Decoupled finite element methods for a fourth-order exterior differential equation

Let $\Omega\subset\mathbb{R}^{d}$ ($d\geq 2$) be a bounded and contractible polytope. Given integer $m$ and a bounded domain $D\subset\mathbb{R}^{d}$, let $H^{m}(D)$ be the standard Sobolev space of functions on $D$. The corresponding norm and semi-norm are denoted by $\|\cdot\|_{m,D}$ and $|\cdot|_{m,D}$, respectively. Set $L^{2}(D)=H^{0}(D)$ with the usual inner product $(\cdot,\cdot)_{D}$. We denote $H_{0}^{m}(D)$ as the closure of $C_{0}^{\infty}(D)$ with respect to the norm $\|\cdot\|_{m,D}$. In case $D$ is $\Omega$, we abbreviate $\|\cdot\|_{m,D}$, $|\cdot|_{m,D}$ and $(\cdot,\cdot)_{D}$ as $\|\cdot\|_{m}$, $|\cdot|_{m}$ and $(\cdot,\cdot)$, respectively. We also abbreviate $\|\cdot\|_{0,D}$ and $\|\cdot\|_{0}$ by $\|\cdot\|_{D}$ and $\|\cdot\|$, respectively. For integer $k\geq 0$, let $\mathbb{P}_{k}(D)$ represent the space of all polynomials in $D$ with the total degree no more than $k$. Set $\mathbb{P}_{k}(D)=\{0\}$ for $k<0$. Let $L_{0}^{2}(D)$ be the space of functions in $L^{2}(D)$ with vanishing integral average values. For a space $B(D)$ defined on $D$, let $B(D;\mathbb{R}^{d}):=B(D)\otimes\mathbb{R}^{d}$ be its vector version. Denote by $h_{D}$ the diameter of $D$. We use $\boldsymbol{n}_{\partial D}$ to denote the unit outward normal vector of $\partial D$, which will be abbreviated as $\boldsymbol{n}$ if not causing any confusion.

We mainly follow the notation set in [4, 6, 7]. For a $d$-dimensional vector space $V$ and a nonnegative integer $j$, define the space ${\rm Alt}^{j}V$ as the space of all skew-symmetric $j$-linear forms. For a multilinear $j$-form $\omega$, its skew-symmetric part

is an alternating form, where $\mathfrak{S}_{j}$ is the symmetric group of all permutations of the set $\{1,\ldots,j\}$, and $\operatorname{sign}(\sigma)$ denotes the signature of the permutation $\sigma$. The exterior product or wedge product of $\omega\in{\rm Alt}^{i}V$ and $\eta\in{\rm Alt}^{j}V$ is given by

where $\otimes$ is the tensor product. It satisfies the anticommutativity law

An inner product on $V$ induces an inner product on ${\rm Alt}^{j}V$ as follows

where the sum is over increasing sequences $\sigma:\{1,\ldots,j\}\to\{1,\ldots,d\}$ and $\{e_{1},\ldots,e_{d}\}$ is any orthonormal basis. If the space $V$ is endowed with an orientation by assigning a positive orientation to some particular ordered basis, the volume form $\textsf{vol}\in{\rm Alt}^{d}V$ is the unique $d$-form characterized by $\textsf{vol}(e_{1},\ldots,e_{d})=1$ for any positively oriented ordered orthonormal basis $e_{1},\ldots,e_{d}$. The Hodge star operator is an isometry of ${\rm Alt}^{j}V$ onto ${\rm Alt}^{d-j}V$ given by

We have

A differential $j$-form $\omega$ is a section of the $j$-alternating bundle, i.e., a map which assigns to each $x\in D$ an element $\omega_{x}\in{\rm Alt}^{j}T_{x}D$, where $T_{x}D$ denotes the tangent space to $D$ at $x$. For a space $B(D)$ defined on $D$, let $B\Lambda^{j}(D)$ be the space of all $j$-forms whose coefficient function belongs to $B(D)$. Notice that $B\Lambda^{0}(D)=B(D)$. The norm $\|\cdot\|_{m,D}$ and semi-norm $|\cdot|_{m,D}$ are extended to space $H^{m}\Lambda^{j}(D)$, and inner product $(\cdot,\cdot)_{D}$ to space $L^{2}\Lambda^{j}(D)$. Define

where $\,{\rm d}$ is the exterior derivative, and the codifferential operator $\delta$ is defined as

