Stability and uniqueness of coupled nonlinear finite element solution for anisotropic diffusion equation with nonlinear capacity term

Abstract

This paper presents the stability of a two-layer coupled discretization fully implicit finite element scheme as well as the uniqueness of its solution. The scheme has been proposed for solving multi-dimensional anisotropic diffusion equations with nonlinear capacity term, and the existence and convergence of its solution have been proved in [Fang et al., J. Comput. Appl. Math. 438 (2024) 115512]. However, the basic theoretical analysis is incomplete, for example, the stability and uniqueness have not been solved yet, which are very important for the application of numerical methods in engineering. In this paper, we further develop the discrete functional analysis techniques to establish a framework with relatively comprehensive theoretical results. Wherein by introducing Ritz projection, rewriting the error equations into equivalent forms, and choosing appropriate test functions, we propose a new inductive argument to overcome the difficulties arising from the coupled nonlinear discretization of the diffusion operator and capacity term. Consequently, on the basis of the existence and convergence properties, we prove for the first time that the nonlinear finite element method is stable, thereby its solution is unique. Numerical examples show that the scheme is stable and has no numerical oscillations compared with the classical Crank–Nicolson scheme.