We discuss a mortar-type P1NC/P0 element method for the incompressible Stokes problem. We prove the inf-sup condition and obtain the optimal error estimate. Meanwhile, we propose a 𝒲-cycle multigrid for solving this discrete problem and prove the optimal convergence of the multigrid method, that is, the convergence rate is independent of the mesh size and mesh level. Finally, numerical experiments are presented to confirm our theoretical results.
1. Introduction
As we all know, the application of viscous incompressible flows is of considerable interest. For example, the design of hydraulic turbines, or rheologically complex flows appears in many processes which are involved in plastics and molten metals. Therefore, in recent decades, many engineers and mathematicians have concentrated their efforts on the Stokes problem, especially the problem that can be handled by the finite element methods. In [1], Girault and Raviart provided a fairly comprehensive treatment of the most recent development in the finite-element method. Some new divergence-free elements were proposed to solve Stokes problem recently (see [2, 3] and others). Due to this development in the finite-element theory, many numerical algorithms were established to solve the Stokes equations. Among these algorithms, multigrid methods and domain decomposition methods for the Stokes equations are very prevalent. In [4], the authors constructed an efficient smoother. Based on the smoother, the multigrid methods have been greatly developed (see [5, 6]). Meanwhile, a FETI-DP method was extended to the incompressible Stokes equations in [7, 8], a BDDC algorithm for this problem was developed too in [9] and others.
In the last twenty years, mortar element methods have attracted much attention and it was first introduced in [10]. This method is a nonconforming domain decomposition method with nonoverlapping subdomains. In mortar finite-element methods, the meshes on adjacent subdomains may not match with each other across the interfaces of the subdomains. The coupling of the finite-element functions on adjacent meshes is done by enforcing the so-called mortar condition across the interfaces ( see [10] for details). There have been considerable researches on the mortar element methods (see [11–13] and others).
In [12], the author discussed the mortar-type conforming element (P2/P1 element) method for the Stokes problem, and then Chen and Huang proposed the mortar-type nonconforming element (Q1rot/Q0 element) method for the problem in [5]. It is well known that the rotated Q1 element is a rectangle element, and it is not a flexible finite element since it is only suitable for the rectangular or L-shape-bounded domain. Moreover, the rotated Q1 element is a quadratic element and is not as convenient as the linear elements in calculating.
In this paper, we apply the mortar element method coupling with P1 nonconforming finite element to the incompressible Stokes problem. The P1 nonconforming finite element is a triangular element and it is suitable for more extensive polygonal domain than the rotated Q1 element. Moreover, owing to its linearity, the computational work is less than the rotated Q1 element. We prove the so-called inf-sup condition and obtain the optimal error estimate. When solving the discrete problem, we also present a 𝒲-cycle multigrid algorithm, but the analysis about the convergence of the multigrid is different from [5]. We only prove that the prolong operator satisfies the criterion which proposed in [14] and we obtain the optional convergence with simpler analysis than that in [5]. Meanwhile, we do some numerical experiments which were realized in [5]. From numerical results, we note that the number of iterations is less than the rotated Q1 element method when achieving the same relative error.
The rest of this paper is organized as follows. In Section 2, we review the Stokes problem and introduce the mortar element method for P1 nonconforming element. Section 3 gives verification of the inf-sup condition and error estimate. The multigrid algorithm and the convergence analysis are given in Sections 4 and 5, respectively. The last section presents some numerical experiments. Throughout this paper, we denote by “C” a universal constant which is independent from the mesh size and level, whose values can differ from place to place.
2. Preliminaries
We only consider the incompressible flow problem, the steady-state Stokes problem, so that we can compare the results with those in [5].
The partial differential equations of the model problem is
(2.1)-Δu+∇p=finΩ,divu=0inΩ,u=0on∂Ω,
where Ω is bounded convex polygonal domain in R2, u represents the velocity of fluid, p is pressure, and f is external force. Define
(2.2)L02(Ω)={q∈L2(Ω)∣∫Ωqdx=0}.
