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该文先对有限元线法导出的二阶常微分方程组问题,建立了有限元分析的精确单元理论,推导出任意点的真解计算公式,再以之为依据给出近似单元的两种单元能量投影(EEP)超收敛公式——简约格式和凝聚格式。简约格式采用线性形函数作为权函数,计算简单方便,具有强超收敛性。凝聚格式则用m次凝聚形函数作为权函数,可使位移和位移导数的超收敛解的各分量均能达到h2m阶的最佳超收敛结果。广泛的数值试验表明,该法是EEP超收敛算法在二阶常微分方程组问题上的成功推广,具有和单个常微分方程问题一致的良好性态。
In this paper, the accurate element theory of finite element analysis is established for the second-order ordinary differential equations derived by the finite element method. The true solution of any point is deduced. Based on it, two kinds of approximate elements are given Cell Energy Projection (EEP) Super Convergence Formulas - Simple and Condensed Formats. The simple form uses the linear form function as the weight function, the calculation is simple and convenient, and has strong super convergence. The coacervate form uses the m-th condensing form function as a weighting function, so that each component of the superconvergence solution of the displacement and displacement derivatives can reach the optimal super-convergence result of h2m order. Extensive numerical experiments show that this method is a successful extension of the EEP superconvergence algorithm for the second-order ordinary differential equations and has the same good behavior as the single ordinary differential equations.