Nonconforming finite element Stokes complexes in three dimensions

Abstract

Two nonconforming finite element Stokes complexes starting from the conforming Lagrange element and ending with the nonconforming $P_1$-$P_0$ element for the Stokes equation in three dimensions are constructed. And commutative diagrams are also shown by combining nonconforming finite element Stokes complexes and interpolation operators. The lower order $\mathbf{H}(\text{grad}\text{curl})$-nonconforming finite element only has $14$ degrees of freedom, whose basis functions are explicitly given in terms of the barycentric coordinates. The $\mathbf{H}(\text{grad}\text{curl})$-nonconforming elements are applied to solve the quad-curl problem, and optimal convergence is derived. By the nonconforming finite element Stokes complexes, the mixed finite element methods of the quad-curl problem are decoupled into two mixed methods of the Maxwell equation and the nonconforming $P_1$-$P_0$ element method for the Stokes equation, based on which a fast solver is discussed. Numerical results are provided to verify the theoretical convergence rates.

26 pages. This paper has been accepted for publication in SCIENCE CHINA Mathematics

References in corpus (2)

• Curl-curl conforming elements on tetrahedra
• Simple curl-curl-conforming finite elements in two dimensions

Cited by in corpus (1)

• Error analysis of a decoupled finite element method for quad-curl problems