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A posteriori error estimates for space-time finite element approximation of quasistatic hereditary linear viscoelasticity problems. (English) Zbl 1076.74056
Summary: We give a space-time Galerkin finite element discretisation of the quasistatic compressible linear viscoelasticity problem as described by an elliptic partial differential equation with a fading-memory Volterra integral. The numerical scheme consists of a continuous Galerkin approximation in space based on piecewise polynomials of degree $$p>0$$ $$(cG(p))$$, with a discontinuous Galerkin piecewise constant $$(dG(0))$$ or linear $$(dG(1))$$ approximation in time. A posteiori Galerkin error estimates are derived by exploiting the Galerkin framework and optimal stability estimates for a related dual backward problem. The a posteriori error estimates are quite flexible; strong $$L_p$$-energy norms of the errors are estimated using time derivatives of the residual terms when the data are smooth, while weak-energy norms are used when the data are non-smooth (in time). We also give upper bounds on the $$dG(0)cG(1)$$ a posteriori error estimates which indicate optimality. However, a complete analysis is not given.

##### MSC:
 74S05 Finite element methods applied to problems in solid mechanics 74D05 Linear constitutive equations for materials with memory 65N15 Error bounds for boundary value problems involving PDEs 65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
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