Conforming and nonconforming VEMs for the fourth-order reaction-subdiffusion equation: a unified framework

We establish a unified framework to study the conforming and nonconforming virtual element methods (VEMs) for a class of time dependent fourth-order reaction-subdiffusion equations with the Caputo derivative. To resolve the intrinsic initial singularity we adopt the nonuniform Alikhanov formula in the temporal direction. In the spatial direction three types of VEMs, including conforming virtual element, C-0 nonconforming virtual element and fully nonconforming Morley-type virtual element, are constructed and analysed. In order to obtain the desired convergence results, the classical Ritz projection operator for the conforming virtual element space and two types of new Ritz projection operators for the nonconforming virtual element spaces are defined, respectively, and the projection errors are proved to be optimal. In the unified framework we derive a prior error estimate with optimal convergence order for the constructed fully discrete schemes. To reduce the computational cost and storage requirements, the sum-of-exponentials (SOE) technique combined with conforming and nonconforming VEMs (SOE-VEMs) are built. Finally, we present some numerical experiments to confirm the theoretical correctness and the effectiveness of the discrete schemes. The results in this work are fundamental and can be extended into more relevant models.