Could the functions in larger space than $L^2$ be approximated by finite element basis functions?

Question

Let $u \in V:=\{v \in L^{1+\alpha}(\Omega): \nabla v \in \mathbf{L}^2(\Omega)\}$, where $0<\alpha<1$.

Clearly, $H^1(\Omega):=\{v \in L^2(\Omega): \nabla v \in \mathbf{L}^2(\Omega)\} \subset V$. As we all know, $u \in H^1(\Omega)$ can be approximated by the linear finite element basis functions.

Could the function $u \in V$ also be approximated by the finite element basis functions?

Answer 1

Yes, this is just a density argument: The finite element functions are dense in $H^1(\Omega)$ (w.r.t. the $H^1$-norm). And $H^1(\Omega)$ is dense in $V$ w.r.t. the $V$-norm. Hence, the finite element functions are dense in $V$.