Higher-order Adaptive Finite Difference Methods for Fully Nonlinear Elliptic Equations
This work provides a new numerical approach for solving fully nonlinear elliptic PDEs, which are challenging to solve in complex geometries, but the results are incremental as they extend existing finite difference techniques.
The paper introduces generalized finite difference methods for fully nonlinear elliptic PDEs that use adaptive meshes and handle complex geometries while ensuring consistency, monotonicity, and convergence. The methods achieve higher-order convergence and demonstrate efficiency and accuracy through computational examples.
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.