The numerical integration for the bilinear form of quadrilateral or hexahedral finite elements can never be exact. It is discovered that the standard Gauss‐Legendre integration is exact if one of the two Q_k polynomials in the bilinear form is a P_k polynomial. Based on this observation, it is proved the Q_k finite elements, under practical numerical integration, retain the optimal order of convergence on general quadrilateral grids in 2 dimensional space, and on general hexahedral grids in 3 dimensional space. Numerical tests are provided, verifying all theoretic findings.

中文翻译：

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四边形和六面体有限元的精确数值积分

四边形或六面体有限元的双线性形式的数值积分永远不可能是精确的。发现如果双线性形式的两个Q _k多项式之一是P _k多项式，则标准的Gauss-Legendre积分是精确的。在此基础上，证明了在实际数值积分下，Q _k个有限元在二维空间中一般的四边形网格和三维空间中一般的六面体网格上都保持了最优​​的收敛顺序。提供了数值测试，验证了所有理论上的发现。