$C^1$ Quintic Splines on Domains Enclosed by Piecewise Conics and Numerical Solution of Fully Nonlinear Elliptic Equations
It provides a novel finite element framework for solving fully nonlinear elliptic equations on curved domains, which is a challenging problem in computational PDEs.
The paper introduces bivariate C1 piecewise quintic finite elements for curved domains enclosed by piecewise conics and demonstrates their effectiveness for solving the Monge-Ampère equation via Böhmer's method, with numerical results showing convergence.
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-Bézier techniques, and demonstrate the effectiveness of these finite elements for the numerical solution of the Monge-Ampère equation over curved domains by Böhmer's method.