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

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

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: May 1, 2018
• 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.
• Contract Date: Nov 30, 2017