THE EFFICIENT ADI GALERKIN FINITE ELEMENT METHODS FOR THE THREE-DIMENSIONAL NONLOCAL EVOLUTION PROBLEM ARISING IN VISCOELASTIC MECHANICS

In this article, we investigate and analyze new methods for the numerical solution of the three-dimensional nonlocal evolution problem aris-ing in viscoelastic mechanics. Then these methods combine Galerkin finite element methods (FEMs) for the spatial discretization with corresponding al-ternating direction implicit (ADI) algorithms, based on the backward Euler (BE) method and Crank-Nicolson (CN) method, respectively, from which, the Riemann-Liouville (R-L) integral term is approximated by relevant convolution quadrature rules. The L2-norm stability and convergence of two ADI Galerkin schemes are proved by the energy argument. Numerical results confirm the predicted space-time convergence orders.