The O(h2)$\mathcal {O}(h^2)$ virtual element method for a nonlinear sub‐diffusion equation involving a non‐smooth potential parameter

Abstract The main goal of this study is to propose an efficient framework for solving time‐fractional nonlinear sub‐diffusion equations involving non‐smooth potential terms. By addressing such potential terms, this work alleviates key constraints in the equations of the existing model, enabling a more comprehensive treatment of associated inverse problems. The study is dedicated to establishing a rigorous theoretical framework utilizing a recently developed numerical methodology, ensuring robust and reliable solutions for such equations. The virtual element method, a generalization of the finite element method designed for polygonal and polyhedral meshes under the Galerkin framework, is employed. A fully discrete scheme is constructed using virtual elements for the spatial domain and the Grünwald–Letnikov equality combined with backward Euler‐convolution quadrature for temporal discretization. The novelty of this work lies in the seamless integration of the Grünwald–Letnikov approximation with the virtual element method to handle time‐fractional nonlinear sub‐diffusion equations with non‐smooth potential parameters, an approach not previously addressed. This combination enables accurate and flexible discretization on general polygonal meshes, particularly suited for complex geometries. We rigorously establish the theoretical proofs for the existence and uniqueness of the discrete solution, along with its optimal convergence properties. Error estimates are derived in both the ‐seminorm and the ‐norm, ensuring a precise assessment of the approximation accuracy. To further validate the theoretical results, we provide numerical experiments that demonstrate the robustness of the proposed method across both convex and non‐convex mesh configurations, highlighting the practical advantages of the virtual element method in fractional‐order settings.