Smoothers Based on Nonoverlapping Domain Decomposition Methods for $H(\mathbf{curl})$ Problems: A Numerical Study

Let $\Omega$ be a domain that is bounded in $\mathds{R}^{3}$. We will work with the $H_{0}(\mathbf{curl};\Omega)$ Hilbert space which consists of vector fields in the space $(L^{2}(\Omega))^{3}$ with curl also in $(L^{2}(\Omega))^{3}$ and vanishing tangential components on the boundary $\partial\Omega$ (cf. [12]). We will consider the following variational problem: Find $\bm{u}\in H_{0}(\mathbf{curl};\Omega)$ such that

where

Here, $(\cdot,\cdot)$ is the standard inner product on $[L_{2}(\Omega)]^{3}$ and we assume that $\alpha$ is nonnegative and $\beta$ is strictly positive. We also assume that $\bm{f}$ is a square integrable vector field on $\Omega$, i.e., $\bm{f}\in(L^{2}(\Omega))^{3}$. In this manuscript, we will provide a multigrid framework for solving our model problem (1).

The model problem (1) is originated from the applications in Maxwell’s equation; see [21]. Relevant fast solvers, such as multigrid methods and domain decomposition methods, for problems connected with $H(\mathbf{curl})$ have been discussed in [14, 15, 17, 4, 24, 25, 9, 10, 16, 11, 26, 18].

It is well-known that the traditional smoothers for solving the scalar elliptic problems do not work well for vector field problems related to $H(\mathbf{curl})$ and $H(\mathrm{div})$; see [8]. Hence, a special smoothing technique is necessary for vector field problems. Function space splitting methods based on Helmholtz type decompositions pioneered by Hiptmair [14, 13] and Hiptmair and Xu [15] have been considered. In [2, 4, 3], an overlapping type domain decomposition preconditioner has been applied. We also note that the author and Brenner considered smoothers associated with nonoverlapping type domain decomposition methods for $H(\mathrm{div})$ problems in [5, 6].

In this paper, we propose a $V$–cycle multigrid method with nonoverlapping domain decomposition smoothers and mainly consider the numerical study that is not covered by theories in [23]. In [23], the author provided the convergence analysis with the assumptions, i.e., convex domain and constant material parameters in (1). We test our method with less strict conditions, e.g., jump coefficients for $\alpha$ and $\beta$, nonconvex domain. We note that our multigrid method is an $H(\mathbf{curl})$ counterpart of the method in [5, 6] and a nonoverlapping alternative of the method in [4], which requires less computational costs when applying the smoother.

The remainder of this paper is structured as follows. In Section 2, the standard way to discretize our model problem using the hexahedral Nédélec element is introduced. We next present our $V$–cycle multigrid algorithm in Section 3. Finally, we provide numerical experiments in Section 4.

We first consider a triangulation $\mathcal{T}_{h}$ of $\Omega$ into hexahedral elements. The lowest order hexahedral Nédélec element [20, 22] has the following form:

on each hexahedral mesh, where the $a_{i}$’s, $b_{i}$’s and $c_{i}$’s are constants. We note that the twelve degrees of freedom can be completely recovered by the average tangential component on each edge of the element. Using the finite elements, we obtain the following discretized problem for (1): Find $\bm{u}_{h}\in W_{h}$ such that

where $W_{h}$ is the Nédélec finite element space of the lowest order.

We define the operator $A:W_{h}\rightarrow W_{h}^{\prime}$ in the following way:

Here, $\langle\cdot,\cdot,\rangle$ is the canonical bilinear form on $W_{h}^{\prime}\times W_{h}$. We then have the following discrete problem:

where $f_{h}\in W_{h}^{\prime}$ defined by

In a multigrid setting, we construct a nested sequence of triangulations, $\mathcal{T}_{0},\mathcal{T}_{1},\cdots,$ starting with the initial triangulation $\mathcal{T}_{0}$ consisting of few hexahedral elements. We assume that $\mathcal{T}_{k}$ is obtained by uniform subdivision from $\mathcal{T}_{k-1}$. We then define $W_{k}$ which is the lowest order Nédélec space related to the $k^{\rm th}$ level mesh and the corresponding discrete problem: Find $\bm{u}_{k}\in W_{k}$ such that

Here, $f_{k}$ is defined by

In order for solving the discrete problem (6) using multigrid methods, we need two essential ingredients, intergrid transfer operators and smoothers. We note that everything else can be constructed in a standard way.

