Fast evaluation and robust error analysis of the virtual element methods for time fractional diffusion wave equation

The article is concerned with and analyzes the α-robust error bound for time-fractional diffusion wave equations with weakly singular solutions. Nonuniform L1-type time meshes are used to handle non-smooth systems, and the sum-of-exponentials (SOEs) approximation for the kernels function is adopted to reduce the memory storage and computational cost. Meanwhile, the virtual element method (VEM), which can deal with complex geometric meshes and achieve arbitrary order of accuracy, is constructed for spatial discretization. Based on the explicit factors and discrete complementary convolution kernels, the optimal error bound of the fully discrete SOEs-VEM scheme in the $L^2$-norm is derived in detail and that is α-robust, i.e., the bounds will not explosive growth while $\alpha \to 2^{-}$. Finally, some numerical experiments are implemented to verify the theoretical results.