A BDDC algorithm for the mortar-type rotated $Q_1$ FEM for elliptic problems with discontinuous coefficients

The method of balancing domain decomposition by constraints (BDDC) was first introduced by Dohrmann in [1]. Mandel and Dohrmann restated the method in an abstract manner, and provided its convergence theory in [2]. The BDDC method is closely related to the dual-primal FETI (FETI-DP) method [3], which is one of dual iterative substructuring methods. Each BDDC and FETI-DP method is defined in terms of a set of primal continuity. The primal continuity is enforced across the interface between the subdomains and provides a coarse space component of the preconditioner. In [4], Mandel, Dohrmann, and Tezaur analyzed the relation between the two methods and established the corresponding theory.

In the last decades, the two methods have been widely analyzed and successfully been extended to many different types of partial differential equations. In [3], the two algorithms for elliptic problems were rederived and a brief proof of the main result was given. A BDDC algorithm for mortar finite element was developed in [5], meanwhile, the author also extended the FETI-DP algorithm to elasticity problems and Stokes problems in [6, 7], respectively. These algorithms are based on locally conforming finite element methods, and the coarse space components of the algorithms are related to the cross-points (i.e., corners), which are often noteworthy points in domain decomposition methods (DDMs). Since the cross-points are related to more than two subregions, thus it is not convenient to design the domain decomposition algorithm.

The BDDC method derives from the Neumann-Neumann domain decomposition method (see [8]). The difference is that the BDDC method applies an additive rather than a multiplicative coarse grid correction, and substructure spaces have some constraints which result in non-singular subproblems. Thus we need not modify the bilinear forms on subdomains, and we can solve each subproblem and coarse problem in parallel.

The rotated $Q_1$ element is an important nonconforming element. It was introduced by Rannacher and Turek in [9] for stokes equations originally, and it is the simplest example of a divergence-stable nonconforming element on quadrilaterals. Since its degree of freedom is integral average on element edge which is not related to the corners, and each degree of freedom on subdomain interfaces is only included in two neighboring subdomains, so it is easy to design the BDDC algorithm.

The mortar technique was introduced in [10]. This method is nonconforming domain decomposition methods with nonoverlapping subdomains. The meshes on different subdomains need not align across subdomain interfaces, and the matching of discretizations on adjacent subdomains is only enforced weakly. This offers the advantages of freely choosing highly varying mesh sizes on different subdomains and is very promising to approximate the problems with abruptly changing diffusion coefficients or local anisotropic.

In this paper, we study the BDDC algorithm for the mortar-type rotated $Q_1$ element for the second order elliptic problem with discontinuous coefficients, where the discontinuities lie only along the subdomain interfaces. Following the technique in [11], we construct an auxiliary discrete space and build our BDDC algorithm on an equivalent auxiliary problem. This approach overcomes the difficulty caused by the mortar condition and simplifies the implementation of the BDDC preconditioning iteration. Furthermore, since the rotated $Q_1$ element is not related to the subdomain’s vertices, we can complete our theoretical analysis conveniently. It is proved that the condition number of the preconditioned operator is independent of the jumps of the coefficients and only depends logarithmically on the ratio between the subdomain size and mesh size. Numerical experiments are presented to confirm our theoretical analysis.

The rest of this paper is organized as follows: in Section 2, we introduce the model problem and the auxiliary problem. Section 3 gives the BDDC algorithm and proposes the BDDC preconditioner. Several technical tools are presented and analyzed in Section 4. In Section 5, we give the proof of the main result. Last section provides numerical experiments. For convenience, the symbols ⪯, ⪰ and ≍ are used, and $x_1⪯y_1$, $x_2⪰y_2$, and $x_3\asymp y_3$ mean that $x_1\le C_1{y}_1$, $x_2\ge C_2{y}_2$, and $c_3{x}_3\le y_3\le C_3{y}_3$ for some constants $C_1$, $C_2$, $C_3$, and $c_3$ that are independent of discontinuous coefficients and mesh size.

$f\in L^2(\mathrm{\Omega })$, the coefficients ${\rho }_i(x)$ ($i=1,\dots ,N$) are piecewise positive constants over ${\mathrm{\Omega }}_i$ ($i=1,\dots ,N$).