The mixed variational formulation of problem (2.1) is to find (u,p)∈(H01(Ω))2×L02(Ω) such that
(2.3)a(u,v)+b(v,p)=〈f,v〉,∀v∈(H01(Ω))2,b(u,q)=0,∀q∈L02(Ω),
where the bilinear formulations a(·,·) on (H01(Ω))2×(H01(Ω))2, b(·,·) on (H01(Ω))2×L02(Ω) and the dual parity 〈·,·〉 on (L2(Ω))2×(L2(Ω))2 are given, respectively, by
(2.4)a(u,v)=∫Ω∇u⋅∇vdx,b(v,q)=-∫Ωdivvqdx,〈f,v〉=∫Ωf⋅vdx.
It is well known that the bilinear form b(·,·) satisfies the inf-sup condition, that is, there exists a positive constant β for any q∈L02(Ω) such that
(2.5)supv∈(H01(Ω))2b(v,q)‖v‖(H1(Ω))2≥β‖q‖L2(Ω).
According to the assumption on Ω and the saddle point theory in [15], we know that if f∈(L2(Ω))2, then there exists a unique solution (u,p)∈(H01(Ω)⋂H2(Ω))2×(L02(Ω)⋂H1(Ω)) satisfying
(2.6)‖u‖(H2(Ω))2+‖p‖H1(Ω)≤C‖f‖(L2(Ω))2.
We now introduce a mortar finite-element method for solving problem (2.3). First, we partition Ω into nonoverlapping polygonal subdomains such that
(2.7)Ω-=⋃i=1NΩ-i,Ωi⋂Ωj=ϕifi≠j.
They are arranged, so that the intersection of Ωi⋂Ωj for i≠j is an empty set or an edge, or a vertex; that is, the partition is geometrically conforming. Denote by γm the common open edge to Ωi and Ωj, then the interface Γ=⋃i=1N∂Ωi∖∂Ω is broken into a set of disjoint open straight segments γm(1≤m≤M), that is,
(2.8)Γ=⋃m=1Mγ-m,γm⋂γn=ϕifm≠n.
By γm(i) we denote an edge of Ωi called mortar and by δm(j) an edge of Ωj that geometrically occupies the same place called nonmortar.
With each Ωi, we associate a quasiuniform triangulation 𝒯h(Ωi) made of elements that are triangles. The mesh size hi is the diameter of largest element in 𝒯h(Ωi). We define h=max1≤i≤Nhi, 𝒯h=⋃i=1N𝒯h(Ωi). Let CR nodal points be the nonconforming nodal points, that is, the midpoints of the edges of the elements in 𝒯h(Ωi). Denote the set of CR nodal points belonging to Ω-i, ∂Ωi and ∂Ω by ΩihCR, ∂ΩihCR and ∂ΩhCR, respectively.
For each triangulation 𝒯h(Ωi) on Ωi, the P1 nonconforming element velocity space and piecewise constant pressure space are defined, respectively, as follows:
(2.9)Xh(Ωi)={vi∈(L2(Ωi))2|vi|τislinear∀τ∈Th(Ωi),viiscontinuousatmidpointofτ,v(mi)=0∀mi∈∂ΩhCR(L2(Ωi))2},Qh(Ωi)={qi∈L2(Ωi)|qi|τisaconstantforτ∈Th(Ωi)}.
Then the product space X~h(Ω)=∏i=1NXh(Ωi) is a global P1 nonconforming element space for 𝒯h on Ω.
For any interface γm=γm(i)=δm(j)(1≤m≤M), there are two different and independent triangulations 𝒯h(γm(i)) and 𝒯h(δm(j)), which produce two sets of CR nodes belonging to γm: the midpoints of the elements belonging to 𝒯h(γm(i)) and 𝒯h(δm(j)) denoted by γm(i)CR and δm(j)CR, respectively.
In order to introduce the mortar condition across the interfaces γm, we need the auxiliary test space Sh(δm(j)) which is defined by
(2.10)Sh(δm(j))={v∈(L2(δm(j)))2∣vispiecewiseconstantonelementsoftriangulationTh(δm(j))}.
For each nonmortar edge δm(j), define the L2-projection operator: Qh,δm(j):(L2(γm))2→Sh(δm(j)) by
(2.11)(Qh,δm(j)v,w)L2(δm(j))=(v,w)L2(δm(j)),∀w∈Sh(δm(j)).
Now we can define the mortar-type P1 nonconforming element space as follows:
(2.12)Xh(Ω)={v∈X~h(Ω)∣v|Ωi∈Xh(Ωi),Qh,δm(j)(v|δm(j))=Qh,δm(j)(v|γm(i)),∀γm=γm(i)=δm(j)⊂Γ(v|δm(j))},
the condition of the equality in (2.12) which the velocity function v satisfies is called mortar condition.
