I am taking a course about the Finite Element Method, and we are now talking about error estimates (a priori error estimates). My lecturer keeps mentioning 'Interpolation' but I do not think I know what that means and why it is useful.
If the error estimate is to measure the distance between the exact solution and the approximate solution, why do we need to measure the distance between the exact solution and its 'Interpolation'?
Background:
Suppose we have a problem $(P)$: Find $u$ in $V$ such that:
$$a(u,v) = l(v), \ \ \ \ \forall v \in V$$
$V$ is a Hilbert space, $a(\cdot,\cdot) : V \times V \to \mathbb{R}$ a bilinear bounded and coercive form and $l(\cdot) : V \to \mathbb{R}$ a linear bounded form. By the Lax-Miligram lemma, we have a unique solution of $(P)$.
Let $\{\phi_i^h\}_{i=1}^N$ be a basis of a finite dimensional space $V_h \subset V$. We consider the problem $(P_h)$ in $V_h$: Find $u_h$ in $V_h$ such that: $$a(u_h,v_h) = l(v_h), \ \ \ \ \forall v_h \in V_h$$ We call
\begin{align}
I_h : V & \to V_h \\ u & \to I_hu
\end{align}
the interpolation of $u$ in $V_h$.
Best Answer
What you try to do in FEM is basically to find an approximation $\bar{\phi}$ of your true solution $\phi$ to your PDE. The approximate solution can be written as $$ \phi(x) \approx \sum_{i=1}^{n}N_i \phi(x_i) = \bar{\phi}. $$ You know only the values of $\phi$ at given points $x_i$. The functions $N_i$ are often refered to as interpolation functions or basis functions, typically the so called hat-function is used in linear finite elements. I am very sure, that your lecturer refers to your solution $\bar{\phi}$ as the interpolation, since it is an interpolation of the real solution $\phi$. The image below [Przybilla 2022] shows how $\bar{\phi}$ approximates $\phi$ with hat-functions.