For simplicity, we only consider the geometrically conforming case, i.e., the intersection between the closure of two different subdomains is empty, or a vertex, or an edge. The subdomains ${\{{\mathrm{\Omega }}_i\}}_{i=1}^N$ together form a coarse partition ${\mathcal{T}}_H(\mathrm{\Omega })$, we denote the diameter of each ${\mathrm{\Omega }}_i$ by $H_i$. Let ${\mathcal{T}}_h({\mathrm{\Omega }}_i)$ be a quasi-uniform partition with the mesh size $O(h_i)$, made up of shape regular rectangles in ${\mathrm{\Omega }}_i$. The resulted partition can be nonmatched across adjacent subdomain interfaces. We denote the sets of edges of the triangulation ${\mathcal{T}}_h({\mathrm{\Omega }}_i)$ in ${\mathrm{\Omega }}_i$ and $\partial {\mathrm{\Omega }}_i$ by ${\mathrm{\Omega }}_{i,h}^e$, $\partial {\mathrm{\Omega }}_{i,h}^e$ respectively, and let ${\mathrm{\Omega }}_{i,h}$, $\partial {\mathrm{\Omega }}_{i,h}$ be the sets of vertices of the triangulation ${\mathcal{T}}_h({\mathrm{\Omega }}_i)$ that are in ${\overline{\mathrm{\Omega }}}_i$, $\partial {\overline{\mathrm{\Omega }}}_i$ respectively.

We denote ${\mathrm{\Gamma }}_{ij}$ the common open edge of ${\mathrm{\Omega }}_i$ and ${\mathrm{\Omega }}_j$, and let $\mathrm{\Gamma }={\bigcup }_{ij}{\mathrm{\Gamma }}_{ij}$. Each ${\mathrm{\Gamma }}_{ij}$ can be regarded as two sides corresponding to the two subdomains ${\mathrm{\Omega }}_i$ and ${\mathrm{\Omega }}_j$. We define one of the sides of ${\mathrm{\Gamma }}_{ij}$ as mortar denoted by ${\gamma }_{m,i}$ and the other one as nonmortar denoted by ${\delta }_{m,j}$, here m represents the indexing of ${\mathrm{\Gamma }}_{ij}$ (see Figure 1). We assume that: (1) the mortar for ${\gamma }_{m,i}={\delta }_{m,j}={\mathrm{\Gamma }}_{ij}$ is chosen by the condition ${\rho }_j⪯{\rho }_i$; (2) there is at least one subdomain which has two mortar sides associated with each cross point; (3) $h_i⪯h_j$, i.e., $h_i/h_j$ is bounded. The first condition used in choosing mortar sides is essential (see the numerical tests in [12]). The last condition is technical but not essential for the convergence analysis. Along each ${\mathrm{\Gamma }}_{ij}$, there are two independent and different 1-D meshes which are denoted by ${\mathcal{T}}_h^i({\gamma }_{m,i})$ and ${\mathcal{T}}_h^j({\delta }_{m,j})$. For each nonmortar side ${\delta }_{m,j}={\mathrm{\Gamma }}_{ij}$, we denote by $M^{h_j}({\delta }_{m,j})\subset L^2({\mathrm{\Gamma }}_{ij})$ an auxiliary test space whose functions are piecewise constant on ${\mathcal{T}}_h^j({\delta }_{m,j})$. We denote by $Q_m$ the $L^2$-orthogonal projection from the $L^2({\mathrm{\Gamma }}_{ij})$ space to the $M^{h_j}({\delta }_{m,j})$ space.

It can easily be shown that $a_h(\cdot ,\cdot )$ is positive definite on ${\mathcal{V}}_h$, which yields the existence and uniqueness of the discrete solution. The error estimate between the discrete and the continuous solution is discussed in [13].

Since the mortar condition depends on both the degrees of freedom on the interfaces and the ones near the interfaces, it is difficult to construct a preconditioner directly for (2.3). To overcome this difficulty, we introduce a new discrete space and an auxiliary problem which is equivalent to problem (2.3).

