QuadrilateralH(div) Finite Elements

Abstract

Abstract. We consider the approximation properties of quadrilateral finite element spaces of vector fields defined by the Piola transform, extending results previously obtained for scalar approximation. The finite element spaces are constructed starting with a given finite dimensional space of vector fields on a square reference element, which is then transformed to a space of vector fields on each convex quadrilateral element via the Piola transform associated to a bilinear isomorphism of the square onto the element. For affine isomorphisms, a necessary and sufficient condition for approximation of order r + 1 in L 2 is that each component of the given space of functions on the reference element contain all polynomial functions of total degree at most r. In the case of bilinear isomorphisms, the situation is more complicated and we give a precise characterization of what is needed for optimal order L 2 -approximation of the function and of its divergence. As applications, we demonstrate degradation of the convergence order on quadrilateral meshes as compared to rectangular meshes for some standard finite element approximations of H(div). We also derive new estimates for approximation by quadrilateral Raviart-Thomas elements (requiring less regularity) and propose a new quadrilateral finite element space which provides optimal order approximation in H(div). Finally, we demonstrate the theory with numerical computations of mixed and least squares finite element approximations of the solution of Poisson's equation.

Finite element differential forms on curvilinear cubic meshes and their approximation properties

“…Necessary and sufficient conditions on the space of reference shape functions were given for multilinearly mapped cubical elements in the case of 0-forms in two and three dimensions in [5] and [18], respectively. For the case of (n − 1)-forms, i.e., H (div) finite elements, such conditions were given in n = 2 dimensions in [6], in the lowest order case r = 1 in three dimensions in [16], and for general r in three dimensions in [9]. Necessary and sufficient conditions for 1-forms in three dimensions (H (curl) elements) were also given in the lowest order case in [16], and a closely related problem for H (curl) elements in 3D was studied in [10].…”

“…The results of [5] were extended to three dimensions in [18]. The case of vector (H (div)) finite elements in two dimensions was studied in [6]. In that case the transformation of the shape functions must be done through the Piola transform and it turns out the same issue arises, but even more strongly, the requirement on the reference shape functions needed to ensure optimal order approximation being more stringent.…”

HexahedralH(div) andH(curl) finite elements

“…In the series of papers [1][2][3], the approximation properties of finite elements on quadrilateral meshes in R 2 was considered. Papers [1,2] examine the case of scalar approximation and paper [3] the approximation of vector functions in the space H(div; Ω) = {v ∈ L 2 (Ω) : div v ∈ L 2 (Ω)} (see also [10]).…”

Stabilized continuous and discontinuous Galerkin techniques for Darcy flow

“…This makes the implementation involved, specially for three dimensional problems. On the other hand, this FE space experiments a loss of accuracy in some meshes (see [2]). Finally, when dealing with a coupled StokesDarcy problem it is hard to find mixed FE methods that are stable for both the Stokes and the Darcy problems (see [1,23]).…”