For the Steklov eigenvalue problem, we establish a type of multigrid discretizations based on the fixed-shift inverse iteration and study in depth its a priori/a posteriori error estimates. In addition, we also propose an adaptive algorithm on the basis of the a posteriori error estimates. Finally, we present some numerical examples to validate the efficiency of our method.
1. Introduction
Due to the wide applications in physical and mechanical field (see, e.g., [1–3]), there has been a lot of research on the numerical methods for Steklov eigenvalue problems; for instance, [4] studied the conforming linear finite element approximation, [5, 6] studied the nonconforming finite elements approximation, [7, 8] discussed a two-grid method of the conforming and nonconforming finite element method based on the inverse iteration, respectively, [9] studied multiscale asymptotic method, [10] studied multilevel method, [11] studied the spectral method, and [12] studied an adaptive algorithm based on the shifted inverse iteration.
In this paper we establish a type of multigrid discretizations based on the fixed-shift inverse iteration for the Steklov eigenvalue problem. The multilevel method in [10] made use of the inverse iteration and the extended finite element method. Compared with [10], our method has less computational complexity since we have no correction step in each iteration. On the other hand, compared with [12], we adopt the fixed-shift and thus avoid selecting appropriate shift to ensure the efficiency of shifted inverse iteration; meanwhile, we also do not face the difficulty of solving an almost singular algebraic system in the shifted inverse iteration.
We analyze elaborately the a priori and the a posteriori error estimates of the method proposed in this paper. Then, based on the a posteriori error estimates we design an adaptive algorithm of fixed-shift inverse iteration type. Moreover, we also compare the performance of three types of multigrid methods. Numerical results illustrate that our method is also an efficient method for solving the Steklov eigenvalue problem.
The rest of this paper is organized as follows. In the subsequent section, some preliminaries needed in this paper are presented. In Section 3, a scheme of the inverse iteration with fixed-shift based on multigrid discretizations is established, and the a priori error estimates are also given. The a posteriori error estimates of the inverse iteration with fixed-shift are analyzed in Section 4. Numerical experiments are presented in the final section.
In this paper, C with or without subscript denotes a constant independent of mesh size and iterative times.
2. Preliminaries
Consider the Steklov eigenvalue problem(1)-Δu+u=0inΩ,∂u∂n=λuon∂Ω,where Ω⊂R2 is a polygonal domain with θ being the largest inner angle of Ω and ∂u/∂n is the outward normal derivative.
We denote the real order Sobolev spaces with norm ∥·∥t and ∥·∥t,∂Ω by Ht(Ω) and Ht(∂Ω), respectively; H0(∂Ω)=L2(∂Ω).
The variational form of (1) is given by the following: find λ∈R and u∈H1(Ω), u≠0, such that(2)au,v=λbu,v,∀v∈H1Ω,where (3)au,v=∫Ω∇u·∇v+uvdx,bu,v=∫∂Ωuvds,ub=bu,u1/2=u0,∂Ω.
As we know, a(·,·) is a symmetric, continuous, and H1(Ω)-elliptic bilinear form on H1(Ω)×H1(Ω). Thus, we use a(·,·) and ·a=a(·,·)=·1 as the inner product and norm on H1(Ω), respectively.
Let H-1/2(∂Ω) be the dual space of H1/2(∂Ω) with norm given by (4)w-1/2,∂Ω=supv∈H1/2∂Ωw,vv1/2,∂Ω, where 〈w,v〉 is the dual product on H-1/2(∂Ω)×H1/2(∂Ω). When w∈L2(∂Ω), 〈w,v〉=b(w,v).
Let {πh} be a family of regular triangulations of Ω with the mesh diameter h, and let Vh⊂H1(Ω) be a space of piecewise polynomials defined on πh. For any w∈H1(Ω), the following conclusion holds:(5)limh→0infv∈Vhw-va=0.
The conforming finite element approximation of (2) is the following: find λh∈R and uh∈Vh, uh≠0, such that(6)auh,v=λhbuh,v,∀v∈Vh.
Define the operators T:H1(Ω)→H1(Ω) and Th:H1(Ω)→Vh⊂H1(Ω) satisfying(7)aTg,v=bg,v,∀v∈H1Ω,(8)aThg,v=bg,v,∀v∈Vh.
Define the Ritz projection Ph:H1(Ω)→Vh by (9)au-Phu,v=0,∀v∈Vh.
From [13], we know that T-Tha→0(h→0); (2) and (6) have the equivalent operator forms Tu=μu and Thuh=μhuh, respectively, where Th=PhT, μ=1/λ, and μh=1/λh.
Suppose that λ and λh are the kth eigenvalue of (2) and (6), respectively, and the algebraic multiplicity of λ is equal to q, λ=λk=λk+1=⋯=λk+q-1. Let M(λ) be the space spanned by all eigenfunctions corresponding to λ and let Mh(λ) be the direct sum of eigenspaces corresponding to all eigenvalues of (6) that converge to λ. Let M^(λ)={v:v∈Mλ,va=1}.
