# Isogeometric Analysis for Surface PDEs with Extended Loop Subdivision

## TL;DR

This paper introduces an isogeometric analysis method for surface PDEs using extended Loop subdivision, achieving high accuracy and efficiency on complex surfaces, and outperforming traditional finite element methods.

## Contribution

The paper develops a novel isogeometric analysis approach based on extended Loop subdivision, providing exact geometry representation and improved accuracy for surface PDEs.

## Key findings

- Second-order accuracy in L2-norm demonstrated
- Outperforms standard linear finite element methods
- Effective on both open and closed surfaces

## Abstract

We investigate the isogeometric analysis for surface PDEs based on the extended Loop subdivision approach. The basis functions consisting of quartic box-splines corresponding to each subdivided control mesh are utilized to represent the geometry exactly, and construct the solution space for dependent variables as well, which is consistent with the concept of isogeometric analysis. The subdivision process is equivalent to the $h$-refinement of NURBS-based isogeometric analysis. The performance of the proposed method is evaluated by solving various surface PDEs, such as surface Laplace-Beltrami harmonic/biharmonic/triharmonic equations, which are defined on different limit surfaces of the extended Loop subdivision for different initial control meshes. Numerical experiments demonstrate that the proposed method has desirable performance in terms of the accuracy, convergence and computational cost for solving the above surface PDEs defined on both open and closed surfaces. The proposed approach is proved to be second-order accuracy in the sense of $L^2$-norm by theoretical and/or numerical results, which is also outperformed over the standard linear finite element by several numerical comparisons.