## Wayne State University Dissertations

WSU Access

January 2021

Dissertation

Ph.D.

Mathematics

Zhimin Z. Zhang

#### Abstract

In this dissertation, we discuss the conforming finite element discretization of high-order equations involving operators such as $(\curl\curl)^2$, $\grad\Delta\div$, and $-\curl\Delta\curl$. These operators appear in various models, such as continuum mechanics, inverse electromagnetic scattering theory, magnetohydrodynamics, and linear elasticity. Naively discretizing these operators and their corresponding eigenvalue problems using the existing $H^2$-conforming element would lead to spurious solutions in certain cases. Therefore, it is desirable to design conforming finite elements for equations containing these high-order differential operators.

The $\curl\curl$-conformity or $\grad\curl$-conformity requires that the tangential component of $\curl \bm u_h$ is continuous. Recall that the N\'ed\'elec element requires only the continuity of the tangential component of $\bm u_h$.Due to the continuity requirement and the naturally divergence-free property of the curl operator, it is challenging to construct $\grad\curl$-conforming elements. We start from the two dimensional case, where $\curl\bm u_h$ is a scalar. Our previous construction \cite{WZZelement} is based on the existing polynomial spaces $Q_{k-1,k}\times Q_{k,k-1}$ and $\mathcal R_k$. The restriction of $k\geq 4$ for a triangular element or $k\geq 3$ for a rectangular element has to be imposed since an interior bubble should be included in the shape function space of $\curl\bm u_h$, and hence the simplest triangular or rectangular element has 24 degrees of freedom. To reduce the degrees of freedom, we resort to the discrete de Rham complex to construct elements. The Poincar\'e operator enables us to tailor the shape function space to our needs (not necessarily the existing polynomial spaces). As a result, we construct a finite element complex, which contains three families of $\grad\curl$-conforming elements without the restriction on polynomial degrees. One of three families is consistent with the previous construction in high-order cases. The lowest-order triangular and rectangular finite elements have only 6 and 8 degrees of freedom, respectively.

Unlike the two-dimensional case, $\curl\bm u_h$ in three dimensions should be a divergence-free vector in the space $H^1\otimes\mathbb V$, which relates the $\curl\Delta\curl$ problems to the Stokes problem. However, it is challenging to construct an inf-sup stable finite element Stokes pair that preserves the divergence-free condition at the discrete level.Neilan \cite{neilan2015discrete} constructed a finite element complex that includes a stable Stokes pair and an $H^1(\curl)$-conforming element on tetrahedral meshes. Based on the same Stokes pair, we construct a finite element complex which contains three families of $\grad\curl$-conforming elements. Compared to the $H^1(\curl)$-conforming elements \cite{neilan2015discrete} which have at least 360 DOFs, our $\grad\curl$-conforming elements have weaker continuity ($\bm u_h$ is in $H(\curl)$ instead of $H^1\otimes\mathbb V$) and thus fewer degrees of freedom. However, our elements still have at least 279 degrees of freedom. Recently, Guzm\'an and Neilan stabilized the lowest-order three dimensional Scott-Vogelius pair by enriching the velocity space with modified Bernardi-Raugel bubbles \cite{guzman2018inf}, which inspires us to use it to construct $\grad\curl$-conforming elements with fewer degrees of freedom. To obtain a family of elements, we first generalize their construction to an arbitrary order by enriching the velocity space with modified face or/and interior bubbles. Then we construct the whole finite element complex which contains three families of $\grad\curl$-conforming elements on tetrahedral meshes. The lowest-order element has only 18 degrees of freedom.

The $\grad\div$-conformity requires that the normal component and divergent of the finite element function $\bm u_h$ are continuous. Since $\div\bm u_h$ is a scalar, the construction of the finite element complex and the $\grad\div$-conforming elements is similar to the $\grad\curl$ elements in two dimensions. The simplest tetrahedral and cubical elements have only 8 and 14 degrees of freedom, respectively.

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