Suppose we are looking at the error estimates for a mixed finite element method which approximates $L_2$ for scalar functions (say, $u$) and $H_{div}$ for vector functions (say, $\mathbf{w}$). Consider 3D case.Usually, error estimates in mesh norms are written (e.g., in Brezzi-Fortin or ant other classical works) for $u$, or its gradient $\mathbf{w} = - \nabla u$ projected to the mesh.My question is how the error (convergence factor) depends on the choice of the projector $\pi_h$ for scalars and $\Pi_h$ for vectors? There are two obvious choices: a point-wise projector and an $L_2$-averaging projector. Here I assume that a point-wise projector is acting on functions which are at least continuous so there is no problem with evaluating them at a single point. An example of a point-wise projector (for lowest order elements):$$(\pi_h u)(i,j,k) = u (x_{1/2}, y_{j+1/2}, z_{k+1/2})$$An example of a scalar $L_2$-averaging projector (for lowest order elements):$$(\pi_h u)(i,j,k) = \frac{1}{mes(K_{i,j,k})} \int_{K_{i,j,k}} u(x,y,z) d\Omega$$Being more precise, in my work I need to estimate $\| \mathbf{w}^0_h - \Pi_h \mathbf{w}^0_{ex}\|_{2,h}$, where $\mathbf{w}^0_h$ is the initial mesh heat flux and $\mathbf{w}^0_{ex}$ is the exact initial flux, i.e. $\mathbf{w}^0_{ex} = - \nabla u^0$.The mesh heat flux is computed by solving $ \mathbf{A} \mathbf{w}^0_h = \mathbf{B} \pi_h u^0$, where again $\pi_h$ is either a point-wise or $L_2$-averaging projector. Here $A$ is the flux mass matrix and $\mathbf{B}$ - discrete (minus) gradient operator, i.e. $\mathbf{B} \sim -\nabla$ for the considered elements.Any comments or references are welcome.
A correct mesh projector in error estimates for Finite Element Method
finite element;error estimation;projection