Let $u_N$ be the Galerkin-Approximation of u and $\dim(V_N)=N$. Suppose that the error $\Vert u_N - u \Vert$ can be developed in a power series $\sum_{i=0}^{\infty}C_i(N^{\alpha})^i$ with $C_i,\alpha\in\mathbb{R}$. I want to derive an approximate calculation of the convergence rate of $\alpha$ under the assumption that $u_N \rightarrow u.$
Unfortunately i don't know how to start. Can someone help please ?
Thanks in advance.