一个非常规高效数值积分方法

An Unconventional High Accuracy Numerical Quadrature Method

利用一个特殊非协调矩形元导出了一个新的使用节点少而代数精度高的非常规数值积分公式.利用有限元方法的分析技巧,在较弱的条件下(即在Sobolev空间模意义下)证明了由此公式导出的复化公式具有与复化Simpson公式和复化Gauss公式一样的收敛阶O(h4).而且在精细剖分下,该公式比后两种积分公式大致节约25%的计算量.最后,通过两个数值算例验证了理论分析的正确性.

In this paper, a new unconventional numerical quadrature with fewer node values is established based on a special nonconforming rectangular finite element. By use of the analysis skills of finite element method it is proved that this numerical quadrature is of the same convergence order O（h^1 ） as that of Simpson and Gaussian rules under weak condition （in the sense of sobolev norm）. At the same time, the computing cost of the new formula is only three quarters of the later two rules. At last, we also give two examples to confirm our theoretical analysis.