Nonconforming virtual element methods for fourth-order nonlinear reaction-diffusion systems: a unified framework and analysis

TL;DR

This paper introduces a unified high-order nonconforming virtual element framework for nonlinear fourth-order reaction-diffusion problems, providing rigorous error analysis and numerical validation across various boundary conditions.

Contribution

It develops a novel, unified error analysis for high-order nonconforming virtual element methods applicable to non-convex domains, with minimal regularity assumptions.

Findings

Optimal error estimates achieved for the schemes.

Numerical experiments validate theoretical results.

Framework applicable to multiple boundary conditions.

Abstract

We develop a unified framework for the design and analysis of high-order nonconforming virtual element methods for nonlinear fourth-order reaction--diffusion problems in two dimensions, with emphasis on clamped, Navier, and Cahn--Hilliard-type boundary conditions. Time discretization is performed using the backward Euler scheme, while the spatial approximation relies on nonconforming virtual element spaces of arbitrary order $k \ge 2$, encompassing both $C^0$-nonconforming and Morley-type methods. A key contribution of this work is the development of a novel and rigorous unified error analysis for these numerical schemes, applicable to domains that are not necessarily convex, differing from the existing literature. By introducing a class of Companion operators, we construct novel Ritz-type projections and derive a new error equation that enables us to obtain optimal error estimates for the scheme under a minimal spatial regularity assumption on the weak solution. Finally, we present numerical experiments on polygonal meshes as applications of the proposed framework, including the extended Fisher--Kolmogorov equation, and a fourth-order model with Cahn--Hilliard-type boundary conditions, which validate the theoretical results and illustrate the performance of the method for the three classes of boundary conditions.