论文标题
用于制定任意顺序半连续矢量有限元的多面模板
Polytopal templates for the formulation of semi-continuous vectorial finite elements of arbitrary order
论文作者
论文摘要
Hilbert Spaces $ H（\ Mathrm {curl}）$和$ h（\ Mathrm {div}）$是在De Rham Complex的上下文中提出的各种问题，以保证适当的问题。因此，构造子空间的构建是制定可行数值溶液的关键步骤。另外，根据域的三胞胎，多项式空间和自由度的标准定义，这项工作旨在引入一种新颖的，简单的方法，即直接构建半连续矢量基础的基础通过多型元素来构建参考元素，该元素通过多层模板和下属的$ h^1 $ h^1 $ h^1 $ -Onconformentormial subspace。然后，通过一致的Piola变换将基本函数从参考元素映射到物理域中的元素。该方法的定义是这样的，即可以独立选择基础$ h^1 $ - 包含子空间，从而允许任意多项式顺序的构造。基本函数是通过将基础与为参考元素的每个多室定义的模板向量乘法来产生的。我们证明了第一类和第二类的nédélec元素Brezzi-Douglas-Marini Elements和Raviart-Thomas元素的无通用结构。在松弛的微态模型中使用两个示例证明了该方法的应用
The Hilbert spaces $H(\mathrm{curl})$ and $H(\mathrm{div})$ are needed for variational problems formulated in the context of the de Rham complex in order to guarantee well-posedness. Consequently, the construction of conforming subspaces 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 constructing semi-continuous vectorial base functions on the reference element via polytopal templates and an underlying $H^1$-conforming polynomial subspace. The base functions are then mapped from the reference element to the element in the physical domain via consistent Piola transformations. The method is defined in such a way, that the underlying $H^1$-conforming subspace can be chosen independently, thus allowing for constructions of arbitrary polynomial order. The base functions arise by multiplication of the basis with template vectors defined for each polytope of the reference element. We prove a unisolvent construction of Nédélec elements of the first and second type, Brezzi-Douglas-Marini elements, and Raviart-Thomas elements. An application for the method is demonstrated with two examples in the relaxed micromorphic model