Since we are dealing with the nested finite element spaces, the natural injection can be used as the coarse-to-fine operator $I_{k-1}^{k}:W_{k-1}\longrightarrow W_{k}$. The fine-to-coarse operator $I_{k}^{k-1}:W_{k}^{\prime}\longrightarrow W_{k-1}^{\prime}$ is then defined by

We now focus on the missing piece, smoother. As we are interested in designing nonoverlapping type domain decomposition smoothers, we borrow the standard two level domain decomposition framework, i.e., the coarse level and the fine level that are associated with $\mathcal{T}_{k-1}$ and $\mathcal{T}_{k}$, respectively.

Before we construct the smoothers, we set up notations for the geometric substructures. Let $\mathcal{E}_{k-1}$, $\mathcal{F}_{k-1}$, and $\mathcal{V}_{k-1}$ be the sets of interior edges, interior faces, and interior vertices of the triangulation $\mathcal{T}_{k-1}$, respectively.

We first consider the interior space. Given any coarse element $T\in\mathcal{T}_{k-1}$, let us define the subspace $W_{k}^{T}$ by

Let $J_{T}$ denote the natural injection from $W_{k}^{T}$ into $W_{k}$. The operator $A_{T}:W_{k}^{T}\rightarrow\left(W_{k}^{T}\right)^{\prime}$ is constructed by

For a coarse edge $E\in\mathcal{E}_{k-1}$, there are four coarse elements, $T_{E}^{i},i=1,2,3,4$, in $\mathcal{T}_{k-1}$ and four coarse faces, $F_{E}^{i},i=1,2,3,4$, in $\mathcal{F}_{k-1}$, that are sharing $E$. The edge space $W_{k}^{E}$ of $W_{k}$ is defined as follow:

Let $J_{E}:W_{k}^{E}\rightarrow W_{k}$ be the natural injection and the operator $A_{E}:W_{k}^{E}\rightarrow\left(W_{k}^{E}\right)^{\prime}$ be defined by

Finally, we define the vertex space $W_{k}^{P}$ of $W_{k}$. For each coarse vertex $P\in\mathcal{V}_{k-1}$, there are eight elements, $T_{P}^{i},i=1,\cdots,8$, in $\mathcal{T}_{k-1}$, twelve faces, $F_{P}^{j},j=1,\cdots,12$, in $\mathcal{F}_{k-1}$, and six edges, $E_{P}^{l},l=1,\cdots,6$, in $\mathcal{E}_{k-1}$, that have the point $P$ in common. We define the vertex space $W_{k}^{P}$ by

The natural injection $J_{P}:W_{k}^{P}\rightarrow W_{k}$ and the operator $A_{P}$ are obtained by a similar way to $J_{E}$ and $A_{E}$, respectively.

We now define two smoothers, the edge-based and the vertex-based preconditioners. The edge-based smoother $M_{E,k}^{-1}$ is constructed as follow:

Similarly, the vertex-based smoother $M_{P,k}^{-1}$ is obtained by

Here, $\eta_{E}$ and $\eta_{P}$ are damping factors and $J_{T}^{t}:W_{k}^{\prime}\rightarrow\left(W_{k}^{T}\right)^{\prime}$, $J_{E}^{t}:W_{k}^{\prime}\rightarrow\left(W_{k}^{E}\right)^{\prime}$, and $J_{P}^{t}:W_{k}^{\prime}\rightarrow\left(W_{k}^{P}\right)^{\prime}$ are the transposes of $J_{T}$, $J_{E}$, and $J_{P}$, respectively. We can decide the damping factors such that

where $\rho\left(M_{E,k}^{-1}A_{k}\right)$ and $\rho\left(M_{P,k}^{-1}A_{k}\right)$ are the spectral radii of $M_{E,k}^{-1}A_{k}$ and $M_{P,k}^{-1}A_{k}$, respectively. We note that the conditions in (15) are satisfied if $\eta_{E}\leq 1/12$ and $\eta_{P}\leq 1/8$.

Putting all together, we can completely determine the multigrid framework. The output $MGV\left(k,g,\bm{z}_{0},m\right)$ of the $k^{\rm th}$ level (symmetric) multigrid $V$-cycle algorithm for $A_{k}\bm{z}=g$, with initial guess $\bm{z}_{0}\in W_{k}$ and $m$ smoothing steps, is defined as follows:

For $k=0$, the result is obtained by using a direct solver:

In order to check the efficiency of the algorithm, we consider the following error propagation operator:

The operator $\mathbb{E}_{k}^{m}$ is affiliated with the error after one $k^{\rm th}$ multigrid sweep with $m$ smoothing steps. For more detail, see [7, Chapter 6].

We note that a part of the finite element discretizations has been implemented with the MFEM library [1, 19]. The codes used for the experiments are available at the repository https://github.com/duksoon-open/MG_ND.

This work was supported by the National Research Foundation of Korea (NRF) grant funded by the Korea government (MSIT) (No. 2020R1F1A1A01072168).