# Non-Conforming Mesh Refinement for High-Order Finite Elements

## TL;DR

This paper introduces a versatile algorithm for non-conforming adaptive mesh refinement in high-order finite element methods, supporting complex geometries, parallel computing, and dynamic adaptation in hydrodynamics simulations.

## Contribution

It presents a novel, general algorithm and data structure for non-conforming mesh refinement applicable to various element types and orders, with efficient parallel implementation.

## Key findings

- Verified correctness through numerical experiments.
- Achieved efficient parallel scaling on billions of elements.
- Successfully integrated into a high-order hydrodynamics solver.

## Abstract

We propose a general algorithm for non-conforming adaptive mesh refinement (AMR) of unstructured meshes in high-order finite element codes. Our focus is on h-refinement with a fixed polynomial order. The algorithm handles triangular, quadrilateral, hexahedral and prismatic meshes of arbitrarily high order curvature, for any order finite element space in the de Rham sequence. We present a flexible data structure for meshes with hanging nodes and a general procedure to construct the conforming interpolation operator, both in serial and in parallel. The algorithm and data structure allow anisotropic refinement of tensor product elements in 2D and 3D, and support unlimited refinement ratios of adjacent elements. We report numerical experiments verifying the correctness of the algorithms, and perform a parallel scaling study to show that we can adapt meshes containing billions of elements and run efficiently on 393,000 parallel tasks. Finally, we illustrate the integration of dynamic AMR into a high-order Lagrangian hydrodynamics solver.