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Parallel-in-space-time, adaptive finite element framework for non-linear parabolic equations

Dyja, Robert; Ganapathysubramanian, Baskar; van der Zee, Kristoffer George

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Authors

Robert Dyja

Baskar Ganapathysubramanian

KRISTOFFER VAN DER ZEE KG.VANDERZEE@NOTTINGHAM.AC.UK
Professor of Numerical Analysis &computational Applied Mathematics



Abstract

We present an adaptive methodology for the solution of (linear and) non-linear time dependent problems that is especially tailored for massively parallel computations. The basic concept is to solve for large blocks of space-time unknowns instead of marching sequentially in time. The methodology is a combination of a computationally efficient implementation of a parallel-in-space time finite element solver coupled with a posteriori space-time error estimates and a parallel mesh generator. While we focus on spatial adaptivity in this work, the methodology enables simultaneous adaptivity in both space and time domains. We explore this basic concept in the context of a variety of time-steppers including Θ-schemes and Backward Difference Formulas. We specifically illustrate this framework with applications involving time dependent linear, quasi-linear and semi-linear diffusion equations. We focus on investigating how the coupled space-time refinement indicators for this class of problems aspect spatial adaptivity. Finally, we show good scaling behavior up to 150,000 processors on the NCSA Blue Waters machine. This conceptually simple methodology enables scaling on next generation multi-core machines by simultaneously solving for large number of time-steps, and reduces computational overhead by locally refining spatial blocks that can track localized features. This methodology also opens up the possibility of efficiently incorporating adjoint equations for error estimators and inverse

Citation

Dyja, R., Ganapathysubramanian, B., & van der Zee, K. G. (in press). Parallel-in-space-time, adaptive finite element framework for non-linear parabolic equations. SIAM Journal on Scientific Computing, 40(3), Article C283-C304. https://doi.org/10.1137/16M108985X

Journal Article Type Article
Acceptance Date Nov 27, 2017
Online Publication Date May 1, 2018
Deposit Date Nov 30, 2017
Publicly Available Date Mar 28, 2024
Journal SIAM Journal on Scientific Computing
Print ISSN 1064-8275
Electronic ISSN 1095-7197
Publisher Society for Industrial and Applied Mathematics
Peer Reviewed Peer Reviewed
Volume 40
Issue 3
Article Number C283-C304
DOI https://doi.org/10.1137/16M108985X
Keywords Parabolic problems; Parallel-in-time; Finite element method; Adaptive mesh refinement
Public URL https://nottingham-repository.worktribe.com/output/930662
Publisher URL https://epubs.siam.org/doi/abs/10.1137/16M108985X
Additional Information ©2018 Society for Industrial and Applied Mathematics. Unauthorized reproduction of this article is prohibited.

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