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AM Seminar: Efficient solvers and high-order finite element methods for next-generation supercomputers

Speaker Name: 
Will Pazner
Speaker Title: 
Sidney Fernbach Postdoctoral Fellow
Speaker Organization: 
Lawrence Livermore National Laboratory
Start Time: 
Monday, February 24, 2020 - 4:00pm
End Time: 
Monday, February 24, 2020 - 5:00pm
BE 372
Abhishek Halder


Modern computer architectures favor the use of high-order discretizations for many problems. These methods feature high accuracy per degree of freedom and high arithmetic intensity, both of which are attractive properties for GPU-based supercomputers, such as LLNL's Sierra. However, the cost of assembling or even storing the matrix associated with these discretizations can be prohibitively expensive, thus necessitating the use of matrix-free methods and solvers. Sum-factorization and related techniques allow for efficient operator evaluation, however solving the resulting large linear systems remains challenging. Furthermore, common preconditioners such as those arising from domain decomposition methods typically require dense blockwise factorizations, which are not possible without an explicit representation of the matrix.

In this talk, I will describe recent work on the development of matrix-free preconditioners and solvers for very high-order finite element discretizations. One approach is based on a sparse, low-order refined discretization which is spectrally equivalent to the high-order discretization.  The resulting preconditioners are robust with respect to the mesh spacing and polynomial degree. These preconditioners can be extended to discontinuous Galerkin methods, for which they are also robust with respect to the penalty parameter. Applications of these methods to radiative transfer and incompressible fluid flow will be discussed. Another approach makes use of the fast diagonalization method with tensor-product approximations, where the preconditioner can be constructed on the GPU.


Will Pazner is the Sidney Fernbach Postdoctoral Fellow at Lawrence Livermore National Laboratory's Center for Applied Scientific Computing. He received his Ph.D. from Brown University in 2018, where he was co-supervised by Prof. Per-Olof Persson and Prof. Chi-Wang Shu. Will was a postdoctoral scholar at UC Berkeley, and is an affiliate of the Mathematics Group at Lawrence Berkeley National Laboratory.

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