# Thread: Convergence of the sequence of discretized solutions of a variational problem

1. ## Convergence of the sequence of discretized solutions of a variational problem

Consider the Galerkin discretization of an abstract variational problem where the Hilbert space V is separable.
http://en.wikipedia.org/wiki/Galerki...stract_problem

Each of the subspaces Vn is generated by the first n terms of a sequence of elements of the separable Hilbert space V. This sequence is such that each of these subspaces will be dense in V. The problem is proving that there is a subsequence of the bounded sequence of discretized solutions , that converges weakly to the solution of the variational abstract problem , then proving that the same subsequence converges strongly and finally proving that the sequence of discretized solutions converges to the solution of the variational abstract problem .

2. Let's see... Let $\displaystyle (u^{(n)})$ be the Galerkin approximation. Identify elements in each subspace $\displaystyle V_n$ with their natural injections into $\displaystyle V$. Coercivity of $\displaystyle a$ implies that $\displaystyle c||u^{(n)}||^2\leq a(u^{(n)},u^{(n)})=f(u^{(n)})\leq ||f||||u^{(n)}||$, which gives that $\displaystyle (u^{(n)})$ is bounded in $\displaystyle V$ and must therefore possess a subsequence, denoted by $\displaystyle (u^{(n)})$ also, to weakly converge to $\displaystyle u\in V$. Since $\displaystyle a(\cdot,v)$ is linear, we have $\displaystyle a(u^{(n)},v)\rightarrow a(u,v)$ which means $\displaystyle u$ is also a solution of the initial problem. Now, to demonstrate the strong convergence of $\displaystyle u^{(n)}$, notice that coercivity and Galerkin orthogonality imply $\displaystyle c||u^{(n)}-u||^2\leq a(u-u^{(n)},u-u^{(n)})=f(u-u^{(n)})=0(n)$.

3. Thank you for the answer. I havenīt been able to prove as yet that, as you state in your answer, for any linear form and for any weakly convergent to u, we will always have that . Can this be infered from the Lax Milgram lemma?

4. Not really. Just the definition of weak convergence.