Consider $P$ a linear partial differential operator in $\Bbb R ^n$. Consider some boundary condition that guarantees a unique solution (if any) of $Pu = 0$.

Let $(u_i)_{i \in I}$ be a net of distributions such that $\lim \limits _{i \in I} P u_i = 0$. Is it possible, then, to conclude that there exist a distribution $u$ such that $\lim \limits _{i \in I} u_i = u$ and $Pu = 0$?

Playing with words, if I have a net of approximate solutions of a PDE, are its elements approximations of a solution of that PDE? That is, can I use them to construct a solution?

(I have chosen to work with distributions since they seem the most convenient, feel free to modify this context suiting your needs. If stronger hypotheses are needed, please say so in comments below and I shall edit my question to reflect this.)