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Numerical analysis of Volterra integro-differential equations for viscoelastic rods and membranes. (English) Zbl 1428.74216

Summary: We consider the initial boundary value problems for a homogeneous Volterra integro-differential equations for viscoelastic rods and membranes in a bounded smooth domain \(\Omega \). The memory kernel of the equation is made up in a complicated way from the (distinct) moduli of stress relaxation for compression and shear, which is challenging to approximate. The literature reported on the numerical solution of this model is extremely sparse. In this paper, we will study the second order continuous time Galerkin approximation for its space discretization and propose a fully discrete scheme employing the Crank-Nicolson method for the time discretization. Then, we derive the uniform long time error estimates in the norm \(L_t^1(0, \infty; L_x^2)\) for the finite element solutions. Some numerical results are presented to illustrate our theoretical error bounds.

MSC:

74S05 Finite element methods applied to problems in solid mechanics
65R20 Numerical methods for integral equations
65M60 Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs
45K05 Integro-partial differential equations
65M15 Error bounds for initial value and initial-boundary value problems involving PDEs
74K10 Rods (beams, columns, shafts, arches, rings, etc.)
74K15 Membranes
74D05 Linear constitutive equations for materials with memory
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