Explicit Bounds of Eigenvalues for Stiffness Matrices by Quadratic Hierarchical Basis Method (Journal of Computational Mathematics, 2003)
The bounds for the eigenvalues of the stiffness matrices in the finite element discretization corresponding to $Lu := - u'' $ with zero boundary conditions by quadratic hierarchical basis are shown explicitly. The condition number of the resulting system behaves like $O(\frac{1}{h})$ where $h$ is the mesh size. We also analyze a main diagonal preconditioner of the stiffness matrix which reduces the condition number of the preconditioned system to $O(1)$.
Keywords: hierarchical basis; multilevel.