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Space-Time Discretizations Enabling Parallel-in-Time Simulations (Section 3.4 Numerical Analysis)

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Technical Report,01 Sep 2015,31 Aug 2018

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Portland State University Portland United States

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Major Goals Many simulations in science and engineering described by partial differential equations demand the use of powerful computational resources and rely on efficient software libraries that can utilize these resources. With the ever increase of data and need for better accuracy in the simulations for processes that evolve in time, the efficient use of computer resources reaches a point when one can take benefit of increased number of parallel processors if the simulations are performed in the combined space-time domain. The latter is challenging since it increases the memory requirement by an order of magnitude since a simulation formulated in a 3D physical space domain has to be run now in a 4D combined space-time domain. The project aims to alleviate this severe memory constraint by developing new discretization techniques in combined space-time domain that utilize dimension reduction which is accurate enough and lead to discrete problems that can be efficiently solved by existing parallel software libraries that are designed to work independently of the dimension of the problem. The above goals can be achieved in two complementary ways one, by designing efficient adaptive mesh refinement AMR discretization procedures in the combined space-time computational domain, and then applying dimension reduction to further achieve memory savings. All this is possible after efficient scalable mesh generation and construction of classes of space-time 4D finite elements are designed, analyzed and made available in scalable libraries. This was the main corner-stone goal of the project. Accomplishments Key accomplishments i A main accomplishment We designed finite element spaces for the whole de Rham sequence in 4D. The 4D finite element spaces are the first ones ever designed that are made publicly available to the research community they are accessible through the public finite element library MFEM, and also through the library 1.

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  • Numerical Mathematics

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