单元能量投影法的数学分析
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摘要
有限元法是解偏微分方程的有效方法之一。但是有限元解的导数一般在单元边界不连续且整体精度不高。因而如何提高有限元解导数的精度成为近年来有限元研究的热点之一。
2004年袁驷教授对于二阶方程两点边值问题基于力学解释提出了所谓的“单元能量投影法”，其基本思想来源于结构力学中的矩阵位移法和有限元数学理论中的投影定理。数值例子显示了良好的效果。2006年单元能量法被推广到四阶两点边值问题的有限元计算中，同样获得了令人满意的效果。但是这一系列很有吸引力的结果均缺少严格的数学分析。
本文主要对二阶方程和四阶方程两点边值问题的单元能量投影法进行数学分析，获得了一系列好的结果。我们的主要贡献是：
1．对于自伴二阶两点边值问题，我们运用投影型插值及强超逼近结果对单元能量投影法导出的一种逐点导数与位移恢复公式进行了细致的分析，准确地指出了它们的收敛精度，这个结果修正了袁驷教授原先报导的结果。
2．对于非自伴二阶两点边值问题，我们导出了准确解在节点上导数的一种表达式，再运用“正交性修正”证明了单元能量投影法节点恢复导数的O(h~(2k))(k≥1)阶超收敛性。这是目前获得的后处理最高阶超收敛结果。
3．对于四阶两点边值问题的单元能量投影法。我们把袁驷教授的结果推广到更一般的方程并对节点恢复弯矩证得了O(h~(2k-2))阶的超收敛精度。这也是目前四阶问题的后处理最高阶超收敛结果。对于恢复剪力，我们证明了O(h~(2k-3))阶的较好结果。
The finite element method is one of the efficient numerical methods to solve partial different equations. But in general the derivatives of the finite element solution are not continuous across the boundary of elements and have low global accuracy. To improve the accuracy of the derivative of the finite element solution became one of the most important research subject recently.
In 2004 for second order two-point boundary value problems Professor Yuan proposed the so called element-energy-projection (EEP) method based on mechanical interpretation. The fundamental idea comes form the strategy in conventional matrix displacement method for skeletal structures and the projection theorem in finite element mathematical theory. The numerical examples show high accuracy of the new method. In 2006, Professor Yuan applied the element energy projection method to the finite element computation of fourth order two point boundary value problems. The numerical examples show that the EEP method also works well for the fourth order problems. But there is no strict mathematical analysis for these very attractive results.
In this thesis we give a mathematical analysis on the element energy projection method. Our main results are as follows:
1. For self-adjoint second order two point boundary value problem, using projection type interpolation and ultra-approximation results we study the pointwise derivative and displacement recovery formula in detail. We obtain their convergence rate exactly. This result corrected Yuan's earlier conclusion.
2. For non-self-adjoint second order two point boundary value problem, we derive formula of derivative of exact solution in nodal point. By virtue of orthogonal correction technique we proved the o(h~(2k)(k≥1)super convergence for the nodal recovery derivative derived by EEP method. This is the highest order superconvergence result for derivative postprocessing at present.
3. For the EEP method of fourth order two point boundary value problem, we extend Yuan's method to more general equations and proved the O(h~(2k-2)) super convergence for recovery bending moments. This is the highest order superconvergence result for fourth problems up to now. For recovery shear forces we proved the super convergence.
2004年袁驷教授对于二阶方程两点边值问题基于力学解释提出了所谓的“单元能量投影法”，其基本思想来源于结构力学中的矩阵位移法和有限元数学理论中的投影定理。数值例子显示了良好的效果。2006年单元能量法被推广到四阶两点边值问题的有限元计算中，同样获得了令人满意的效果。但是这一系列很有吸引力的结果均缺少严格的数学分析。
本文主要对二阶方程和四阶方程两点边值问题的单元能量投影法进行数学分析，获得了一系列好的结果。我们的主要贡献是：
1．对于自伴二阶两点边值问题，我们运用投影型插值及强超逼近结果对单元能量投影法导出的一种逐点导数与位移恢复公式进行了细致的分析，准确地指出了它们的收敛精度，这个结果修正了袁驷教授原先报导的结果。
2．对于非自伴二阶两点边值问题，我们导出了准确解在节点上导数的一种表达式，再运用“正交性修正”证明了单元能量投影法节点恢复导数的O(h~(2k))(k≥1)阶超收敛性。这是目前获得的后处理最高阶超收敛结果。
3．对于四阶两点边值问题的单元能量投影法。我们把袁驷教授的结果推广到更一般的方程并对节点恢复弯矩证得了O(h~(2k-2))阶的超收敛精度。这也是目前四阶问题的后处理最高阶超收敛结果。对于恢复剪力，我们证明了O(h~(2k-3))阶的较好结果。
The finite element method is one of the efficient numerical methods to solve partial different equations. But in general the derivatives of the finite element solution are not continuous across the boundary of elements and have low global accuracy. To improve the accuracy of the derivative of the finite element solution became one of the most important research subject recently.
