Second-order BDF ADI Galerkin finite element method for the evolutionary equation with a nonlocal term in three-dimensional space

Abstract

In this work, we propose and analyze a new method for the solution of the three-dimensional evolutionary equation with a nonlocal term. Then the method combines Galerkin finite element methods (FEMs) for the spatial discretization with an alternating direction implicit (ADI) algorithm based on the second-order backward differentiation formula (BDF2), where the Riemann-Liouville (R-L) integral term is approximated via second-order convolution quadrature (CQ) rule. The L2-norm stability and convergence are proved. Numerical results confirm the predicted space-time convergence rates.