International Conference on Spectral and High Order Methods
12th - 16th July 2021 | Vienna, Austria
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plenary 8: Per-Gunnar Martinsson
fast direct solvers and high order discretizations
University of Texas at Austin, United States of America
That the linear systems arising upon the discretization of elliptic PDEs can be solved very efficiently is well-known, and many successful iterative solvers with linear complexity have been constructed (multigrid, Krylov methods, etc). Interestingly, it has also been demonstrated that it is often possible to directly compute an approximate inverse to the coefficient matrix in linear (or close to linear) time. The talk will survey some recent work in the field and will argue that fast direct solvers have several attractive properties, including improved stability and robustness, and dramatic improvements in speed in certain environments.
More specifically, the talk will explore the interplay between high order discretizations and direct solvers. This nexus presents challenges, since the execution time of direct solvers in many cases deteriorates as the discretization order increases. However, it also presents compelling opportunities, in particular in the context of developing high accuracy solvers for elliptic problems with oscillatory solutions.
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