In 2004 for second order two-point boundary value problems Professor Yuan proposed the so called element-energy-projection (EEP) method based on mechanical interpretation. The fundamental idea comes form the strategy in conventional matrix displacement method for skeletal structures and the projection theorem in finite element mathematical theory. The numerical examples show high accuracy of the new method. In 2006, Professor Yuan applied the element energy projection method to the finite element computation of fourth order two point boundary value problems. The numerical examples show that the EEP method also works well for the fourth order problems. But there is no strict mathematical analysis for these very attractive results.
In this thesis we give a mathematical analysis on the element energy projection method. Our main results are as follows:
1. For self-adjoint second order two point boundary value problem, using projection type interpolation and ultra-approximation results we study the pointwise derivative and displacement recovery formula in detail. We obtain their convergence rate exactly. This result corrected Yuan's earlier conclusion.
2. For non-self-adjoint second order two point boundary value problem, we derive formula of derivative of exact solution in nodal point. By virtue of orthogonal correction technique we proved the o(h~(2k)(k≥1)super convergence for the nodal recovery derivative derived by EEP method. This is the highest order superconvergence result for derivative postprocessing at present.
3. For the EEP method of fourth order two point boundary value problem, we extend Yuan's method to more general equations and proved the O(h~(2k-2)) super convergence for recovery bending moments. This is the highest order superconvergence result for fourth problems up to now. For recovery shear forces we proved the super convergence.
引文
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[2] Zienkiewicz, O.C., Cheung, Y.K. The finite element method in structural and continuum mechanics. McGraw-Hill, 1967.
[3] Douglas, Jr.J., Dupont, T. Some superconvergence results for Galerkin methods for the approximate solution of two-point boundary value problems. In: Topics in Numerical Analysis (Proc. Roy. Irish Acad. Conf., univ. coll., Dublin, 1972), London, Academic Press, 1973, 89-92.
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[5] Lin, Q., Yan, N.N., Zhou, A.H. A rectangle test for interpolated finite elements. In: Proc. Syst. Sci. & Syst. Eng., Great wall Culture Pub. Co., Hong Kong, 1991, 217-229.
[6] Zlamal, M. Some superconvergonce results in the finite element method. In Mathematical aspects of finite element methods (Pro. Conf. Consiglio Naz. delle Ricerche (C. N. R.), Rome, 1975). Lecture Notes in Math., 606, Springer, Berlin, 1977, 353-362.
[7] 陈传淼，朱起定．有限元法的一种新估计及应力佳点定理．湘潭大学自然科学通讯，1978，1：10-20．
[8] 陈传淼．有限元方法及其提高精度的分析．长沙：湖南科学技术出版社，1982．
[9] 朱起定，林群．有限元超收敛理论．长沙：湖南科技出版社，1989．
[10] 林群，朱起定．有限元的预处理与后处理理论．上海：上海科学技术出版社，1994
[11] Wahlbin, L.B. Superconvergence in Galerkin finite element methods. Lecture Notes in Mathematics 1605, Springer, 1995.
[12] 陈传淼，黄云清．有限元高精度理论．长沙：湖南科学技术出版社，1995．
[13] 林群,严宁宁.高效有限无构造与分析.保定:河北大学出版社,1996．
[14] Ainsworth,M.,Oden,J.T.A posteriori error estimation in finite element analysis.Wiley lnterscience,New York,2000．
[15] 陈传淼.有限元超收敛构造理论.长沙:湖南科学技术出版社,2001．
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