In Galerkin methods, we seldom can measure the accuracy of an approximation by tracking the value of the residual. For example, take the wave equation:
$ u_{tt} = u_{xx}$, $(x,t) \in (0,L) \times (0,T)$ with approximation $u(x,t) \approx u_N(x,t) = \sum_{i=1}^N\phi_i(x)q_i(t)$
The residual for this approximation is defined as follows:
$R(x,t,N) = \sum_{i=1}^N\phi_i''(x)q_i(t) - \phi_i(x)q_i''(t)$
for which we expect $R(x,t) = 0$ everywhere for $(x,t) \in (0,L) \times (0,T)$. The error metric in this case will be
$ \epsilon(N) = \int_0^T \int_0^L R(x,t,N)^2 dx dt $
However, in the Finite Element method, developing the above integral, the term $\phi_i''(x)$ persists. This term is not well defined in the FEM as we have $\phi_i(x)$ continuous and differentiable once. Is there a way to estimate the residual without necessitating $\phi_i''(x)$?
My Work
I understand that in the FEM, we are mostly concerned with the error metric $||u-u_N||$ rather than the error in residual. The other solution will be to compute $u_\infty$ (ie, $N$ very large, assuming convergence occurs) and use the error metric $||u_\infty-u_N||$ instead. However, in my case, as I increase $N$, I do not necessarily keep my initial conditions constant. My understanding is the $u_\infty$ is computed for given initial and boundary conditions while, for myself, only the boundary conditions are given and the initial conditions are found such that the solution is periodic $u_N(x,t) = u(x,t+T)$. So the initial conditions are not given per-se. I would still prefer if there existed a way to compute the residual.