Matrix decomposition algorithms for arbitrary order C⁰ tensor product finite element systems

Matrix decomposition algorithms (MDAs) are fast direct methods for the solution of systems of linear algebraic equations which arise in the approximation of Poisson's equation on the unit square using various techniques such as finite difference, spline collocation and spectral methods. The attraction of MDAs is that they employ fast Fourier transforms and require O(N² log N) operations on an N × N uniform partition of the unit square. In this paper, MDAs are formulated for the solution of the finite element Galerkin equations arising when spaces of C⁰ piecewise polynomials of degree k ≥ 3 are employed. Results of numerical experiments exhibit the expected optimal global convergence rates and superconvergence phenomena.