Denote by $H_{0}\Lambda^{j}(D)$ the subspace of $H\Lambda^{j}(D)$ with vanishing trace. We can identify $H_{0}^{m}\Lambda^{d}(D)$ and $H_{0}\Lambda^{d-1}(D)$ as $H_{0}^{m}(D)\cap L_{0}^{2}(D)$ and $H_{0}(\operatorname{div},D)$, respectively. The spaces $L^{2}\Lambda^{j}(D)$ and $H^{*}\Lambda^{j}(D)$ are to be interpreted as the trivial space $\{0\}$ for $j\geq d+1$, whereas $\mathbb{P}_{k}\Lambda^{d+1}(D)$ is to be interpreted as $\mathbb{R}$. We will abbreviate the Sobolev space $B(D)$ as $B$ when $D=\Omega$ if not causing any confusion.

For sufficient smooth $(j-1)$-form $\omega$ and $j$-form $\mu$, it holds the integration by parts

For $(d-1)$-dimensional face $F$ of $\partial D$, we have

Let $\kappa$ be the Koszul operator mapping $\mathbb{P}_{k-1}\Lambda^{d-j+1}(D)$ to $\mathbb{P}_{k}\Lambda^{d-j}(D)$, then for a contractible domain $D$ it holds the decomposition [6, 4]

This implies $\mathbb{P}_{0}\Lambda^{j}(D)=\delta\mathbb{P}_{1}\Lambda^{j+1}(D)$. Set $\mathbb{P}_{k}^{-}\Lambda^{j}(D):=\,{\rm d}\mathbb{P}_{k}\Lambda^{j-1}(D)\oplus\kappa\mathbb{P}_{k-1}\Lambda^{j+1}(D)$. We have $\mathbb{P}_{k}^{-}\Lambda^{0}(D)=\mathbb{P}_{k}\Lambda^{0}(D)$ and $\mathbb{P}_{k}^{-}\Lambda^{d}(D)=\mathbb{P}_{k-1}\Lambda^{d}(D)$. For a $0$-form $v$, $\,{\rm d}v=\nabla v\cdot\,{\rm d}\boldsymbol{x}$ with $\,{\rm d}\boldsymbol{x}=(\,{\rm d}x_{1},\ldots,\,{\rm d}x_{d})^{\intercal}$. If not causing any confusion, we will use $\nabla v$ to represent $\,{\rm d}v$ for $0$-form $v$.

Let $\{\mathcal{T}_{h}\}_{h>0}$ be a regular family of simplicial meshes of $\Omega\subset\mathbb{R}^{d}$, where $h=\max_{T\in\mathcal{T}_{h}}h_{T}$. For $\ell=0,1,\ldots,d-1$, denote by $\Delta_{\ell}(\mathcal{T}_{h})$ and $\Delta_{\ell}(\mathring{\mathcal{T}}_{h})$ the set of all subsimplices and all interior subsimplices of dimension $\ell$ in the partition $\mathcal{T}_{h}$, respectively. For a simplex $T$, we let $\Delta_{\ell}(T)$ denote the set of subsimplices of dimension $\ell$. For a subsimplex $f$ of $\mathcal{T}_{h}$, let $\mathcal{T}_{f}$ be the set of all simplices in $\mathcal{T}_{h}$ sharing $f$. Denote by $\omega_{T}$ the union of all the simplices in the set $\{\mathcal{T}_{\texttt{v}}\}_{\texttt{v}\in\Delta_{0}(T)}$. For integer $k\geq 0$, define $\mathbb{P}_{k}(\mathcal{T}_{h}):=\mathbb{P}_{k}\Lambda^{0}(\mathcal{T}_{h})$ with

Let $Q_{k,T}:L^{2}\Lambda^{j}(T)\rightarrow\mathbb{P}_{k}\Lambda^{j}(T)$ for $T\in\mathcal{T}_{h}$ be the $L^{2}$-orthogonal projection operator.

