The aggregated unfitted finite element method on parallel tree-based adaptive meshes

TL;DR

This paper introduces a scalable parallel adaptive finite element scheme combining the aggregated finite element method with mesh refinement, effectively handling cut cell problems and demonstrating robustness and efficiency in large-scale benchmarks.

Contribution

It presents a novel distributed-memory implementation of an unfitted finite element scheme with a two-step construction algorithm, enabling efficient parallel adaptive mesh refinement.

Findings

Demonstrates optimal mesh adaptation and robustness to cut location.

Shows parallel efficiency on classical benchmarks.

Facilitates large-scale, error-driven mesh adaptation.

Abstract

In this work, we present an adaptive unfitted finite element scheme that combines the aggregated finite element method with parallel adaptive mesh refinement. We introduce a novel scalable distributed-memory implementation of the resulting scheme on locally-adapted Cartesian forest-of-trees meshes. We propose a two-step algorithm to construct the finite element space at hand by means of a discrete extension operator that carefully mixes aggregation constraints of problematic degrees of freedom, which get rid of the small cut cell problem, and standard hanging degree of freedom constraints, which ensure trace continuity on non-conforming meshes. Following this approach, we derive a finite element space that can be expressed as the original one plus well-defined linear constraints. Moreover, it requires minimum parallelization effort, using standard functionality available in existing large-scale finite element codes. Numerical experiments demonstrate its optimal mesh adaptation capability, robustness to cut location and parallel efficiency, on classical Poisson $hp$-adaptivity benchmarks. Our work opens the path to functional and geometrical error-driven dynamic mesh adaptation with the aggregated finite element method in large-scale realistic scenarios. Likewise, it can offer guidance for bridging other scalable unfitted methods and parallel adaptive mesh refinement.