## Superconvergence analysis and extrapolation of nonconforming mixed finite element approximation for nonlinear viscoelastic wave equation.(Chinese. English summary)Zbl 1363.74077

Summary: A nonconforming mixed finite element method for nonlinear viscoelastic wave equation is studied based on a new mixed variational form. By utilizing the properties of interpolation on the element, high accuracy analysis, derivative delivery techniques with respect to time $$t$$ and mean-value theorem, two special properties of $$EQ_1^{\mathrm {rot}}$$ element: (a) the consistency error is of order $$O(h^2)$$, which is one order higher than its interpolation error $$O(h)$$; (b) the interpolation operator is equivalent to its Ritz-projection operator, the superclose properties and the global superconvergence with order $$O(h^2)$$ for the primitive solution $$u$$ in broken $$H^1$$ norm and the intermediate variable $$p$$ in $$L^2$$ norm are obtained through interpolated postprocessing approach, respectively. Furthermore, by constructing a suitable auxiliary problem, the extrapolation result with higher order $$O(h^3)$$ for $$u$$ and $$p$$ are derived through Richardson scheme.

### MSC:

 74S05 Finite element methods applied to problems in solid mechanics 74J30 Nonlinear waves in solid mechanics 65M60 Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs 65M12 Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs