β-robust virtual element method for distributed order time-fractional reaction-diffusion equation with variable coefficients

In this paper, a distributed-order time-fractional equation with variable coefficients on polygonal meshes is considered. Initially, the distributed-order time-fractional derivative is converted into a multi-term time-fractional derivative. The L1 scheme on graded meshes is then employed to handle the singularity of the time-fractional derivative. For variable coefficients in the spatial direction, modified approximated bilinear forms of the virtual element method are constructed to maintain optimal convergence. Utilizing the beta-robust discrete fractional Gronwall inequality, beta-robust stability and optimal error estimates of the fully discrete scheme in the L-2-norm are established. These results remain valid and stable as beta -> 1(-). Numerical results are presented to illustrate the sharpness of the theoretical findings.