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A nonconforming mixed element analysis for nonlinear strongly damped wave equations. (Chinese. English summary) Zbl 1363.65168

Summary: With the help of \(EQ_1^{rot} + Q_{10}\times Q_{01}\) element, a nonconforming mixed finite element method for nonlinear strongly damped wave equation is investigated by utilizing high accuracy analysis and two special properties of \(EQ_1^{rot}\) element: (a) the consistency error is of order \(O(h^2)\) which is one order higher than its interpolation error; (b) the interpolation operator is equivalent to its Ritz-projection operator, the super-close and the global super-convergence results with order \(O(h^2)\) for the primitive solution \(u\) in broken \(H^1\)-norm and flux variable \(\vec{p}\) in \(L^2\)-norm are obtained through interpolated postprocessing approach respectively for semi-discrete scheme. At the same time, the super-close results with order \(O(h^2 +\tau^2)\) are obtained through constructing a new fully-discrete scheme.

MSC:

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
35L70 Second-order nonlinear hyperbolic equations
65M15 Error bounds for initial value and initial-boundary value problems involving PDEs
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