Ludmil Zikatanov
Penn State
Abstract: In this talk we shall present a number of new results on the subspace correction method. This method provides an abstract framework for a class of divide and conquer techniques (such as the multigrid method) for the solution of linear and nonlinear partial differential equations. We shall first report on an ongoing project on a fast Poisson solver for elliptic equations discretized on general unstructured grids, some new algebraic multigrid techniques developed for this solver and their applications to a couple of practical problems including lattice materials and powder compaction simulations. In addition, we shall also report on some preliminary results from our ongoing investigation on efficient numerical methods for liquid-crystal modelling. The convergence rate estimate for various subspace correction methods mentioned above amounts to, at least for the linear problem, the norm estimate of the the product of non-expensive operators in Hilbert space. As the second part of the talk, we shall present a new theory for the estimate of this product of operators, which is given by an elegant identity. Most known convergence estimates in the literature can be obtained as a simple consequence of this sharp estimate. In addition, we expect that this new theory can be used to design more efficient algorithms, such as algebraic multigrid methods. Finally, we shall point out the intimate relationship between the subspace correction method with another very popular class of methods, namely the method of alternating projections.