非线性粘弹性波动方程的非协调混合元超收敛分析和外推

Superconvergence Analysis and Extrapolation of Nonconforming Mixed Finite Element Apporximation for Nonlinear Viscoelastic Wave Equation

研究了非线性粘弹性波动方程在新混合元格式下非协调混合有限元方法.利用插值理论、高精度分析、平均值理论和对时间t的导数转移的技巧,借助于EQ_1^(rot)元所具有的两个性质:(a)其相容误差为O(h^2)阶比它的插值误差高一阶;(b)插值算子与Ritz投影等价,以及插值后处理技术,分别导出了原始变量u的H^1模和中间变量p的L^2模下O(h^2)阶超逼近性质和整体超收敛.进一步,通过构造适当的辅助问题,运用Richordson外推格式,得到了更高精度O(h^3)阶外推结果.

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, tow special properties of EQ1^rot element： （a） the consistency error is of order O（h^2） which is one order higher than its interpolation errorO（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 Hlnorm 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.