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

TL;DR

This paper introduces a massively parallel adaptive finite element framework for solving non-linear parabolic equations by solving large space-time blocks simultaneously, enabling efficient scalability and adaptivity in both space and time.

Contribution

It presents a novel parallel-in-space-time finite element method with adaptive refinement and error estimation, scalable to 150,000 processors for complex time-dependent problems.

Findings

Achieved good scaling up to 150,000 processors on Blue Waters.

Demonstrated effective space-time adaptivity for diffusion equations.

Reduced computational overhead through optimized memory and communication strategies.

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. This methodology enables, in principle, simultaneous adaptivity in both space and time (within the block) domains. We explore this basic concept in the context of a variety of time-steppers including $\Theta$-schemes and Backward Differentiate 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 affect spatial adaptivity. Finally, we show good scaling behavior up to 150,000 processors on the Blue Waters machine. This is achieved by careful usage of memory via block storage and non-zero formats along with lumped communication for matrix assembly. This methodology enables scaling on next generation multi-core machines by simultaneously solving for large number of time-steps, and reduces computational overhead by 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 design problems, since blocks of space-time are simultaneously solved and stored in memory.