A virtual element scheme for the time-fractional parabolic PDEs over distorted polygonal meshes
Zaffar Mehdi Dar, M. Chandru
Abstract
We extend the virtual element method to the two-dimensional time-fractional parabolic PDE, characterized by a fractional derivative of order α ∈ ( 0 , 1 ) in time. To illustrate the working of this fractional virtual element scheme, a numerical investigation of the following time-fractional problem over distorted polygonal meshes is conducted. c D t α u ( z , t ) − Δ u = f ( z , t ) in z ∈ Ω , t ∈ ( 0 , T ] , where Ω is a spatial domain, α is fractional order, and t is time variable. Our methodology is based on the fundamental technical component, fractional version of the Grunwald–Letnikov approximation. We prove the method’s well-posedness, that is the approximate solution’s existence and uniqueness. The fully discrete scheme inherently maintains stability and consistency by leveraging the discrete maximal regularity and the energy projection operator. The convergence in the L 2 -norm and H 1 -norm over distorted mesh configuration is validated by numerical results, underlining the practical effectiveness of the proposed method.