Accurate numerical integration for the quadrilateral and hexahedral finite elements. Issue 3 (16th December 2020)

• Record Type: Journal Article
• Authors: Xie, Ziqing; Zhang, Shangyou
• Is Part Of: Numerical methods for partial differential equations. Volume 37:Issue 3(2021)
• Page Start: 2446
• Page End: 2457
• Publication Date: 2020-12-16
• Subjects: finite element -- quadrature -- quadrilateral grids -- hexahedral grids; Differential equations, Partial -- Numerical solutions -- Periodicals; 515.353
• DOI: 10.1002/num.22723
• Languages: English
• ISSNs: 0749-159X

Abstract

Abstract: 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.