# Space-time balancing domain decomposition

## TL;DR

This paper introduces two-level space-time domain decomposition preconditioners for parabolic PDEs, demonstrating weak scalability in both space and time, and achieving efficient parallel solutions on large core counts.

## Contribution

The work extends balancing domain decomposition by constraints to space-time problems, proposing new preconditioners with proven scalability and efficiency for large-scale parallel computations.

## Key findings

- Preconditioners are weakly scalable in time and space.
- Achieves constant iteration counts for larger problems.
- Demonstrates excellent wall clock time scalability on thousands of cores.

## Abstract

In this work, we propose two-level space-time domain decomposition preconditioners for parabolic problems discretized using finite elements. They are motivated as an extension to space-time of balancing domain decomposition by constraints preconditioners. The key ingredients to be defined are the sub-assembled space and operator, the coarse degrees of freedom (DOFs) in which we want to enforce continuity among subdomains at the preconditioner level, and the transfer operator from the sub-assembled to the original finite element space. With regard to the sub-assembled operator, a perturbation of the time derivative is needed to end up with a well-posed preconditioner. The set of coarse DOFs includes the time average (at the space-time subdomain) of classical space constraints plus new constraints between consecutive subdomains in time. Numerical experiments show that the proposed schemes are weakly scalable in time, i.e., we can efficiently exploit increasing computational resources to solve more time steps in the same {total elapsed} time. Further, the scheme is also weakly space-time scalable, since it leads to asymptotically constant iterations when solving larger problems both in space and time. Excellent {wall clock} time weak scalability is achieved for space-time parallel solvers on some thousands of cores.