Virtual element approximation and BDF2 time-discrete scheme for a partial integro-differential equation with a singular Abel's kernel

This work focuses on developing a virtual element framework for analyzing nonlinear partial integro-differential equations with singular Abel-type kernels. To address the singularity in the integral term, we propose two strategies: one based on linear interpolation over a uniform temporal mesh and another employing a graded mesh. Notably, while the uniform mesh fails to retain second-order temporal accuracy, the graded mesh successfully achieves it. For spatial discretization, we adopt the virtual element method and derive rigorous error estimates for the semi-discrete scheme. Nonlinear terms are approximated via a second-order temporal discretization. Following this, we establish a fully-discrete scheme. We rigorously establish the unconditional stability and convergence rates of the proposed method in separate theorems. Finally, we validate the theoretical results through numerical experiments on various domains.