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

3. Thank you for the answer. I haven´t 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.