Abstract
In mixed finite element approximations of Hodge Laplace problems associated with the de Rham complex, the exterior derivative operators are computed exactly, so the spatial locality is preserved. However, the numerical approximations of the associated coderivatives are nonlocal and can be regarded as an undesired effect of standard mixed methods. For numerical methods with local coderivatives, a perturbation of low order mixed methods in the sense of variational crimes has been developed for simplicial and cubical meshes. In this paper we extend the low order method to all high orders on cubical meshes using a new family of finite element differential forms on cubical meshes. The key theoretical contribution is a generalization of the linear degree, in the construction of the serendipity family of differential forms, and this generalization is essential in the unisolvency proof of the new family of finite element differential forms.
Highlights
In this paper we consider finite element methods for the Hodge Laplace problems of the de Rham complex where both the approximation of the exterior derivative and the associated coderivative are spatially local operators
The locality of the coderivative operator is not fulfilled in standard mixed methods for these problems
Pursuing numerical methods with local coderivative is related to the development of various numerical methods for the Darcy flow problems
Summary
In this paper we consider finite element methods for the Hodge Laplace problems of the de Rham complex where both the approximation of the exterior derivative and the associated coderivative are spatially local operators. The locality of the coderivative operator is not fulfilled in standard mixed methods for these problems (cf [8, 9]). Pursuing numerical methods with local coderivative is related to the development of various numerical methods for the Darcy flow problems. To discuss this local coderivative property in a more familiar context, let us consider the mixed form of a model second-order elliptic equation with the vanishing Dirichlet boundary condition: Find (σ, u) ∈ H(div, Ω) × L2(Ω) such that (1.1). Key words and phrases. perturbed mixed methods, local constitutive laws
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