Finite element preconditioning for legendre spectral collocation approximations to elliptic equations and systems

Spectral collocation approximations based on Legendre-Gauss-Lobatto (LGL) points for Helmholtz equations as well as for the linear elasticity system in rectangular domains are studied. The collocation method is set up in a variational fashion according to which Neumann boundary conditions are fulfilled in a weak sense through a penalty method on the boundary residue. This approach yields better accuracy than the classical collocation method, which enforces the Neumann conditions in a strong form. The arising system is then preconditioned by a matrix built up starting from piecewise bilinear finite element discretization of the same boundary value problem at the LGL grid. The eigenvalues of the preconditioned matrix Asp have positive real part, and the condition number of Asp is asymptotically independent of the size of Asp, ranging between 1.39 for the fully Dirichlet problem and 1.58 for the Neumann one, with intermediate values attained for the case of mixed boundary value problems. Iterative procedures of gradient type for the preconditioned system (the minimal residual Richardson method and the recently introduced Bi-CGSTAB method are applied here) are shown to be very effective for any kind of boundary conditions. The application of the preconditioners in the framework of domain decomposition is also discussed.