# Coupling parallel adaptive mesh refinement with a nonoverlapping domain decomposition solver

## TL;DR

This paper investigates how adaptive mesh refinement impacts the performance of a parallel domain decomposition solver, specifically BDDC, demonstrating scalability and acceptable convergence even with large, complex problems.

## Contribution

It presents a prototype implementation combining adaptive mesh refinement with a parallel BDDC solver, analyzing effects of mesh adaptivity and disconnected subdomains on convergence and scalability.

## Key findings

- Refined meshes and disconnected subdomains slightly reduce BDDC convergence.
- The three-level BDDC solver maintains good scalability up to thousands of cores.
- A large problem with over 10^9 unknowns was successfully solved on 2048 cores.

## Abstract

We study the effect of adaptive mesh refinement on a parallel domain decomposition solver of a linear system of algebraic equations. These concepts need to be combined within a parallel adaptive finite element software. A prototype implementation is presented for this purpose. It uses adaptive mesh refinement with one level of hanging nodes. Two and three-level versions of the Balancing Domain Decomposition based on Constraints (BDDC) method are used to solve the arising system of algebraic equations. The basic concepts are recalled and components necessary for the combination are studied in detail. Of particular interest is the effect of disconnected subdomains, a typical output of the employed mesh partitioning based on space-filling curves, on the convergence and solution time of the BDDC method. It is demonstrated using a large set of experiments that while both refined meshes and disconnected subdomains have a negative effect on the convergence of BDDC, the number of iterations remains acceptable. In addition, scalability of the three-level BDDC solver remains good on up to a few thousands of processor cores. The largest presented problem using adaptive mesh refinement has over 10^9 unknowns and is solved on 2048 cores.