# Higher-order Adaptive Finite Difference Methods for Fully Nonlinear Elliptic Equations

## TL;DR

This paper presents innovative higher-order adaptive finite difference methods for solving fully nonlinear elliptic PDEs, emphasizing flexibility, efficiency, and convergence on complex geometries.

## Contribution

It introduces generalized finite difference schemes on adaptive, piecewise Cartesian meshes with boundary points, enabling higher-order accuracy and efficient computation.

## Key findings

- Methods demonstrate high accuracy in computational examples.
- Adaptive meshes improve solution flexibility for complex geometries.
- Algorithms efficiently compute non-traditional finite difference stencils.

## Abstract

We introduce generalised finite difference methods for solving fully nonlinear elliptic partial differential equations. Methods are based on piecewise Cartesian meshes augmented by additional points along the boundary. This allows for adaptive meshes and complicated geometries, while still ensuring consistency, monotonicity, and convergence. We describe an algorithm for efficiently computing the non-traditional finite difference stencils. We also present a strategy for computing formally higher-order convergent methods. Computational examples demonstrate the efficiency, accuracy, and flexibility of the methods.