I'm a little stuck trying to estimate a condition number in FEM context. I would like to prove the following:
Consider the stiffness matrix $K$ for piecewise linear functions on a quasi-uniform mesh in one dimension of the Laplace equation. Prove that the condition number of $K$ is bounded by $\mathcal{O}(h^{−2})$. (Hint: use inverse estimates.)
The matrix is given by $A = \begin{pmatrix} 2 & -1 & 0 & 0 \\ -1 & 2 & -1 & 0 \\ 0 & -1 & 2 & -1 & 0 & 0 \\ \ldots & \ddots & & & \ddots \\ \ldots & & & 0 & -1 & 2 & -1 \\ \ldots & & & & 0 & -1 & 2 \end{pmatrix}$
I do not know how to start to be honest, because i do not know how to estimate the smallest and largest eigenvalues. In dimension one i think that i can get explicitly the eigenvalues and eigenvectors, but i would like to estimate it, because is more general.
Thanks you very much.