The global P0 element pressure space on Ω is defined by
(2.13)Qh(Ω)={q∈L02(Ω)∣q|Ωi∈Qh(Ωi)}.
We now establish the discrete system for problem (2.3) based on the mixed finite-element spaces Xh(Ω)×Qh(Ω).
We first define the following formulations:
(2.14)ahi(uhi,vhi)=∑τ∈Th(Ωi)∫τ∇uhi⋅∇vhidx,∀uhi,vhi∈Xh(Ωi),bhi(vhi,phi)=-∑τ∈Th(Ωi)∫τdiv vhi⋅phidx,∀vhi∈Xh(Ωi),∀phi∈Qh(Ωi).
Let
(2.15)ah(uh,vh)=∑i=1Nahi(uh,vh),bh(vh,ph)=∑i=1Nbhi(vh,ph).
Then the discrete approximation of problem (2.3) is to find (uh,ph)∈Xh(Ω)×Qh(Ω) such that
(2.16)ah(uh,vh)+bh(vh,ph)=〈f,vh〉,∀vh∈Xh(Ω),bh(uh,qh)=0,∀qh∈Qh(Ω).
In the next section, we prove that the discrete problem (2.16) has a unique solution and we obtain error estimate.
3. Existence, Uniqueness, and Error Estimate of the Discrete Solution
According to the Brezzi theory, the well-posedness of problem (2.16) depends closely on the characteristics of both bilinear forms ah(·,·) and bh(·,·). We equip the space Xh(Ω) with the following norm:
(3.1)‖v‖h2:=∑i=1N‖v‖h,i2,‖v‖h,i2:=ahi(v,v).
We can find in [1] that the local space family {Xh0(Ωi),Qh0(Ωi)} is div-stable; that is, there exists a constant β~ independent of hi such that
(3.2)supv~h∈Xh0(Ωi)bh(v~h,q~h)‖v~h‖h,i≥β~∥q~h∥L2(Ωi),∀q~h∈Qh0(Ωi),
where Xh0(Ωi)={v∈Xh(Ωi)∣v(mi)=0,∀mi∈∂Ωi,hCR},Qh0(Ωi)=Qh(Ωi)⋂L02(Ωi).
In order to prove that the global space family Xh(Ω)×Qh(Ω) is div-stable, it is necessary to define the global spaces as
(3.3)Q˘h(Ω)={q˘=∏i=1Nq˘i∈RN,(qˇ,1)=∑i=1Nqˇi|Ωi|=0}.
We first prove that the family {Xh(Ω),Qˇh(Ω)} is div-stable.
Lemma 3.1.
The following inf-sup condition holds:
(3.4)supvh∈Xh(Ω)bh(vh,qˇ)‖vh‖h≥βˇ‖qˇ‖L2(Ω)∀qˇ∈Qˇh(Ω),
where the constant βˇ does not depend on h.
Proof.
We decompose the space (H01(Ω))2 by (H01(Ω))2=∏i=1NV(Ωi)(V(Ωi)=(H01(Ω))2|Ωi) and define a local interpolation operator πi: V(Ωi)→Xh(Ωi) as
(3.5)πiv(mi)=1|ei|∫eivds,
where ei is an edge of τ∈𝒯h(Ωi), mi is the midpoint of ei. Then we can define a global interpolation operator π:(H01(Ω))2→X~h(Ω) as follows:
(3.6)πv=(π1v1,π2v2,…,πNvN),vi=v|Ωi,∀v∈(H01(Ω))2.
Define the operator Ξh,δm(j): X~h(Ω)→X~h(Ω) by
(3.7)(Ξh,δm(j)v)(mi)={Qh,δm(j)(v|γm(i)-v|δm(j))(mi),mi∈δm(j)CR,0,otherwise.
We can deduce that for any v∈(H01(Ω))2, there exists a vh*∈Xh(Ω) satisfying
(3.8)b(v-vh*,qˇ)=0.
