非正则三角剖分对于有限元收敛性的影响  

INFLUENCE OF IRREGULAR TRIANGULATION ON THE FINITE ELEMENT CONVERGENCE

作　　者：田明[1] 

机构地区：[1]中国民用航空学院理学院,天津300300

出　　处：《石油化工高等学校学报》2004年第4期83-86,共4页Journal of Petrochemical Universities

基　　金：辽宁省科学技术基金资助 (0 0 10 84 )

摘　　要：网格剖分是目前有限元理论中的一个前沿性研究课题 ,随着超级计算机不断发展 ,大规模的科学计算已经成为可能 ,然而经典的有限元方法中的区域剖分仍是需要从理论方法上加以关注的问题。冯康曾经证明了基于三角形单元进行线性Lagrange插值时的误差估计 ,而StrangG则指出了有限元方法剖分过程中不能采用大钝角单元 ,综合这两种结论 ,就有了两种剖分方式 ,即小锐角单元剖分及大钝角单元剖分。关于小锐角单元剖分的可行性研究结果 ,对于实际问题的计算有着重要意义及深刻影响。首先分析了小锐角单元及大锐角单元情形下二次Lagrange插值的特征。以三角形单元外接圆半径作为收敛指标证明了平面多角形区域泊松边值问题的二次La grange有限元误差估计 ,数值计算实例也反映了这一分析结论。The mesh dissection is a proceeding study subject in the finite element theory at present. With the super computer developing continuously, it is possible to do large-scale scientific computation. However, the mesh dissection in the classical finite element theory still needs to be given more attention. FENG Kang proved the error estimates for linear Lagrange interpolation based on the triangular element. Strang G pointed out that it wasn't permitted to use obtuse angle element in the triangulation. Combined with the above two conclusions, there are two kinds of dissections, one is acute angle triangulation, the other is obtuse angle triangulation. The results for the acute angle triangulation will have important meaning and profound influences on the real problems. First, the property of 2’degree Lagrange interpolation under acute angle element and obtuse angle element is analyzed, and then the finite element error estimates to the Poisson boundary problem in the plane polygon domain is proved. The analysis conclusion is also verified by the example of numerical computation. The definite theoretical basis is given for the computation reality of acute angle triangulation to a certain extent.

关 键 词：插值 误差估计 有限元 

分 类 号：O241.3[理学—计算数学]