Stability of the high-order exponential finite difference method for two-dimensional nonlinear elliptic problems via discrete maximum principle

This work analyzes the consistency, stability, and convergence of the high-order exponential finite difference method (EFDM) applied to two-dimensional nonlinear elliptic problems under Dirichlet boundary conditions on a domain Ω =a, ba, b. The approach employs a nine-point stencil on the discretization grid to approximate the Laplacian operator, increasing the order of accuracy relative to conventional five-point schemes. Through a Taylor series expansion truncated up to the sixth order, it is shown that the local truncation error is of fourth order, ensuring the consistency of the discretization. Additionally, the discrete maximum principle is established for the proposed problem, and an a priori estimate for nine-point operators is derived, which rigorously guarantees the scheme's stability and provides a bound for the global error. It is concluded that the high-order exponential method is consistent, stable, and highly efficient for solving the analyzed class of problems.