to have a detour over $\mathcal I_h\,u$ in the triangle inequality then?
$$\|u-u_h\|_{U}\le \|u-\mathcal I_h\,u\|_U+\|\mathcal I_h\,u-u_h\|_{U_h}$$
The next aspect is that the degrees of freedom on simplices are defined as integrals over sub-simplex traces. They just need to be unisolvent (to form a basis for the dual space) but most authors seem to use them like the resulting functionals are also a dual basis to the finite element basis functions. The necessity of the sub-simplex restriction is not quite clear to me. Hiptmair (1999) - Canonical construction of finite elements includes this as a locality requirement in the very definition of degrees of freedom
For the local ansatz spaces on the reference element we adopt the notation $\mathcal X^l_k(\hat T)$, where $l$ stands for the order of the differential forms and $k\in\mathbb N_0$ is related to the degree of the polynomials.
[...]
Lemma 7. Let $\hat f$ denote an $(n - 1)$-dimensional "face" of the reference simplex $\hat T$. Then the trace of $\omega\in\mathcal X^l_k(\hat T)$ on $\hat f$ belongs to the space $\omega\in\mathcal X^l_k(\hat f)$.
[...]
- Unisolvence: ... is a basis of the dual space $\mathcal X^l_k(\hat T)'$
- Invariance: ... remain invariant under canonical transformations of differential forms accompanying a transformation of the reference simplex.
- Locality: For any face $\hat f$ of the simplex $\hat T$ and $\omega\in \mathcal X^l_k(\hat T)$ the trace $\omega_{|\hat f}$ is uniquely determined by certain degrees of freedom that depend only on $\omega_{|\hat f}$.
There is a canonical interpolant $\mathcal I_h$ being defined by directly using the degrees of freedom to map $u$ to coefficients and then multiply these with the corresponding basis elements to obtain a function $\mathcal I_h\,u\in U_h$. Authors now complain that because some of the degrees of freedom are restricted to subsimplices, it is not possible to bound $\|u-\mathcal I_h\,u\|_{U}$ when $u$ has low regularity.
Couldn't we just use the $U$-inner product here to obtain a dual basis of $U_h$ in the case that $U$ is an inner product space such as $L^2$? I mean not for the degrees of freedom but for the interpolant. This would be equivalent to a best approximation then, I guess. There is the Clément interpolant construction which seems to do something similar by multiplying a hat function of some node with the best $u$-matching polynomial up to order $r$ over all simplices that share this node. What are the benefits of this construction over the orthogonal projection / best approximation?
To sum up, my questions are
- Do we still have a best approximation (the element) when $U$ is not an inner product space?
- What is the high level reason for the introduction of interpolation operators besides that everyone does it like that? (is it necessary or just helpful to obtain error bounds?)
- Why don't we use the $L^2$ inner product directly to define quasi-interpolants?
- What is the difference of an interpolant and a quasi-interpolant?
Update: after reading a bit more, I found some hints
- "a best approximation of X by a closed subspace $C\subseteq X$ exists and is unique if $X$ is a Hilbert space, a uniformly convex Banach space, or more generally, a reflexive space." (*)
- It seems to be a strategy to obtain global error bounds from local error bounds: "the Lagrange interpolation operator $\mathcal I^1_h$ possesses numerous local interpolation properties. In other words, one observes that the interpolation error is controlled elementwise before it is controlled on a global level. This observation serves as a motivation for the introduction of local interpolation operators." (*)
- People use the $L^2$ projection in that manner so it might be that it does not possess some important properties. Commutation with the exterior derivative and with local transformations (pullbacks) and also preservation of boundary conditions might be desirable here. Still, this is just a guess.
- It seems that the nomer interpolation is used in a generalized sense and does not imply the exact reproduction of function values at specific points anymore so some authors use the prefix quasi to account for this. "The term “interpolation” is used here in a broad sense, since the degrees of freedom (dofs) are not necessarily point evaluations. For the interpolation operator to be useful, one needs to extend the domain of the linear forms" (*)