For each $v\in {\mathcal{V}}_h$, we define an element $\tilde{v}={\prod }_{i=1}^N{\tilde{v}}_i\in X_h(\mathrm{\Omega })$ that satisfies the following conditions:

where $\overline{v}\in L^2({\gamma }_{m,i})$ is a piecewise constant function on elements of ${\mathcal{T}}_h^i({\gamma }_{m,i})$ such that $\overline{v}{|}_e=\frac{1}{|e|}{\int }_e{v}_i{|}_{{\gamma }_{m,i}}ds$ for any $e\in {\mathcal{T}}_h^i({\gamma }_{m,i})$. Note that the average value of $\tilde{v}$ on $e\in {\mathcal{T}}_h^j({\delta }_{m,j})$ can be calculated by (2.5).

For the two related spaces ${\mathcal{V}}_h$ and ${\tilde{\mathcal{V}}}_h$, we have the following result.

Lemma 2.1 ([12])

From the above lemma, we only need to construct a preconditioner for the operator ${\tilde{A}}_h$.

In this section, we introduce our BDDC preconditioner for problem (2.7) and describe the BDDC algorithm.

here $X_h^0({\mathrm{\Omega }}_i)=\{v\in X_h({\mathrm{\Omega }}_i):{\int }_{e}vds=0,\mathrm{\forall }e\in \partial {\mathrm{\Omega }}_{i,h}^e\}$. Let $X_h(\partial {\mathrm{\Omega }}_i)={\mathcal{H}}_i(X_h({\mathrm{\Omega }}_i))$. We define ℋ as a corresponding piecewise harmonic operator on the auxiliary space ${\tilde{\mathcal{V}}}_h$ by $\mathcal{H}{|}_{{\mathrm{\Omega }}_i}={\mathcal{H}}_i$.

We define a space ${\tilde{X}}_h(\mathrm{\Gamma })=\{v\in {\prod }_{i=1}^N{X}_h(\partial {\mathrm{\Omega }}_i):{\int }_{{\gamma }_{m,i}}v{|}_{{\mathrm{\Omega }}_i}ds={\int }_{{\delta }_{m,j}}v{|}_{{\mathrm{\Omega }}_j}ds,\mathrm{\forall }{\gamma }_{m,i}={\delta }_{m,j}\subset \mathrm{\Gamma }\}$. The space ${\tilde{X}}_h(\mathrm{\Gamma })$ is between ${\tilde{\mathcal{V}}}_h(\mathrm{\Gamma })$ and ${\prod }_{i=1}^N{X}_h(\partial {\mathrm{\Omega }}_i)$, and our BDDC preconditioner is mainly constructed on this space.

As we know, the technical aspect in DDMs is that the preconditioner includes a coarse problem which can enhance the convergence. In view of the characteristic of the space ${\tilde{X}}_h(\mathrm{\Gamma })$, we select the standard coarse space ${\mathcal{V}}_H(\mathrm{\Omega })$ which is the rotated $Q_1$ finite element space associated with the coarse partition ${\mathcal{T}}_H(\mathrm{\Omega })$, and it satisfies primal constraints on subdomain interfaces.

Denote ${\tilde{\mathcal{V}}}_{\mathrm{\Delta }}(\mathrm{\Gamma })={\prod }_{i=1}^N{\tilde{\mathcal{V}}}_{\mathrm{\Delta }}({\mathrm{\Gamma }}_i)$. The coarse space and the product space ${\tilde{\mathcal{V}}}_{\mathrm{\Delta }}(\mathrm{\Gamma })$ play an important role in the description and analysis of our iterative method.

Define an extension operator $R_i^T:X_h(\partial {\mathrm{\Omega }}_i)\to {\tilde{\mathcal{V}}}_h(\mathrm{\Gamma })$ as

We also need to define another prolongation operator $E_i:X_h({\mathrm{\Omega }}_i)\to {\tilde{\mathcal{V}}}_h$ as follows:

In what follows, we describe our BDDC preconditioning algorithm, we apply the basic framework of additive Schwarz method (or parallel subspace correction method [14]). From the decomposition (3.1), we only need to choose appropriate subspace solvers.

