Finite-Element Preconditioning for Pseudospectral Solutions of Elliptic Problems

A preconditioning technique for pseudospectral solutions of elliptic problems based on quadrangular finite-element algorithms is analyzed, which exhibits excellent convergence properties. The pseudospectral technique is implemented through a collocation grid based on Gauss–Lobatto quadrature nodes associated to the Jacobi orthogonal polynomials. Various types of basis functions are used in the finite-element preconditioner (i.e., low-order Lagrange or cubic Hermite elements). Dirichlet and Neumann problems are investigated in one- and two-space dimensions. Numerical results show that the eigenvalue spectrum of the iteration matrix is inside the unit circle and even, close to zero for a wide range of operators. This property ensures convergence until roundoff error level in a few iterations. The differences between finite-element and finite-difference preconditioning are analyzed. Finally, the application of the algorithm to a problem exhibiting geometric induced singularities is discussed.