# Scale FEM-Basis by mesh-width

Let me explain my setting: I want to solve a second order, linear, elliptic PDE on some domain $\Omega\subset\mathbb{R}^2$ open and bounded. I discretize my domain using quadrilateral elements with equidistant mesh-size $h$ in both directions, that are aligned to the axes. My FE-Space is $$V_h = \{u:\Omega\to\mathbb{R}|\,u_{|Q} = p(x)q(y),\,\text{ for some } p,q\in\mathcal{P}_1,\,\forall Q\in\mathcal{Q}_h\}$$

In the method I will be using, I am 'correcting' the basis functions chosen for $V_h$ in order to get piecewise constants. When estimating the error done in this projection, I directly get (no estimates done, so this cannot result in other terms) something of the form $$\|\nabla\partial\varphi_i\|_{L^2(Q)}$$ $\varphi_i$ being one of the basis function for $V_h$ and $Q\in\mathcal{Q}_h$ some element.

The problem is now the following: The standard, tensor product basis for a quadrilateral $Q\in\mathcal{Q}_h$ has for one of the local nodes the form $$\varphi_i(x,y)= \frac{x}{h}\frac{y}{h}$$ and the second derivative of this function scales like $\tfrac{1}{h^2}$. This basically kills my approach where I need a good estimate of the above norm, also if $h\to 0$.

My Idea: Why not scale the basis functions? If I drop the assumption that $\phi_i(v_i) = 1$ where $v_i$ is the local vertex my basis function corresponds to, precisely $$\tilde\varphi_i(x,y)=xy,$$ obviously this solves the current problem, since the second derivative of this function scales as 1.

My question: Since the values of the above basis function $\tilde\varphi_i$ are now not in the range $[0,1]$ anymore, but in $[0,h^2]$, will this cause problems when solving the linear system later?

I think the matrix will eventually get singular due to machine precision if $h$ is small enough.

What do you think? Maybe you have even tried this, or know some reference where this is addressed..

Thank you very much for all your responses or hints!

Denis

Could you explain a little bit what are you trying to estimate? FE basis functions in the "unscaled" coordinates usually are chosen to have a local support and, say, equal to 1 at a vertex and already vanish at the neighboring vertices of the mesh in your case. So, essentially their gradients have a factor $\frac{1}{h}$.
If you want to change the basis functions and scale them by a factor of $h$ than you will essentially multiply your system, both left and righthand sides by this factor.
• Okay let me conclude again shortly. I need good estimates for the second derivatives of my basis functions. This is really the main problem I need to solve in order for my method to work. Especially I don't want a factor $1/h^2$ when estimating the norms of the second derivatives. Till now I was using the standard nodal basis, so especially the functions have value 1 at one vertex. If I now 'flatten' them to have value $h^2$ at the same vertex then the second derivatives are good. As you said: The LHS is then multiplied by $(h^2)^2$ and the RHS by $h^2$. Maybe smb. knows about the tradeoff?