Finite volume element method for two-dimensional fractional subdiffusion problems
This work provides a rigorous numerical analysis for a finite volume method applied to fractional subdiffusion problems, which is incremental for researchers in numerical PDEs.
The paper proposes and analyzes a finite volume element method for two-dimensional fractional subdiffusion equations, achieving optimal error estimates in L∞(L2) norm and convergence rates of O(h² + k^{1+α}). Numerical experiments confirm optimal convergence and show that theoretical regularity assumptions are pessimistic.
In this paper, a semi-discrete spatial finite volume (FV) method is proposed and analyzed for approximating solutions of anomalous subdiffusion equations involving a temporal fractional derivative of order $α\in (0,1)$ in a two-dimensional convex polygonal domain. Optimal error estimates in $L^\infty(L^2)$- norm is shown to hold. Superconvergence result is proved and as a consequence, it is established that quasi-optimal order of convergence in $L^{\infty}(L^{\infty})$ holds. We also consider a fully discrete scheme that employs FV method in space, and a piecewise linear discontinuous Galerkin method to discretize in temporal direction. It is, further, shown that convergence rate is of order $O(h^2+k^{1+α}),$ where $h$ denotes the space discretizing parameter and $k$ represents the temporal discretizing parameter. Numerical experiments indicate optimal convergence rates in both time and space, and also illustrate that the imposed regularity assumptions are pessimistic.