Remark 3.1 The bilinear form on the coarse space can be different from that on substructure space, here we only use the exact solvers. On each subdomain, we avoid the possible singularity of local subproblem and we need not modify the bilinear forms.

We have the following main result.

where $H/h={max}_i(H_i/h_i)$.

In this section we state and prove a few technical lemmas necessary for the proof of Theorem 3.1. Our theoretical analysis is based on the substructuring theory of conforming elements.

For the operator $I_{\mathcal{E}}^0$, we have the following result.

Lemma 4.1 ([15])

Remark 4.1 The above lemma is related to vertex-edge-face arguments in substructuring methods, in view of the characteristic for the rotated $Q_1$ element, here the results only concern the inequalities for faces.

Let $V^{h/2}({\mathrm{\Omega }}_i)$ be the conforming element space of bilinear continuous functions on the partition ${\mathcal{T}}_{h/2}({\mathrm{\Omega }}_i)$ which is constructed by joining the midpoints of the edges of elements of ${\mathcal{T}}_h({\mathrm{\Omega }}_i)$. We now introduce a local equivalence map ${\mathcal{M}}_i:X_h({\mathrm{\Omega }}_i)\to V^{h/2}({\mathrm{\Omega }}_i)$ as follows (cf. [13]).

Definition 4.2 Given $v\in X_h({\mathrm{\Omega }}_i)$, we define ${\mathcal{M}}_{i}v\in V^{h/2}({\mathrm{\Omega }}_i)$ by the values of ${\mathcal{M}}_{i}v$ at the vertices of the partition ${\mathcal{T}}_{h/2}({\mathrm{\Omega }}_i)$.

where the sum is taken over all edges $e_i$ with the common vertex P, $e_i\in \partial E_i$, $E_i\in {\mathcal{T}}_h({\mathrm{\Omega }}_i)$.

where $e_l\in \partial E_1\cap \partial {\mathrm{\Omega }}_i$ and $e_r\in \partial E_2\cap \partial {\mathrm{\Omega }}_i$ are the left and right neighbor edges of P, $E_1,E_2\in {\mathcal{T}}_h({\mathrm{\Omega }}_i)$. If P is a vertex of ${\mathrm{\Omega }}_i$, then $E_1=E_2$.

where $u_{ij}\in {\tilde{\mathcal{V}}}_{\mathrm{\Delta }}({\mathrm{\Gamma }}_i)$, and for any $e\in {\mathrm{\Gamma }}_{ij}^e$,${\int }_e{u}_{ij}ds/|e|={\int }_e{u}_{i}ds/|e|$; for any $e\in \partial {\mathrm{\Omega }}_{i,h}^e\setminus {\mathrm{\Gamma }}_{ij}$, ${\int }_e{u}_{ij}ds/|e|=0$.

where ${\mathcal{H}}_i$ is a piecewise bilinear conforming element harmonic operator, and we have used the minimal energy property of discrete harmonic functions. □

In the proof of Theorem 3.1 we use the abstract framework of ASM methods (see [16]), we need to prove three assumptions. Assumption II follows from the standard coloring argument, we only need to prove Assumption I and Assumption III.

First we show the following stability of the decomposition.

Lemma 5.1 (Assumption I)

where we have used the fact ${\sum }_{i=1}^N{R}_i^{T}u=u$, $\mathrm{\forall }u\in {\tilde{\mathcal{V}}}_h(\mathrm{\Gamma })$. Hence $u_i\in {\tilde{\mathcal{V}}}_{\mathrm{\Delta }}({\mathrm{\Gamma }}_i)$ and the equality (5.1) holds.

So (5.4)-(5.6) lead to (5.2). □

Next we state the local stability as follows.

Lemma 5.2 (Assumption III)

Proof To prove (5.8) we first introduce a function ${\theta }_m={\prod }_{i=1}^N{\theta }_{m,i}\in {\tilde{\mathcal{V}}}_h$ associated with a mortar side ${\gamma }_{m,i}\subset \mathrm{\Gamma }$, which satisfies the following: