Least-squares spectral element preconditioners for fourth order elliptic problems

In this paper, we propose preconditioners for the system of linear equations that arise from a discretization of fourth order elliptic problems in two and three dimensions (d=2,3) using spectral element methods. These preconditioners are constructed using separation of variables and can be diagonalized and hence easy to invert. For second order elliptic problems this technique has proven to be successful and performs better than other preconditioners in the framework of least-squares methods. We show that these preconditioners are spectrally equivalent to the quadratic forms by which we approximate them. Numerical results for the condition number reflects the effectiveness of the preconditioners.