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

Mao, Fengmei; Luo, Juan; Wang, Junjun

Math. Pract. Theory 45, No. 9, 205-218 (2015).

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

Keywords:

nonlinear viscoelastic wave equation; nonconforming finite element; new mixed finite element; superconvergence; extrapolation