# A High Resolution PDE Approach to Quadrilateral Mesh Generation

## TL;DR

This paper introduces a high-order spectral element method for generating high-resolution quadrilateral meshes in complex 2D domains, leveraging spectral convergence and advanced field-guided techniques.

## Contribution

It presents a novel high-order PDE-based approach using spectral elements and open source tools for precise quadrilateral mesh generation without post-processing.

## Key findings

- Spectral convergence achieves high accuracy in guiding fields.
- Corner handling with DG allows flexible discretization.
- Meshes have naturally curved quadrilaterals without additional curvature adjustments.

## Abstract

We describe a high order technique to generate quadrilateral decompositions and meshes for complex two dimensional domains using spectral elements in a field guided procedure. Inspired by cross field methods, we never actually compute crosses. Instead, we compute a high order accurate guiding field using a continuous Galerkin (CG) or discontinuous Galerkin (DG) spectral element method to solve a Laplace equation for each of the field variables using the open source code Nektar++. The spectral method provides spectral convergence and sub-element resolution of the fields. The DG approximation allows meshing of corners that are not multiples of $\pi/2$ in a discretization consistent manner, when needed. The high order field can then be exploited to accurately find irregular nodes, and can be accurately integrated using a high order separatrix integration method to avoid features like limit cycles. The result is a mesh with naturally curved quadrilateral elements that do not need to be curved a posteriori to eliminate invalid elements. The mesh generation procedure is implemented in the open source mesh generation program NekMesh.