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 three-dimensional nonlocal evolution problem arising in viscoelastic mechanics. Then these combine Galerkin finite element (FEMs) spatial discretization with corresponding alternating direction implicit (ADI) algorithms, based on backward Euler (BE) method Crank-Nicolson (CN) method, respectively, from which, Riemann-Liouville (R-L) integral term is approximated by relevant convolution quadrature rules. The $ L^2 $-norm stability convergence two ADI schemes are proved energy argument. Numerical results confirm predicted space-time orders.