It is obvious that δh(λ)≤λσ(h)≤Cρ(h). It follows from Lemma 3.3 in [14] that (11)σh⟶0h⟶0.
By using the trace theorem we have(12)v0,∂Ω≤v1/2,∂Ω≤C1va,∀v∈H1Ω.
Moreover, if v∈H1(Ω) and w∈H1/2(∂Ω) we know that 〈v,w〉=b(v,w)≤v0,∂Ωw0,∂Ω and, consequently,(13)v-1/2,∂Ω≤v0,∂Ω≤C1va,∀v∈H1Ω.
For any g∈H1(Ω), Thg∈H1(Ω). Taking v=Thg in (8) we deduce (14)aThg,Thg=bg,Thg≤g-1/2,∂ΩThg1/2,∂Ω≤C1g-1/2,∂ΩThga, and thus we get(15)Thga≤C1g-1/2,∂Ω.
The following lemmas are needed in our analysis.
Lemma 1.
Let (λ,u) be an eigenpair of (2); then for any v∈H1(Ω) with va=1, the Rayleigh quotient R(v)=a(v,v)/vb2 satisfies(16)Rv-λ=v-ua2vb2-λv-ub2vb2.
Proof.
See page 699 of [13].
Lemma 2.
For any nonzero u,v∈H1(Ω), (17)uua-vvaa≤2u-vaua,uua-vvaa≤2u-vava.
Proof.
See [15].
Lemma 3.
Let λ and λh be the kth eigenvalue of (2) and (6), respectively. Then for any eigenfunction uh corresponding to λh with uha=1, there exist u∈M(λ) and h0>0 such that if h≤h0,(18)uh-ua≤C2δhλ,(19)uh-u0,∂Ω≤C2ρhδhλ,(20)uh-u-1/2,∂Ω≤C2σhδhλ;for any u∈M^(λ), there exists uh∈Mh(λ) such that if h≤h0,(21)u-uha≤C3δhλ,u-uh-1/2,∂Ω≤C3σhδhλ,where constants C2 and C3 are positive and only depend on λ.
Proof.
See page 699 of [13] and Lemma 3.7 and (3.29b) of [14].
If u∈M(λ), v∈H1(Ω), va=1, and v-ua≤4λC1-1, then by Lemma 2 we have (22)v-uuaa≤2v-ua≤2λC1-1,v-uua0,∂Ω≤C1v-uuaa≤12λ.From (2) we have u/ua0,∂Ω=1/λ; then (23)v0,∂Ω≥uua0,∂Ω-v-uua0,∂Ω≥12λ.
Hence, from Lemma 1 we get(24)Rv-λ≤4λ1+λC12v-ua2.
Denote (25)C4=4λ1+λC12,
and then when va=1 and v-ua≤4λC1-1, (24) becomes(26)Rv-λ≤C4v-ua2.
Since (6) implies λh=R(uh), then combining (26) and (18) we deduce that(27)0≤λh-λ≤C4uh-ua2≤C4C22δh2λ.
3. A Priori Error Estimates of the Inverse Iteration with Fixed-Shift
Let Vhi0∞ be a family of conforming finite element spaces that satisfy Vh0=VH, Vhi⊂Vhi+1⊂H1(Ω) (i=0,1,…), and σhi→0(i→∞). Referring to [16], we establish the following scheme of the inverse iteration with fixed-shift based on multigrid discretizations.
Scheme 4 (the inverse iteration with fixed-shift based on multigrid discretizations).
Given the iterative times l and i0. Execute the following.
Step 1.
Solve (2) on VH: find (λH,uH)∈R×VH such that uHa=1 and(28)auH,v=λHbuH,v,∀v∈VH.
Step 2.
Let uh0⇐uH, λh0⇐λH, i⇐1.
Step 3.
Solve a linear system on Vhi: find u′∈Vhi such that(29)au′,v-λhi-1bu′,v=buhi-1,v,∀v∈Vhi;
set uhi=u′/u′a.
Step 4.
Compute the Rayleigh quotient (30)λhi=auhi,uhibuhi,uhi.
Step 5.
If i>i0, then λhi0⇐λhi-1, i⇐i+1; turn to Step 6; else, i⇐i+1, and return to Step 3.
Step 6.
Solve a linear system on Vhi: find u′∈Vhi such that(31)au′,v-λhi0bu′,v=buhi-1,v,∀v∈Vhi;
set uhi=u′/u′a.
Step 7.
Compute the Rayleigh quotient(32)λhi=auhi,uhibuhi,uhi.
Step 8.
If i=l, then output (λhl,uhl) and stop; else, i⇐i+1, and return to Step 6.
Let (λH,uH) be the kth eigenpair of (28); then (λhl,uhl) derived from Scheme 4 is the kth eigenpair approximation of (2).
In the following analysis, we also denote (λH,uH)=(λk,H,uk,H) and (λhl,uhl)=(λkhl,ukhl).
Now, we will analyze the a priori error estimates of Scheme 4.
Denote dist(u,S)=infv∈Su-va.
Our analysis makes use of the following lemma (see Lemma 4.1 in [16]) for the shifted inverse iteration method. Let (λk,uk) and (λk,h,uk,h) denote the kth eigenpair of (2) and (6), respectively, and μk=1/λk, μk,h=1/λk,h, M(μk)=M(λk), and Mh(μk)=Mh(λk).
Lemma 5.
Let (μ0,u0) be an approximation for (μk,uk), where μ0 is not an eigenvalue of Th, and u0∈Vh with u0a=1. Suppose that
dist(u0,Mh(μk))≤1/2;
|μ0-μk|≤ρ/4 and |μj,h-μj|≤ρ/4, for j=k-1,k,k+q(j≠0), where ρ=minμj≠μk|μj-μk| is the separation constant of the eigenvalue μk;
Let δ0 be a positive constant satisfying the following inequalities:(35)max1,C2,C3δ0≤min12,4λjC1-1,(36)4C1C1C4δ02+λkC1δ0+λkC3δ02+C1C5q1/2C4C22δ02≤12,(37)δ0λk-δ0λk≤ρ4,δ0≤λk2,(38)C4C22λj2δ02≤ρ4,j=k-1,k,…,k+q,j≠0.
Condition 6.
There exists u¯∈M^(λk) for j=k-1,k,k+q(j≠0) such that (39)ukhl-1-u¯a≤δ0,λ0-λk≤δ0,δhlλj≤δ0,σhl≤δ0, where λ0 is an approximate eigenvalue of λk, ukhl-1 is an approximate eigenfunction obtained by Scheme 4, and ρ is the separation constant of the eigenvalue μk=1/λk.
Let the eigenvectors {uj,hl}kk+q-1 be an orthonormal basis of Mhl(λk) with respect to a(·,·), and denote(40)u∗=∑j=kk+q-1aukhl,uj,hluj,hl.From Lemma 3, we know that there exist eigenvectors {uj0}kk+q-1⊂M(λk) making uj,hl-uj0 satisfy (18), (19), and (20). Let(41)uk=∑j=kk+q-1aukhl,uj,hluj0,and then uk∈M(λk) and(42)uk-u∗=∑j=kk+q-1aukhl,uj,hluj0-uj,hl.
To estimate the error, we split(43)ukhl-uk=ukhl-u∗-uk-u∗.
Now, we will analyze the first term ukhl-u∗.
Theorem 7.
Let (λkhl,ukhl) be an approximate eigenpair obtained by Scheme 4 with λ0=λkhi0. Assume that Lemma 3 and Condition 6 hold; then(44)ukhl-u∗a≤C04λ0-λkλkhl-1-λk+ukhl-1-u¯-1/2,∂Ω+σhlδhlλk,l≥1,where C0 is independent of mesh parameters and l.
Proof.
We use Lemma 5 to complete the proof. First, we will verify that the conditions of Lemma 5 are satisfied.
From Lemma 3, we know that, for any given u¯∈M^(λk), there exists u~k,hl∈Mhl(λk) such that(45)u¯-u~k,hla≤C3δhlλk,(46)u¯-u~k,hl-1/2,∂Ω≤C3σhlδhlλk,where(47)u~k,hl=∑i=kk+q-1αiui,hl,u~k,hla≤C5.Select μ0=1/λ0 and u0=λkhl-1Thlukhl-1/λkhl-1Thlukhl-1a. Then, by (15) and (13) we have (48)λkhl-1Thlukhl-1-u~k,hla=λkhl-1Thlukhl-1-∑i=kk+q-1λi,hlThlαiui,hla≤C1λkhl-1ukhl-1-∑i=kk+q-1λi,hlαiui,hl-1/2,∂Ω=C1λkhl-1ukhl-1-λkukhl-1+λkukhl-1-λku¯+λku¯-λku~k,hl+λku~k,hl-∑i=kk+q-1λi,hlαiui,hl-1/2,∂Ω≤C1C1λkhl-1-λk+λkukhl-1-u¯-1/2,∂Ω+λku¯-u~k,hl-1/2,∂Ω+∑i=kk+q-1λk-λi,hlαiui,hl-1/2,∂Ω, noting that u~k,hla≥u¯a-u¯-u~k,hla≥1-C3δhl(λk)≥1-C3δ0≥1/2; then using Lemma 2, (46), the Cauchy-Schwartz inequality, (26), Condition 6, and (36) we obtain(49)distu0,Mhlλk≤u0-u~k,hlu~k,hlaa≤2u~k,hlaλkhl-1Thlukhl-1-u~k,hla≤4C1C1λkhl-1-λk+λkukhl-1-u--1/2,∂Ω+λkC3σhlδhlλk+C1C5q1/2C4C22δhl2λk≤4C1C1C4δ02+λkC1δ0+λkC3δ02+C1C5q1/2C4C22δ02≤12,and then Condition (C1) in Lemma 5 holds.
By using the same arguments in [16], it is clear that the other two conditions in Lemma 5 are valid.
Hence, we see that the conditions of Lemma 5 hold.
Then, by the same proof method in [16], we derive that (50)ukhl-u∗a≤16ρC12λk2λ0-λk+2λk2C4C22δhl2λkC1λkhl-1-λk+λkukhl-1-u¯-1/2,∂Ω+λkC3σhlδhlλk+C1C5q1/2C4C22δhl2λk.Noting that the constants C1, C2, C3, C4, C5, and ρ are independent of mesh parameters and l and Condition 6 holds, then based on the above inequality we conclude that there exists a positive constant C0 that is independent of mesh parameters and l such that (44) holds. And we can have minC0/4,C0C1/4>q1/2C2. The proof is completed.
Next, we will analyze the error uk-u∗.
Theorem 8.
The error uk-u∗ satisfies(51)uk-u∗a≤C04δhlλk,(52)uk-u∗-1/2,∂Ω≤C0C14σhlδhlλk.
Proof.
The estimates (51) and (52) can be obtained by the proof arguments in [16].
Based on the above two theorems, we now analyze the a priori error estimates of Scheme 4.
Condition 9.
For any given β0,β0′∈(0,1), there exist 0<β0≤βi<1 and 0<β0′≤βi′<1(i=1,2,…) such that δhi(λk)=βiδhi-1(λk) and σ(hi)=βi′σ(hi-1), respectively, σhi→0(i→∞).
In the practice Condition 9 is not a restrictive condition. For example, let πhi be obtained from πhi-1 via regular refinement (producing 4 congruent elements) such that hi=(1/2)hi-1; then, when M(λk)⊂H1+γ(Ω) and {Tf:f∈H1(Ω)}⊂H1+γ(Ω) we have δhi(λk)≈(1/2)γδhi-1(λk) and σ(hi)≈(1/2)γσ(hi-1) (see [10]), where γ=1 if Ω is convex and 0<γ<1 if Ω is concave.
Theorem 10.
Let (λkhl,ukhl) be an approximate eigenpair obtained by Scheme 4. Suppose that Condition 9 holds; then there exist uk∈M(λk) and H0>0 such that if H≤H0 it is valid that(53)ukhl-uka≤C0δhlλk,(54)ukhl-uk-1/2,∂Ω≤C0C1σhlδhlλk,(55)λkhl-λk≤C4C02δhl2λk,l≥i0.
Proof.
We only prove the result (54) since (53) and (55) can be proved analogously by referring to [16].
The proof is completed by using induction, Theorems 7 and 8. Note that δHλk≤λkσH→0(H→0); then there exists a proper small H0>0 such that if H≤H0, Lemma 3 and the following inequalities hold:(56)C0δHλk≤δ0,C4C02δH2λk≤δ0,δHλj≤δ0,σH≤δ0,(57)C42C04δH2λkλk1β0′1β0+C4C03C1δH2λk1β0′1β0≤1,where j=k-1,k,k+q(j≠0).
When l=i0, it is easy to know that (53)–(55) are valid (see [12, 16]). Suppose that Theorem 10 holds for l-1; that is, there exists u¯∈M(λk) such that (58)ukhl-1-u¯a≤C0δhl-1λk,ukhl-1-u¯-1/2,∂Ω≤C0C1σhl-1δhl-1λk,λkhl-1-λk≤C4C02δhl-12λk.Then we infer from (56) that the conditions of Theorem 7 hold.
From Theorems 7 and 8 we get(59)ukhl-uk-1/2,∂Ω≤C0C12λ0-λkλkhl-1-λk+ukhl-1-u¯-1/2,∂Ω+σhlδhlλk,l≥1.Therefore, for l, from (59) we derive that (60)ukhl-uk-1/2,∂Ω≤C0C12C42C04δhi02λkδhl-12λk+C4C03C1δhi02λkσhl-1δhl-1λk+σhlδhlλk≤C0C12C42C04δhi02λkλk1βl′1βl+C4C03C1δhi02λk1βl′1βl+1σhlδhlλk≤C0C12C42C04δH2λkλk1β0′1β0+C4C03C1δH2λk1β0′1β0+1σhlδhlλk, which together with (57) we get (54) immediately.
4. A Posteriori Error Estimates of the Inverse Iteration with Fixed-Shift
Based on the work of [4, 12, 17–19], in this section, we will discuss the a posteriori error estimates of Scheme 4 for the Steklov eigenvalue problem.
Consider the boundary value problem corresponding to (2): find w∈H1(Ω) such that(61)aw,v=bf,v,∀v∈H1Ω,and its finite element approximation states: find wh∈Vh such that(62)awh,v=bf,v,∀v∈Vh.
For any element T∈πh with diameter hT, we denote by ET the set of edges, and (63)E=⋃T∈πhET. We decompose E=EΩ∪EΓ, where EΩ and EΓ refer to interior edges and edges on the boundary Γ=∂Ω, respectively. For each l∈EΩ, we choose an arbitrary unit normal vector nl and denote the two triangles sharing this edge by Tin and Tout, where nl points outwards Tin.
For vh∈Vh we set (64)∂vh∂nll=∇vh∣Tout·nl-∇vh∣Tin·nl. Let (65)λ^k,hl=1q∑j=kk+q-1λj,hl. For each l∈E we define the jump residual:(66)Jlukhl=12∂ukhl∂nlll∈EΩ,λkhlukhl-∂ukhl∂nll∈EΓ.Now, the local error indicator is defined as(67)ηTukhl=hT2ukhl0,T2+∑l∈ETlJlukhl0,l21/2,and then the global error estimator is given by(68)ηΩukhl=∑T∈πhηT2ukhl1/2.Substituting u∗ for ukhl, we can get the definitions of Jl(u∗), ηT(u∗), and ηΩ(u∗) similarly.
Now, we will estimate the error e=uk-u∗.
From [4, 12], we give the following two lemmas among which Lemma 11 provides the global upper bound of e, while Lemma 12 provides the local lower bound of e.
Lemma 11.
The error e=uk-u∗ satisfies(69)ea≤C6ηΩu∗+σhlδhlλk.
Lemma 12.
The error e=uk-u∗ satisfies the following:
(a) For T∈πh, if ∂T∩Γ=∅, then(70)ηTu∗≤C7e1,T∗,where T∗ denotes the union of T and the triangles sharing an edge with T.
(b) For T∈πh, if ∂T∩Γ≠∅, then(71)ηTu∗≤C8e1,T+∑l∈ET∩EΓl1/2λkuk-λ^k,hlu∗0,l.
Next, we will analyze the error ukhl-u∗.
Theorem 13.
Suppose that the conditions of Theorem 10 are satisfied; then(72)ukhl-u∗a≤C04σhlδhlλk.
Proof.
Note that δHλk≤λkσH→0(H→0); then there exists a proper small H0>0 such that if H≤H0, the following inequality holds:(73)C42C04δH2λkλk1β0′1β0+C4C03C1δH2λk1β0′1β0+C4C02δH2λk≤1.From (44), Theorem 10, and Condition 9, we have (74)ukhl-u∗a≤C04C42C04δhi02λkδhl-12λk+C4C03C1δhi02λkσhl-1δhl-1λk+C4C02δhi02λkσhlδhlλk≤C04C42C04δhi02λkλk1βl′1βl+C4C03C1δhi02λk1βl′1βl+C4C02δhi02λkσhlδhlλk≤C04C42C04δH2λkλk1β0′1β0+C4C03C1δH2λk1β0′1β0+C4C02δH2λkσhlδhlλk, which together with (73) yields (72) immediately.
We give the following lemma by referring to [12] (see Lemma 3.4 in [12]).
Lemma 14.
Suppose that the conditions of Theorem 10 are satisfied; then(75)ηTu∗-ηTukhl≤C9δhl2λku∗1,T+ukhl-u∗1,T,(76)ηΩu∗-ηΩukhl≤C10δhl2λku∗a+ukhl-u∗a.
In the following discussion, combining Lemmas 11, 12, and 14 and Theorem 13, we give the global upper bound and the local lower bound of the error.
Theorem 15.
Suppose that the conditions of Theorem 10 are satisfied; then there exists uk∈M(λk) such that(77)uk-ukhla≤C6ηΩukhl+R1,where R1=C6C10δhl2(λk)u∗a+(C6+C0/4+C6C10(C0/4))σ(hl)δhl(λk).
Proof.
Select uk∈M(λk) which is given by (41); then from Lemma 11, Theorem 13, and (76) we get(78)uk-ukhla≤uk-u∗a+u∗-ukhla≤C6ηΩu∗+σhlδhlλk+C04σhlδhlλk≤C6ηΩukhl+C10δhl2λku∗a+ukhl-u∗a+σhlδhlλk+C04σhlδhlλk≤C6ηΩukhl+R1.The proof is completed.
It is obvious that R1 is a higher order term. Hence, we obtain that ηΩ(ukhl) is a global reliable error indicator of uk-ukhla.
Theorem 16.
Under the conditions of Theorem 10, there exists uk∈M(λk) such that the following hold:
(a) For T∈πhl, if ∂T∩Γ=∅, then(79)ηTukhl≤C7uk-ukhl1,T∗+R2,where R2=(C7+C9)ukhl-u∗1,T∗+C9δhl2(λk)u∗1,T.
(b) For T∈πhl, if ∂T∩Γ≠∅, then(80)ηTukhl≤C8uk-ukhl1,T+R3,where (81)R3=C8+C9ukhl-u∗1,T+C9δhl2λku∗1,T+C8∑l∈ET∩EΓl1/2λkuk-λ^k,hlu∗0,l.
Proof.
We can prove the desired results by using the proof method of Theorem 3.4 in [12].
According to Remark 3.1 in [4] and Remark 3.2 in [12] we know that the term ∑l∈ET∩EΓl1/2λkuk-λ^k,hlu∗0,l is a higher order term. From Theorem 13, we know that ukhl-u∗1,T∗ and ukhl-u∗1,T are also higher order terms. And it is obvious that δhl2(λk)u∗1,T is a higher order term. Therefore, from (79) and (80) we know that ηT(ukhl) is an efficient local error indicator of uk-ukhl1,T∗ and uk-ukhl1,T.
In the following theorem, we give the estimate for approximate eigenvalue.
Theorem 17.
Suppose that the conditions of Theorem 10 are satisfied; then(82)λkhl-λk=OηΩ2ukhl.
Proof.
From Theorem 10 and Lemma 1 it is easy to prove that (83)λkhl-λk=Oukhl-uka2; then combining with Theorem 16 and (77), we can get the desired result (82).
5. Numerical Experiments
In this section we first give an adaptive algorithm of the Rayleigh quotient iteration type and establish an adaptive algorithm of fixed-shift inverse iteration type for the Steklov eigenvalue problem.
The following Algorithm 1 of the Rayleigh quotient iteration type refers to Algorithm 4.3 in [12] or Algorithm 6.1 in [16].
Algorithm 1.
Choose parameter 0<ω<1.
Step 1.
Pick any initial mesh πh0.
Step 2.
Solve (2) on πh0 for discrete solution (λh0,uh0).
Step 3.
Let l⇐0,λ0⇐λh0.
Step 4.
Compute the local indicators ηT(uhl).
Step 5.
Construct π^hl⊂πhl by Marking Strategy E and ω.
Step 6.
Refine πhl to get a new mesh πhl+1 by Procedure REFINE.
Step 7.
Find u′∈Vhl+1 such that(84)au′,v-λ0bu′,v=buhl,v,∀v∈Vhl+1;denote uhl+1=u′/u′a and compute the Rayleigh quotient(85)λhl+1=auhl+1,uhl+1buhl+1,uhl+1.
Step 8.
Let λ0⇐λhl+1, l⇐l+1 and go to Step 4.
Marking Strategy E. Give parameter 0<ω<1.
Step 1.
Construct a minimal subset π^hl of πhl by selecting some elements in πhl such that(86)∑T∈π^hlηT2uhl≥ωηΩ2uhl.
Step 2.
Mark all the elements in π^hl.
ηT(uhl) and ηΩ(uhl) are defined by (67) and (68) with ukhl and λkhl replaced by uhl and λhl, respectively.
Note that when |λ0-λ| is too small, (84) is an almost singular linear equation. Although it has no difficulty in solving (84) numerically (see [12]), one would like to think of selecting a proper integer i0≥0 to establish the following adaptive algorithm.
Algorithm 2.
Choose parameter 0<ω<1.
Steps 1–7. Execute Steps 1–7 of Algorithm 1.
Step 8. If l<i0,λ0⇐λhl+1,l⇐l+1, go to Step 4; else l⇐l+1, go to Step 4.
Marking Strategy E in Algorithm 2 is the same as that in Algorithm 1.
Now, we will implement some numerical experiments to validate our theoretical analysis and show the efficiency of Algorithm 2 with i0=0. We use MATLAB 2012 together with the package of Chen [20] to solve Examples 1, 2, and 3, and we take ω=0.5.
For reading conveniently, we use the following notations in our tables:
λkhl(m): the kth eigenvalue derived from the lth iteration obtained by Algorithm m (m=1,2).
λkhlm-λk: the error of λkhl(m) obtained by Algorithm m (m=1,2).
Nk,l(m): the degrees of freedom of the lth iteration for λkhl(m) (m=1,2).
CPUk,l(m)(s): the CPU time(s) from the program starting to calculate result of the lth iteration appearing by using Algorithm m (m=1,2).
Example 1.
We use Algorithms 1 and 2 to compute the approximations of the 1st and the 2nd eigenvalue of (1) with the triangle linear finite element on Ω=[0,1]×[0,1]. The numerical results are listed in Table 1.
Since the exact eigenvalues are unknown, we use λ1≈0.24007908542 and λ2≈1.49230313453 obtained by the spectral element method (see [21]) as the reference eigenvalues. We show the error curves and the a posteriori estimators obtained by two algorithms for λ1 and λ2 in Figure 1. It can be seen from Figure 1 that the error curves are approximately parallel to the line with slope -1, which indicates that Algorithm 2 achieves the optimal convergence rate of O(Nl-1) as well as Algorithm 1.
Observing the numerical results in Table 1, we can find that when the degrees of freedom are almost the same, the approximate eigenvalues obtained by Algorithm 2 are nearly as accurate as those obtained by Algorithm 1 and their CPU time are roughly the same.
The 1st and the 2nd eigenvalues of Example 1 obtained by Algorithms 1 and 2 with H=2/32.
k
l
Nk,l(1)
λkhl(1)
CPUk,l(1)
Nk,l(2)
λkhl(2)
CPUk,l(2)
1
5
5700
0.24008040
0.54
5700
0.24008040
0.57
1
10
27344
0.24007936
1.60
27343
0.24007936
1.66
1
15
117001
0.24007915
7.67
117001
0.24007915
7.82
1
19
408971
0.24007910
29.06
408971
0.24007910
29.25
1
20
509032
0.24007910
39.88
509032
0.24007910
40.07
1
21
764069
0.24007909
55.90
764069
0.24007909
55.96
2
5
4526
1.49245505
0.51
4529
1.49245448
0.54
2
12
37855
1.49231870
2.31
37941
1.49231871
2.33
2
18
223337
1.49230601
15.77
224277
1.49230600
15.89
2
24
1329617
1.49230362
109.68
1334272
1.49230362
109.90
2
25
1805637
1.49230347
151.36
1812726
1.49230347
151.02
2
26
2439573
1.49230337
211.29
2447212
1.49230337
210.12
The curves of error and the a posteriori error estimators of two algorithms for the 1st (a) and 2nd (b) eigenvalues on the square domain.
Example 2.
We use Algorithms 1 and 2 to compute the approximations of the 1st and the 3rd eigenvalue of (1) with the triangle linear finite element on Ω=([0,1]×[0,1/2])∪([0,1/2]×[1/2,1]). The numerical results are presented in Table 2.
In Figure 2 we depict the error curves and the a posteriori estimators obtained by two algorithms for λ1 and λ3. Here we use λ1≈0.18296423687 and λ3≈1.68860048358 obtained by the spectral element method (see [21]) as the reference eigenvalues. It can be seen from Figure 2 that the error curves are approximately parallel to the line with slope -1, which indicates that Algorithm 2 achieves the optimal convergence rate of O(Nl-1) as well as Algorithm 1.
It also can be seen from Table 2 that when the degrees of freedom are the same, one can use Algorithms 1 and 2 to get the same accurate approximations with nearly the same CPU time.
The 1st and the 3rd eigenvalues of Example 2 obtained by two algorithms with H=2/32.
k
l
Nk,l(1)
λkhl(1)
CPUk,l(1)
Nk,l(2)
λkhl(2)
CPUk,l(2)
1
5
4369
0.18296573
0.46
4369
0.18296573
0.45
1
12
35119
0.18296446
2.00
35119
0.18296446
1.97
1
18
227719
0.18296426
14.90
227719
0.18296426
14.80
1
21
557370
0.18296425
39.77
557370
0.18296425
40.10
1
22
738108
0.18296425
55.04
738108
0.18296425
55.43
1
23
1105016
0.18296424
78.27
1105016
0.18296424
78.83
3
5
3690
1.68889444
0.49
3690
1.68889444
0.52
3
12
31089
1.68863207
1.79
31089
1.68863207
1.84
3
18
183089
1.68860633
11.84
183089
1.68860633
12.08
3
24
1098418
1.68860134
81.42
1098418
1.68860134
83.23
3
25
1485695
1.68860109
112.52
1485695
1.68860109
114.26
3
26
2002315
1.68860097
155.64
2002315
1.68860097
156.76
The curves of error and the a posteriori error estimators of two algorithms for the 1st (a) and 3rd (b) eigenvalues on the L-shaped domain.
Example 3.
We use Algorithms 1 and 2 to compute the approximations of the 1st and the 5th eigenvalue of (1) with the triangle linear finite element on Ω={(x1,x2):x1+x2<1}∖{(x1,x2):0≤x1≤1,x2=0}. The numerical results are presented in Table 3.
Since the exact eigenvalues are unknown, we compute the approximations of two exact eigenvalues of (1): λ1≈0.23957338768 and λ5≈1.41238071918 by the standard adaptive algorithm (see, e.g., [22]) with the degrees of freedom of more than 5000000. We show the curves of the error and the a posteriori estimators obtained by two algorithms for λ1 and λ5 in Figure 3. We can see from Figure 3 that the error curves are approximately parallel to the line with slope -1, which indicates that Algorithm 2 achieves the optimal convergence rate of O(Nl-1) as well as Algorithm 1.
From the numerical results in Table 3, we can conclude that Algorithm 2 is also an efficient approach like Algorithm 1 for solving the Steklov eigenvalue problem.
The 1st and 5th eigenvalues of Example 3 obtained by two algorithms with H=2/32.
k
l
Nk,l(1)
λkhl(1)
CPUk,l(1)
Nk,l(2)
λkhl(2)
CPUk,l(2)
1
5
9391
0.23957586
0.66
9391
0.23957586
0.75
1
10
42645
0.23957397
2.45
42645
0.23957397
2.56
1
15
189550
0.23957352
12.52
189550
0.23957352
12.72
1
19
634556
0.23957342
48.00
634556
0.23957342
47.09
1
20
860490
0.23957341
66.32
860490
0.23957341
65.38
1
21
1131274
0.23957340
91.04
1131274
0.23957340
90.05
5
5
9842
1.41254843
0.70
9842
1.41254843
0.74
5
12
77005
1.41241115
4.53
77005
1.41241115
4.68
5
18
453543
1.41238410
31.53
453543
1.41238410
31.74
5
21
1092642
1.41238245
82.23
1092642
1.41238245
82.74
5
22
1497488
1.41238165
112.92
1497488
1.41238165
113.82
5
23
1993327
1.41238104
155.36
1993327
1.41238104
156.14
The curves of error and the a posteriori error estimators of two algorithms for the 1st (a) and 5th (b) eigenvalues on slit domain.
Example 4.
We use the method in [10] (see Algorithms 4.1 and 7.2 there) to compute the numerical eigenvalues of (1) on [0,1]×[0,1], ([0,1]×[0,1/2])⋃([0,1/2]×[1/2,1]), and {(x1,x2):x1+x2<1}∖{(x1,x2):0≤x1≤1,x2=0}, respectively, and list the associated results in Tables 4–6 which are denoted by λkhl(3) and CPUk,l(3).
From Tables 4–6 we can see that, with the same degrees of freedom Nk,l, our method uses less CPU time to obtain the same accurate approximations, especially for multiple eigenvalue λ2 on [0,1]×[0,1], comparing with the one in [10].
The results of Example 4 on Ω=[0,1]×[0,1].
k
l
Nk,l
λkhl(3)
CPUk,l(3)
λkhl(2)
CPUk,l(2)
1
1
1089
0.240088481
0.08
0.240088481
0.18
1
2
4225
0.240081438
0.17
0.240081438
0.22
1
3
16641
0.240079674
0.48
0.240079674
0.43
1
4
66049
0.240079233
1.92
0.240079233
1.45
1
5
263169
0.240079122
8.92
0.240079122
6.48
1
6
1050625
0.240079095
38.36
0.240079095
32.00
2
1
1089
1.492905398
0.11
1.492905378
0.19
2
2
4225
1.492454269
0.23
1.492454267
0.24
2
3
16641
1.492340958
0.65
1.492340958
0.46
2
4
66049
1.492312593
2.67
1.492312593
1.47
2
5
263169
1.492305499
12.19
1.492305499
6.57
2
6
1050625
1.492303726
56.05
1.492303726
32.00
The results of Example 4 on Ω=([0,1]×[0,1/2])∪([0,1/2]×[1/2,1]).
k
l
Nk,l
λkhl(3)
CPUk,l(3)
λkhl(2)
CPUk,l(2)
1
1
833
0.182975157
0.04
0.182975157
0.19
1
2
3201
0.182966980
0.11
0.182966980
0.22
1
3
12545
0.182964924
0.31
0.182964924
0.36
1
4
49665
0.182964409
1.25
0.182964409
1.06
1
5
197633
0.182964280
5.88
0.182964280
4.60
1
6
788481
0.182964248
26.02
0.182964248
21.61
3
1
833
1.690165085
0.05
1.690165013
0.20
3
2
3201
1.688996545
0.12
1.688996536
0.22
3
3
12545
1.688700132
0.34
1.688700131
0.36
3
4
49665
1.688625481
1.27
1.688625481
1.04
3
5
197633
1.688606742
5.95
1.688606742
4.34
3
6
788481
1.688602046
26.11
1.688602046
20.20
The results of Example 4 on Ω=(x1,x2):x1+x2<1∖(x1,x2):0≤x1≤1,x2=0.
k
l
Nk,l
λkhl(3)
CPUk,l(3)
λkhl(2)
CPUk,l(2)
1
1
2145
0.239589697
0.12
0.239589697
0.22
1
2
8385
0.239577621
0.28
0.239577621
0.31
1
3
33153
0.239574482
0.96
0.239574482
0.77
1
4
131841
0.239573668
4.03
0.239573668
2.99
1
5
525825
0.239573457
17.43
0.239573457
13.80
1
6
2100225
0.239573403
78.31
0.239573403
71.35
5
1
2145
1.413086665
0.12
1.413086553
0.20
5
2
8385
1.412557485
0.28
1.412557472
0.29
5
3
33153
1.412424609
0.91
1.412424608
0.68
5
4
131841
1.412391332
3.92
1.412391332
2.77
5
5
525825
1.412383004
17.35
1.412383004
12.79
5
6
2100225
1.412380921
79.20
1.412380921
64.99
Competing Interests
The authors declare that there are no competing interests regarding the publication of this paper.
Acknowledgments
This work was supported by the National Natural Science Foundation of China (no. 11201093) and the Science and Technology Foundation of Guizhou Province of China (LKS[2013]06).
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