$C^1$ Quintic Splines on Domains Enclosed by Piecewise Conics and Numerical Solution of Fully Nonlinear Elliptic Equations

TL;DR

This paper develops $C^1$ piecewise quintic finite element spaces on curved domains bounded by conics, and demonstrates their effectiveness in solving the Monge-Ampère equation numerically.

Contribution

It introduces new finite element spaces on curved domains and applies them to solve fully nonlinear elliptic equations with high accuracy.

Findings

Effective finite element spaces for curved domains

Successful numerical solution of Monge-Ampère equation

Demonstrated convergence and accuracy

Abstract

We introduce bivariate $C^1$ piecewise quintic finite element spaces for curved domains enclosed by piecewise conics satisfying homogeneous boundary conditions, construct local bases for them using Bernstein-B\'ezier techniques, and demonstrate the effectiveness of these finite elements for the numerical solution of the Monge-Amp\`ere equation over curved domains by B\"ohmer's method.