Unconditionally energy stable ESAV-VEM schemes with variable time steps for the time fractional Allen-Cahn equation

For the time fractional Allen-Cahn equation, integration of the exponential scalar auxiliary variable (ESAV) time discretization with the virtual element method (VEM) spatial discretization leads to ESAV-VEM schemes on polygonal meshes. The resulting ESAV-VEM algorithms are linear and effective for the completely decoupled computations of the phase variable u and the auxiliary variable r. Based on the positive definiteness of variable-step discrete convolution kernels relevant to L1 and L1-CN time discretizations, the corresponding unconditionally energy boundedness results are proven on arbitrary nonuniform time meshes. By means of the discrete gradient structure of the temporal discretization operator, the discrete energy dissipation laws are developed via a unified framework. Moreover, the discrete energy decay nature coincides with the classical analogies as the fractional order tends to one. Finally, a series of numerical experiments are presented to verify the accuracy and efficiency of the proposed method.