Nonconforming finite element Stokes complexes in three dimensions

Two nonconforming finite element Stokes complexes ended 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 $\boldsymbol H(\textrm{grad}~\textrm{curl})$-nonconforming finite element only has $14$ degrees of freedom, whose basis functions are explicitly given in terms of the barycentric coordinates. The $\boldsymbol H(\textrm{grad}~\textrm{curl})$-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 is 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 developed.Comment: 20 page

MultiGrid Preconditioners for Mixed Finite Element Methods of the Vector Laplacian

Due to the indefiniteness and poor spectral properties, the discretized linear algebraic system of the vector Laplacian by mixed finite element methods is hard to solve. A block diagonal preconditioner has been developed and shown to be an effective preconditioner by Arnold et al. (Acta Numer 15:1–155, 2006). The purpose of this paper is to propose alternative and effective block diagonal and approximate block factorization preconditioners for solving these saddle point systems. A variable V-cycle multigrid method with the standard point-wise Gauss–Seidel smoother is proved to be a good preconditioner for the discrete vector Laplacian operator. The major benefit of our approach is that the point-wise Gauss–Seidel smoother is more algebraic and can be easily implemented as a black-box smoother. This multigrid solver will be further used to build preconditioners for the saddle point systems of the vector Laplacian. Furthermore it is shown that Maxwell’s equations with the divergent free constraint can be decoupled into one vector Laplacian and one scalar Laplacian equation

MultiGrid Preconditioners for Mixed Finite Element Methods of the Vector Laplacian

Due to the indefiniteness and poor spectral properties, the discretized linear algebraic system of the vector Laplacian by mixed finite element methods is hard to solve. A block diagonal preconditioner has been developed and shown to be an effective preconditioner by Arnold et al. (Acta Numer 15:1–155, 2006). The purpose of this paper is to propose alternative and effective block diagonal and approximate block factorization preconditioners for solving these saddle point systems. A variable V-cycle multigrid method with the standard point-wise Gauss–Seidel smoother is proved to be a good preconditioner for the discrete vector Laplacian operator. The major benefit of our approach is that the point-wise Gauss–Seidel smoother is more algebraic and can be easily implemented as a black-box smoother. This multigrid solver will be further used to build preconditioners for the saddle point systems of the vector Laplacian. Furthermore it is shown that Maxwell’s equations with the divergent free constraint can be decoupled into one vector Laplacian and one scalar Laplacian equation