In fact, we can set vh*=πv+∑m=1MΞh,δm(j)(πv). Obviously vh*∈Xh(Ω) and
(3.9)b(v-vh*,qˇ)=-∑i=1N∑τ∈Th(Ωi)∫τdiv(v-vh*)qˇdx=-∑i=1N∑τ∈Th(Ωi)∫∂τ(v-vh*)⋅nqˇds=-∑τ∈Th∫∂τ(v-πv)⋅nqˇds+∑τ∈Th∫∂τ∑m=1MΞh,δm(j)(πv)⋅nqˇds=∑j=1M∫δm(j)Qh,δm(j)((πv)|γm(i)-(πv)|δm(j))⋅nqˇjds=∑j=1M∫δm(j)((πv)|γm(i)-(πv)|δm(j))⋅nqˇjds=∑j=1M∫δm(j)(v|γm(i)-v|δm(j))⋅nqˇjds=0.
On the other hand
(3.10)‖v*‖h≤‖πv‖h+‖Ξh,δm(j)πv‖h,
by norm equivalence we have
(3.11)‖πv‖h2=∑τ|πv|H1(τ)2≤C∑τ(πv(mi)-πv(mj))2=C∑τ(1|ei|∫eivds-1|ej|∫ejvds)2=C∑τ(1|ei|∫ei(v-v-)ds-1|ej|∫ej(v-v-)ds)2≤C∑τ(1|ei|2(∫ei(v-v-)ds)2+1|ej|2(∫ej(v-v-)ds)2),
where mi,mj are the midpoints of the edges of τ, and v- is the integral average of v in τ, by Hölder inequality, trace theorem, and Friedrichs’ inequality we can get
(3.12)1|ei|2(∫ei(v-v-)ds)2≤1|ei|∫ei(v-v-)2ds≤Ch-1∫∂τ(v-v-)2ds≤C(h-2∫τ(v-v-)2dx+|v-v-|H1(τ)2)≤C|v|H1(τ)2,
and combining (3.11), we obtain
(3.13)‖πv‖h≤C‖v‖h.
Using norm equivalence we derive
(3.14)‖Ξh,δm(j)πv‖h2≤C∑mi∈δm(j)CR(Ξh,δm(j)πv(mi))2=C∑mi∈δm(j)CR(Qh,δm(j)((πv)|γm(i)-(πv)|δm(j))(mi))2≤Ch-1‖Qh,δm(j)((πv)|γm(i)-(πv)|δm(j))(mi)‖0,γm2≤Ch-1‖(πv)|γm(i)-(πv)|δm(j)‖0,γm2≤Ch-1(‖(πv)|γm(i)-v|δm(j)‖0,γm2+‖v|δm(j)-(πv)|δm(j)‖0,γm2)∶=Ch-1(K1+K2).
From trace theorem and (3.13), it follows that
(3.15)K2≤Ch‖v‖h,j2.
So we only need to estimate K1. Owing to v∈(H01(Ω))2, we then obtain
(3.16)‖(πv)|γm(i)-v|δm(j)‖0,γm2=‖(πv)|γm(i)-v|γm(i)‖0,γm2≤Ch‖v‖h,i2.
The bounds in (3.15) and (3.16) lead to
(3.17)‖Ξh,δm(j)πv‖h2≤C(‖v‖h,i2+‖v‖h,j2),
which together with (3.13) and (3.17) give
(3.18)‖v*‖h≤C‖v‖(H1(Ω))2.
Since {(H01(Ω))2,L02(Ω)} is div-stable, following (3.8) and (3.18), by Fortin rules, we have completed the proof of Lemma 3.1
Now we recall the following Brezzi theory about the existence, uniqueness, and error estimate for the discrete solution.
Theorem 3.2.
The bilinear forms ah(·,·) and bh(·,·) have the following properties:
ah(·,·) is continuous and uniformly elliptic on the mortar-type P1 nonconforming space Xh(Ω), that is,
(3.19)ah(uh,vh)≤‖uh‖h‖vh‖h,∀uh,vh∈Xh(Ω),ah(vh,vh)≥C‖vh‖h2,∀vh∈Xh(Ω);
bh(·,·) is also continuous on the space family Xh(Ω)×Qh(Ω), that is,
(3.20)bh(vh,q)≤‖vh‖h‖q‖L2(Ω),∀vh∈Xh(Ω),q∈Qh(Ω);
the family {Xh(Ω),Qh(Ω)} satisfies the inf-sup condition, that is, there exists a constant β that does not depend on h of triangulation such that
(3.21)supv∈Xh(Ω)bh(v,q)‖v‖h≥β‖q‖L2(Ω),∀q∈Qh(Ω),
so the problem (2.16) has a unique solution, and if one lets (u,p), (uh,ph) be the solution of (2.3) and (2.16), respectively, where (u,p)∈(H01(Ω))2×L02(Ω), u|Ωk∈(H2(Ωk))2, p|Ωk∈H1(Ωk), then
(3.22)‖u-uh‖h+‖p-ph‖L2(Ω)≤C∑k=1Nhk(‖u‖(H2(Ωk))2+‖p‖H1(Ωk)). Proof.
The statements of Brezzi theory are that the properties (3.19)–(3.21) lead to the existence, uniqueness, and error estimate of the discrete solution. In [16], it is proven that ah(·,·) is continuous on Xh(Ω) and is elliptic with a constant uniformly bounded. Furthermore, it is straightforward that bh(·,·) is continuous on Xh(Ω)×Qh(Ω). The point that needs verification is a uniform inf-sup condition (3.21), or equivalently that the family {Xh(Ω)×Qh(Ω)} is div-stable.
Using local inf-sup condition (3.2) and the above lemma, arguing as the proof in Proposition 5.1 of [12], we have the global inf-sup condition (3.21).
4. Numerical Algorithm
In this section, we present a numerical algorithm, that is, the 𝒲-cycle multigrid method for the discrete system (2.16), and we prove the optional convergence of the multigrid method. We use a simpler and more convenient analysis method than that in [5].
In order to set the multigrid algorithm, we need only to change the index h of the partition 𝒯h in Section 2 to be k, and let 𝒯1 be the coarsest partition. By connecting the opposite midpoints of the edges of the triangle, we split each triangle of 𝒯1 into four triangles and we refine the partition 𝒯1 into T2. The partition 𝒯2 is quasi-uniform of size h2=h1/2. Repeating this process, we get a sequence of the partition 𝒯k(k=1,2,…,L), each quasi-uniform of size hk=h1/2k-1.
As in Section 2, with the partition 𝒯k, we define the mortar P1 nonconforming element velocity space and P0 element pressure space as Xk and Qk, respectively. We can see that Xk(k=1,2,…,L) are nonnested, and Qk(k=1,2,…,L) are nested. Furthermore, we denote the P1 nonconforming element product space on Ω by X~k.
Let {φki} be the basis of Xk, and let {ψki} be the basis of Qk. For any vk∈Xk, qk∈Qk, we have the corresponding vector v_k=(v_k,i) and q_k=(q_k,i). We introduce the matrice Ak, Bk, and f_k having the entries ak,ij=a(φki,φkj),bk,ij=b(φki,ψkj), and f_k,i=(f,φki), respectively. Then at level k, the problem (2.16) is equivalent to
(4.1)(AkBkTBk0)(u_kp_k)=(f_k0).
In the following of this section, we introduce our multigrid method; the key of this method is the intergrid transfer operator.
We first define the intergrid transfer operator on the product space, Lk-1k:X~k-1→X~k(4.2)Lk-1kv(mi)={v(mi),mi∈κ,κ∈Tk-1,12(v|κ1(mi)+v|κ2(mi)),mi∈∂κ1⋂∂κ2,κ1,κ2∈Tk,0,mi∈∂Ω,
where κ, κi(i=1,2) is the partition of 𝒯k-1, 𝒯k respectively, mi∈Ωk,iCR(1≤i≤N).
Then we define the intergrid operator on the mortar P1 nonconforming element velocity space, Rk-1k:Xk-1→Xk(4.3)Rk-1kv=Lk-1kv+∑m=1MΞk,δm(j)Lk-1kv,
where Ξk,δm(j) is defined as (3.7).
On the P0 element pressure space, we apply the natural injection operator Jk-1k: Qk-1→Qk, that is,
(4.4)Jk-1k=I.
Therefore, our prolongation operator on velocity space and pressure space can be written as
(4.5)Ik-1k=[Rk-1k,Jk-1k].
Multigrid Algorithm
If k=1, compute the (u1,p1) directly. If k≥2, do the following three steps.
Step 1.
Presmoothing: for j=0,1,…,m1-1, solving the following problem:
(4.6)(u_kj+1p_kj+1)=(u_kjp_kj)-(αkIkBkTBk0)-1×{(AkBkTBk0)(u_kjp_kj)-(f_k0)},
where αk is a real number which is not smaller than the maximal eigenvalue of Ak.
Step 2.
Coarse grid correction: find (u~k-1,p~k-1)∈Xk-1×Qk-1, such that
(4.7)ak-1(u~k-1,vk-1)+bk-1(vk-1,p~k-1)=〈f,Rk-1kvk-1〉-ak(ukm1,Rk-1kvk-1)-bk(Rk-1kvk-1,pkm1),∀vk-1∈Xk-1,bk-1(u~k-1,qk-1)=0,∀qk-1∈Qk-1.
Compute the approximation (uk-1*,pk-1*) by applying μ≥2 iteration steps of the multigrid algorithm applied to the above equations on level k-1 with zero starting value. Set
(4.8)ukm1+1=ukm1+Rk-1kuk-1*,pkm1+1=pkm1+pk-1*.
Step 3.
Postsmoothing: for j=0,1,…,m2-1 solving following problem:
(4.9)(u_km1+j+2p_km1+j+2)=(u_km1+j+1p_km1+j+1)-(αkIkBkTBk0)-1×{(AkBkTBk0)(u_km1+j+1p_km1+j+1)-(f_k0)},
then, (ukm1+m2+1,pkm1+m2+1) is the result of one iteration step.
For convenience, at level k the problem (2.16) can be written as followes: find (uk,pk)∈Xk×Qk such that
(4.10)Lh,k((uk,pk);(vk,qk))=Fk((vk,qk)),∀(vk,qk)∈Xk×Qk.
Since Lh,k((uk,pk);(vk,qk)) is a symmetric bilinear form on Xk×Qk, there is a complete set of eigenfunctions (ϕkj,ψkj), which satisfy
(4.11)Lh,k((uk,pk);(vk,qk))=λj[(ϕkj,vk)0+h2(ψkj,qk)0],∀(vk,qk)∈Xk×Qk,(vk,qk)=∑jcj(ϕkj,ψkj).
In order to verify that our multigrid algorithm is optimal, we need to define a set of mesh-dependent norms. For each k≥0 we equip Xk×Qk with the norm
(4.12)‖|(v,q)|‖0,k=‖(v,q)‖0,k=(‖v‖L2(Ω)2+hk2‖q‖L2(Ω)2)1/2=((v,v)k+hk2(q,q)k)1/2,
and define
(4.13)‖|(vk,qk)|‖s,k={∑j|λj|s|cj|2}1/2,‖v‖k2=∑τ(∇v,∇v)k.
For our multigrid algorithm, we have the following optional convergence conclusion.
Theorem 4.1.
If (u,p) and (uhi,phi)(0≤i≤m+1) are the solutions of problems (2.16) and (4.10), respectively, then there exists a constant 0<γ<1 and positive integer m, all are independent of the level number k, such that
(4.14)‖|(u,p)-(ukm+1,pkm+1)|‖0,k≤γ‖|(u,p)-(uk0,pk0)|‖0,k.
To prove this theorem, we give in the next section two basic properties for convergence analysis of the multigrid, that is, the smoothing property and approximation property.
5. Proof of Theorem <xref ref-type="statement" rid="thm4.1">4.1</xref>
From the standard multigrid theory, the 𝒲-cycle yields a h-independent convergence rate based on the following two basic properties.
We first show the smoothing property. By [[12] Theorem 5.1], we have the following.
Lemma 5.1 (smoothing property).
Assume that λmax(Ak)≤αk≤Cλmax(Ak), if the number of smoothing steps is m, then
(5.1)‖|(uhm-uh,phm-ph)|‖2,k≤Ch-2m‖uh0-uh‖L2(Ω).
The property has been proved in [11].
Next, we prove the approximation property. We just apply the following conclusion in [14], which can simplify the complexity of theoretical analysis.
Lemma 5.2.
If the prolongation operator Ik-1k defined in (4.5) satisfies the following criterion, Then, the approximation property in multigrid method holds and the multigrid algorithm converges optimally.
where (u,p)∈(H01(Ω)∩H2(Ω))2×(L02(Ω)∩H1(Ω)) is the solution of (2.3) with the force term f∈(L2(Ω))2 and (uk-1,pk-1), (uk,pk) are the mixed finite element approximation of (u,p) at levels k-1 and k, respectively.
This lemma has been proved in [14].
Lemma 5.3 (approximation property).
Let (Ik-1k)*:Xk×Qk→Xk-1×Qk-1(k≥1) be defined as follows:
(5.2)Lk-1((Ik-1k)*(vk,qk),(vk-1,qk-1)),=Lk((vk,qk),Ik-1k(vk-1,qk-1)),∀(vk-1,qk-1)∈Xk-1×Qk-1,(vk,qk)∈Xk×Qk.
Then one has
(5.3)‖|(v,q)-Ik-1k(Ik-1k)*(v,q)|‖0,k≤Chk2‖|(v,q)|‖2,k,∀(v,q)∈Xk×Qk.
Proof.
By Lemma 5.2, we only need to prove our prolongation operator Ik-1k that satisfies (A.1), (A.2), and (A.3).
For any v∈Xk-1, the inequality (A.1) holds. In fact
(5.4)‖v-Rk-1kv‖L2(Ω)≤‖v-Lk-1kv‖L2(Ω)+‖∑m=1MΞk,δm(j)Lk-1kv‖L2(Ω),
by Lemma 5.2 in [14], we can get
(5.5)‖v-Lk-1kv‖L2(Ω)≤Chk‖v‖k-1,
by norm equivalence, we deduce
(5.6)‖Ξk,δm(j)Lk-1kv‖L2(Ω)2≤hk2∑mik∈δk,m(j)CR(Ξk,δm(j)Lk-1kv)2(mik)=hk2∑mik∈δk,m(j)CRQk,δm(j)((Lk-1kv)|γm(i)-(Lk-1kv)|δm(j))2(mik)≤Chk‖Qk,δm(j)((Lk-1kv)|γm(i)-(Lk-1kv)|δm(j))‖0,γm2≤Chk‖(Lk-1kv)|γm(i)-(Lk-1kv)|δm(j)‖0,γm2≤Chk(‖(Lk-1kv)|γm(i)-v|δm(j)‖0,γm2+‖v|δm(j)-(Lk-1kv)|δm(j)‖0,γm2.0)=Chk(K1+K2).
Using trace theorem and (5.5), we have
(5.7)K2≤Chk‖v‖k-1,j2.
Owing to v∈Xk-1, then
(5.8)‖(Lk-1kv)|γm(i)-v|δm(j)‖0,δm(j)2≤2‖(Lk-1kv)|γm(i)-Qk-1,δm(j)(v|γm(i))‖0,γm(i)2+2‖Qk-1,δm(j)(v|δm(j))-v|δm(j)‖0,δm(j)2.
The second term of the above inequality can be estimated as follows:
(5.9)‖Qk-1,δm(j)(v|δm(j))-v|δm(j)‖0,γm2=∑e∈Tk-1(δm(j))∫e(v-Qev)2ds,
where Qe is the L2 orthogonal projection onto one-dimensional space which consists of constant functions on an element e, and e is an edge of E which is in the triangulation Tk-1. Using the scaling argument in [17], for any constant c we have
(5.10)∫e(v-Qev)2ds≤∫e(v-c)2ds≤Chk∫e^(v^-c)2ds^≤Chk‖v^-c‖1,E^2≤Chk|v^|1,E^2≤Chk|v|1,E2,
which combining with (5.9) gives
(5.11)‖Qk-1,δm(j)(v|δm(j))-v|δm(j)‖0,γm≤Chk1/2∥v∥k,j.
For the first term of the right side of (5.8), we have
(5.12)‖(Lk-1kv)|γm(i)-Qk-1,δm(j)(v|γm(i))‖0,γm2=‖(Lk-1kv)|γm(i)-v|γm(i)+v|γm(i)+Qk-1,δm(j)(v|γm(i))‖0,γm2≤2‖(Lk-1kv-v)|γm(i)‖0,γm(i)2+2‖v|γm(i)-Qk-1,δm(j)(v|γm(i))‖0,γm(i)2=F1+F2.
Trace theorem and (5.5) give
(5.13)F1≤Chk‖Lk-1kv‖k,i2≤Chk‖v‖k-1,i2.
For F2, by trace theorem and the approximation of the operator Qk-1,δm(j), we have
(5.14)F2≤Chk‖v‖k-1,i2,
which together with (5.4)–(5.13), gives (A.1).
