The efficient ADI Galerkin finite element methods for the three-dimensional nonlocal evolution problem arising in viscoelastic mechanics. (English) Zbl 1529.65082
Summary: In this article, we investigate and analyze new methods for the numerical solution of the three-dimensional nonlocal evolution problem arising in viscoelastic mechanics. Then these methods combine Galerkin finite element methods (FEMs) for the spatial discretization with corresponding alternating 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 \(L^2\)-norm stability and convergence of two ADI Galerkin schemes are proved by the energy argument. Numerical results confirm the predicted space-time convergence orders.
MSC:
| 65M60 | Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs |
| 65M06 | Finite difference methods for initial value and initial-boundary value problems involving PDEs |
| 65N30 | Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs |
| 65D30 | Numerical integration |
| 65M12 | Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs |
| 65M22 | Numerical solution of discretized equations for initial value and initial-boundary value problems involving PDEs |
| 76A10 | Viscoelastic fluids |
| 74H15 | Numerical approximation of solutions of dynamical problems in solid mechanics |
| 74D05 | Linear constitutive equations for materials with memory |
| 35Q74 | PDEs in connection with mechanics of deformable solids |
Keywords:
three-dimensional nonlocal evolution equation; ADI Galerkin methods; convolution quadrature rules; stability and convergenceReferences:
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