# Distributed-memory parallelization of the aggregated unfitted finite element method

## TL;DR

This paper presents a scalable distributed-memory implementation of the aggregated unfitted finite element method (AgFEM), enabling efficient large-scale solutions with standard algebraic multigrid preconditioners on complex 3D problems.

## Contribution

It introduces a parallel distributed-memory implementation of AgFEM that handles large problem sizes using standard solvers, contrasting with previous customized approaches.

## Key findings

- Successfully solved 300 million degrees of freedom problems on 16,000 cores.
- Achieved efficient weak scaling on complex 3D domains.
- Demonstrated compatibility with standard algebraic multigrid preconditioners.

## Abstract

The aggregated unfitted finite element method (AgFEM) is a methodology recently introduced in order to address conditioning and stability problems associated with embedded, unfitted, or extended finite element methods. The method is based on removal of basis functions associated with badly cut cells by introducing carefully designed constraints, which results in well-posed systems of linear algebraic equations, while preserving the optimal approximation order of the underlying finite element spaces. The specific goal of this work is to present the implementation and performance of the method on distributed-memory platforms aiming at the efficient solution of large-scale problems. In particular, we show that, by considering AgFEM, the resulting systems of linear algebraic equations can be effectively solved using standard algebraic multigrid preconditioners. This is in contrast with previous works that consider highly customized preconditioners in order to allow one the usage of iterative solvers in combination with unfitted techniques. Another novelty with respect to the methods available in the literature is the problem sizes that can be handled with the proposed approach. While most of previous references discussing linear solvers for unfitted methods are based on serial non-scalable algorithms, we propose a parallel distributed-memory method able to efficiently solve problems at large scales. This is demonstrated by means of a weak scaling test defined on complex 3D domains up to 300M degrees of freedom and one billion cells on 16K CPU cores in the Marenostrum-IV platform. The parallel implementation of the AgFEM method is available in the large-scale finite element package FEMPAR.