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

TL;DR

This work develops a unified framework for high-order nonconforming virtual element methods for nonlinear fourth-order reaction–diffusion problems in 2D, accommodating clamped, Navier, and Cahn–Hilliard-type boundary conditions. By introducing Companion and Ritz-type operators, the authors obtain optimal error estimates under minimal spatial regularity on nonconvex domains, for both semi- and fully-discrete schemes. The theory is instantiated with C^0-nonconforming and Morley-type VEMs on polygonal meshes and validated through numerical experiments that confirm the predicted convergence rates across boundary conditions. The approach extends the applicability of NCVEMs to a broad class of fourth-order nonlinear problems, including pattern formation and viscous-flow contexts, and sets the stage for future extensions to more complex systems such as CH dynamics and stream-function formulations in fluid dynamics.

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.

Figures (4)

• Figure 1: DoFs of the $C^0$-nonconforming (top) and Morley-type (bottom) VEMs on a pentagonal element for $k=2,3,4$. Dots, squares, arrows, and diamonds correspond to $\mathbf{(D1)}$, $\mathbf{(D2)}$ ($i=0,1$), and $\mathbf{(D3)}$, respectively.
• Figure 2: Schematic representation of the construction of the functional $\widehat{A}_{\varphi}$ defined in \ref{['def:functional']}.
• Figure 3: Example of the fourth families of meshes used: $\Omega^1_h$, $\Omega^2_h$, $\Omega^3_h$ and $\Omega^4_h$.
• Figure 4: Test 1. Snapshots of the exact and numerical solutions at different time instants for the Extended Fisher–Kolmogorov equation with ${\bf CP}$ BCs, using the fully-discrete VE scheme \ref{['fully:dis:schm']}, $\alpha_1=\alpha_2=1$, and mesh $\Omega^1_h$, with $h=1/32$ and $\Delta=10^{-2}$.

Theorems & Definitions (44)

• Theorem 1.1
• Remark 1.1
• Definition 2.1: DoFs-tuple
• Definition 2.2: Associated local DoFs
• Definition 2.3
• Remark 2.1
• Example 2.1
• Definition 2.4: Associated global DoFs
• Remark 2.2
• Definition 2.5