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# Polytopal templates for the formulation of semi-continuous vectorial finite elements of arbitrary order

Oct 2022

The Hilbert spaces $H(\mathrm{curl})$ and $H(\mathrm{div})$ are needed forvariational problems formulated in the context of the de Rham complex in orderto guarantee well-posedness. Consequently, the construction of conformingsubspaces is a crucial step in the formulation of viable numerical solutions.Alternatively to the standard definition of a finite element as per Ciarlet,given by the triplet of a domain, a polynomial space and degrees of freedom,this work aims to introduce a novel, simple method of directly constructingsemi-continuous vectorial base functions on the reference element via polytopaltemplates and an underlying $H^1$-conforming polynomial subspace. The basefunctions are then mapped from the reference element to the element in thephysical domain via consistent Piola transformations. The method is defined insuch a way, that the underlying $H^1$-conforming subspace can be chosenindependently, thus allowing for constructions of arbitrary polynomial order.The base functions arise by multiplication of the basis with template vectorsdefined for each polytope of the reference element. We prove a unisolventconstruction of N\'ed\'elec elements of the first and second type,Brezzi-Douglas-Marini elements, and Raviart-Thomas elements. An application forthe method is demonstrated with two examples in the relaxed micromorphic model

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