线法二阶常微分方程组有限元分析的EEP超收敛计算

EEP SUPER-CONVERGENT COMPUTATION IN FEM ANALYSIS OF FEMOL SECOND ORDER ODES

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。

To solve second order ordinary differential equations （ODEs） derived from Finite Element Method of Lines （FEMOL）, the exact element theory in FEM analysis is established, and formulas for exact solutions at any point are derived. Combined with Element Energy Projection （EEP） method, two EEP super-convergent schemes, simplified form and condensed form, are proposed for approximate elements. The simplified form uses linear shape functions as the test function, which is simple and convenient with certain degree of super-convergence. The condensed form uses condensed shape functions of degree rn as the test function, which is capable of producing optimal O（h^2m） super-convergence for both displacements and displacement derivatives at any point on each element. Numerical experiments show that the proposed EEP super-convergent strategy is a successful extension of the EEP method to second order ODEs in FEMOL with all advantages in single ODE problems being reserved.