In this paper, we use “$\lesssim\cdot\cdot\cdot$” to mean that “$\leq C\cdot\cdot\cdot$”, where $C$ is a generic positive constant independent of $h$, which may have different values in different forms. And $A\eqsim B$ equivalents to $A\lesssim B$ and $B\lesssim A$.

Let $0\leq j\leq d-1$. The weak formulation of problem (1) is to find $u\in H_{0}^{\rm Gd}\Lambda^{j}$ and $\lambda\in H_{0}\Lambda^{j-1}/\,{\rm d}H_{0}\Lambda^{j-2}$ such that

We will apply the framework in [12] to decouple the variational formulation (9) into lower order differential equations.

Recall the de Rham complex [17, 6, 4]

which is exact on the contractible domain $\Omega$. Applying the tilde operation in [13], the de Rham complex (10) implies the exact de Rham complex

where $H^{-1}(\delta,\Lambda^{j+1}):=\{\omega\in H^{-1}\Lambda^{j+1}:\delta\omega\in H^{-1}\Lambda^{j}\}$. We understand $H^{-1}(\delta,\Lambda^{d})=L_{0}^{2}(\Omega)$. With the de Rham complex (11), we build up the following commutative diagram

A mixed formulation based on the commutative diagram (12) is to find $(\gamma,u)\in H^{-1}(\delta,\Lambda^{j+1})\times(H_{0}\Lambda^{j}/\,{\rm d}H_{0}\Lambda^{j-1})$ such that

where $(\gamma,\beta)_{-1}:=-\langle\Delta^{-1}\gamma,\beta\rangle_{H_{0}^{1}\Lambda^{j+1}\times H^{-1}\Lambda^{j+1}}=-\langle\gamma,\Delta^{-1}\beta\rangle_{H^{-1}\Lambda^{j+1}\times H_{0}^{1}\Lambda^{j+1}}$. By introducing the new variable $\phi=-\Delta^{-1}\gamma\in H_{0}^{1}\Lambda^{j+1}$, the unfolded formulation is to find $(\gamma,u,\phi)\in H^{-1}(\delta,\Lambda^{j+1})\times(H_{0}\Lambda^{j}/\,{\rm d}H_{0}\Lambda^{j-1})\times H_{0}^{1}\Lambda^{j+1}$ such that

According to the Helmholtz decomposition (14), we can set $\gamma=\,{\rm d}w-\delta p$ and $\beta=\,{\rm d}\chi-\delta q$ with $w,\chi\in H_{0}\Lambda^{j}/\,{\rm d}H_{0}\Lambda^{j-1}$ and $p,q\in L^{2}\Lambda^{j+2}$. Then the formulation (18) is decomposed as follows [12, Section 3.2]: find $u,w\in H_{0}\Lambda^{j}/\,{\rm d}H_{0}\Lambda^{j-1}$, $\phi\in H_{0}^{1}\Lambda^{j+1}$ and $p\in L^{2}\Lambda^{j+2}/\delta H^{*}\Lambda^{j+3}$ such that

We further employ the Lagrange multiplier to deal with the constraint in the quotient spaces $H_{0}\Lambda^{j}/\,{\rm d}H_{0}\Lambda^{j-1}$ and $L^{2}\Lambda^{j+2}/\delta H^{*}\Lambda^{j+3}$, which induces the following variational formulation: find $u,w\in H_{0}\Lambda^{j}$, $\lambda,z\in H_{0}\Lambda^{j-1}/\,{\rm d}H_{0}\Lambda^{j-2}$, $\phi\in H_{0}^{1}\Lambda^{j+1}$, $p\in L^{2}\Lambda^{j+2}$ and $r\in H^{*}\Lambda^{j+3}/\delta H^{*}\Lambda^{j+4}$ such that

The well-posedness of problem (19c)-(19d) is related to complex (20).

Indeed, complex (20) is a distinct representation of the following complex

Thanks to complex (20), we can regard $(\,{\rm d},\delta)$ as a generalized divergence operator, and problem (19c)-(19d) as a generalized Stokes equation.