I am investigating the following standard elliptic PDE with mixed Dirichlet-Neumann boundary condition:
$-\Delta u=f$ on $\Omega$; $u=0$ on $\Gamma_D$; $\left<n,\nabla u=g\right>$ on $\Gamma_N$
Computationally, we could find the weak solution using finite-dimensional test function space $V$ by requiring:
$\int_\Omega \left<\nabla u,\nabla v\right> d\Omega = \int_\Omega fvd\Omega+ \int_{\Gamma_N}gvds\quad\forall v\in V$
I am wondering whether the weak solution $u$ is continuous on the function space. In other words, if there is another function space $V'$ and I have a functional $d$ that is the metric of function space: $d(V,V')$, would the solution $u$ be continuous on $d$? As a practical example in engineering, consider FEM mesh discreting the PDE on a mesh, if I perturb a vertex of the mesh, would $u$ change continuously?