Obviously, (A.2) naturally holds so we only need to prove (A.3).
By proof of Lemma 5.2 in [14], we can see that
(5.15)‖|(uk,pk)-Ik-1k(uk-1,pk-1)|‖0,k≤‖uk-Lk-1kuk-1‖0,k+‖∑m=1MΞk,δm(j)Lk-1kuk-1‖0,k+hk2‖pk-Jk-1kpk-1‖0,k≤Chk2(‖u‖H2(Ω)+‖p‖H1(Ω))+‖∑m=1MΞk,δm(j)Lk-1kuk-1‖0,k.
Arguing as (5.6), we obtain
(5.16)‖Ξk,δm(j)Lk-1kuk-1‖0,k2≤hk2∑mik∈δk,m(j)CRΞk,δm(j)(Lk-1kuk-1)2(mik)=hk2∑mik∈δk,m(j)CR(Qk,δm(j)((Lk-1kuk-1)|γm(i)-(Lk-1kuk-1)|δm(j)))2(mik)≤Chk‖Qk,δm(j)((Lk-1kuk-1)|γm(i)-uk|γm(i)+uk|δm(j)-(Lk-1kuk-1)|δm(j))‖0,k2≤Chk(‖(Lk-1kuk-1)|γm(i)-uk|γm(i)‖0,γm2+‖(Lk-1kuk-1)|δm(j)-uk|δm(j)‖0,γm2)=Chk(K1+K2).
By (5.15) and trace theorem, we get that
(5.17)K1≤Chk3‖u‖H2(Ωi)2,K2≤Chk3‖u‖H2(Ωj)2,
together with (5.15), (A.3) has been proved, and we have completed the proof of Lemma 5.3.
6. Numerical Results
In this section, we present some numerical results to illustrate the theory developed in the earlier sections. The examples are as same as those in [5], so that we can compare the conclusion with the mortar rotated Q1 element method.
Here we deal with Ω=(0,1)2. We choose the exact solution of (2.1) as
(6.1)u1=2x2(1-x)2y(1-y)(1-2y),u2=-2x(1-x)(1-2x)y2(1-y)2,
for the velocity and p=x2-y2 for the pressure.
For simplicity, we decompose Ω into two subdomains: Ω1=(0,1)×(0,1/2) as nonmortar domain and Ω2=(0,1)×(1/2,1) as mortar domain. The sizes of the coarsest grid are denoted by h1,1 and h1,2, respectively (see Figure 1). The test concerns the convergence of the 𝒲-cycle multigrid algorithm. In what follows, k denotes the level, Nu and Np are the number of the unknowns of the velocity and pressure, the norm ∥·∥0,d is the usual Euclidean norm of a vector which is equivalent to ∥·∥h. iter(m1,m2) denotes the number of iterations to achieve the relative error of residue less than 10-3, where m1 and m2 are the presmoothing steps, and the postsmoothing steps respectively, and the initial approximative solution for the iteration is zero. The numerical results are presented in Tables 1 and 2.
Error estimate for the mortar element method with h1,1=1/4 and h1,2=1/6.
k
Nu
Np
∥u̲-u̲k∥0,d
∥p-pk∥L2(Ω)
1
178
51
0.0772974
0.164013
2
668
207
0.0455288
0.133103
3
2584
831
0.0242334
0.0733174
4
10160
3327
0.0123836
0.037679
5
40288
13311
0.00619081
0.0195834
Iterative numbers for the 𝒲-cycle with h1,1=1/4 and h1,2=1/6.
k
2
3
4
5
iter(4,4)
9
8
8
9
iter(5,5)
8
8
7
7
The coarsest mesh with h1,1=1/4 and h1,2=1/6.
From Table 1, we can see that the errors of the mortar element method for the velocity and the pressure are small, which demonstrates Theorem 3.2.
From Table 2, we can see that the convergence for the 𝒲-cycle multigrid algorithm is optimal; that is, the number of iterations is independent of the level number k. Meanwhile, we note that the number of iterations is less than the rotated Q1 element method in [5] when achieving the same relative error.
Acknowledgment
The authors would like to express their sincere thanks to the referees for many detailed and constructive comments. The work is supported by the National Natural Science Foundation of China under Grant 11071124.
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