Explicit Bounds of Eigenvalues for Stiffness Matrices by Quadratic Hierarchical Basis Method

Abstract

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 $O(1)$.