Explicit a posteriori and a priori error estimation for the finite element solution of Stokes equations
Xuefeng Liu, Mitsuhiro Nakao, Chun'guang You, Shin'ichi Oishi
aa r X i v : . [ m a t h . NA ] J un Explicit a posteriori and a priori error estimation for thefinite element solution of Stokes equations
Xuefeng LIU · Mitsuhiro NAKAO · Chun’guang YOU · Shin’ichi OISHI
June 4, 2020
Abstract
For the Stokes equation over 2D and 3D domains, explicit a posteriori and a priori error estimation are novelly developed for the finite element solution.The difficulty in handling the divergence-free condition of the Stokes equation issolved by utilizing the extended hypercircle method along with the Scott-Vogeliusfinite element scheme. Since all terms in the error estimation have explicit values,by further applying the interval arithmetic and verified computing algorithms,the computed results provide rigorous estimation for the approximation error.As an application of the proposed error estimation, the eigenvalue problem of theStokes operator is considered and rigorous bounds for the eigenvalues are obtained.The efficiency of proposed error estimation is demonstrated by solving the Stokesequation on both convex and non-convex 3D domains.
Keywords
Stokes equation · a posteriori error estimation · a priori errorestimation · finite element method · hypercircle method · eigenvalue problem Mathematics Subject Classification (2010) · · The first author is supported by Japan Society for the Promotion of Science, Grant-in-Aidfor Scientific Research (B) 16H03950, 20H01820 and Grant-in-Aid for Scientific Research(C) 18K03411. The second author is supported by Grant-in-Aid for Scientific Research (C)18K03434. The last author is supported by JST CREST Grant Number JPMJCR14D4, Japan.Xuefeng LIUGraduate School of Science and Technology, Niigata University, Niigata 950-2181, JapanE-mail: xfl[email protected] NAKAOFaculty of Science and Engineering, Waseda University, Tokyo 169-8555, JapanE-mail: [email protected]’guang YOUHe attended this research when he was a Ph.D. student at Academy of Mathematics andSystems Science, Chinese Academy of Sciences, China.E-mail: [email protected]’ichi OISHIFaculty of Science and Engineering, Waseda University, Tokyo 169-8555, JapanE-mail: [email protected] Xuefeng LIU et al.
The error estimation theory for approximate solutions to the Stokes equation is oneof fundamental problems in numerical analysis for fluid simulation. For example,in the approach of investigating the solution to the Navier–Stokes equation byverified computing, an explicit error estimation for the approximate solution tothe Stokes equation is desired. In [14], by providing the a priori error estimationfor the Stokes equation, Watanabe–Yamamoto–Nakao developed an algorithm toverify the solution existence for a stationary Navier–Stokes equation over a 2Dsquare domain. However, for general 2D domains and further 3D domains, the apriori error estimation for Stokes equation is not yet available, which remains to bethe bottleneck problem for the solution verification of the Navier–Stokes equation.The main difficulty in the a priori error estimation is due to the divergence-freecondition required in the Stokes equation. The classical study on the numericalsolutions to Stokes equation usually involves bounded but unknown constants inthe error estimation terms; see, for example, the pioneer work in [13]. For 2Ddomains, the Korn inequality (see, e.g., [11]) has been utilized to construct the a priori error estimation for star-shaped 2D domain. However, for a 2D domainwith general shape, and the domains in 3D space, e.g. a cube, it is still an openproblem to give explicit values for the constant in Korn’s inequality.This paper is an approach to solve the bottleneck problem in the solutionverification of the Navier–Stokes equation. We apply the Scott-Vogelius type finiteelement method (FEM) [3, 17, 16] to obtain a divergence-free approximation to theStokes equation and then propose an explicit a priori error estimation for the Stokesequation over general 2D and 3D domains. In our proposed error estimation, theidea of the hypercircle method has been utilized to take the advantage of thedivergence-free property of the approximate solution and further construct theexplicit error estimation. The hypercircle method, also named by the Prage-Syngetheorem, has been used in the error estimation for the Poisson equation (see [6, 9,7]); the error estimation here can be regarded as a direct extension of the resultof Liu–Oishi [9].The features of proposed method can be summarized as follows: – By combining the extended hypercircle method and the Scott-Vogelius FEMscheme, one can obtain explicit error estimation for the finite element solu-tion to the Stokes equation. Since all terms in the error estimation have ex-plicit values, by further applying the verified computation technique, the com-puted rigorous results can be further applied in the solution verification for theNavier–Stokes equation. – The proposed method only utilizes the H information of the weak solutions,which enables the application of the error estimation to Stokes equations overgeneral 2D and 3D non-convex domains, in which cases, the solution usuallycontains singularity.As an application of the explicit error estimation proposed in this paper, weconsider the eigenvalue problem of the Stokes equation and provide explicit lowerand upper bounds for the eigenvalues of the Stokes operator. See the detaileddiscussion in § § § § xplicit error estimation for the finite element solution of Stokes equations 3 FEM spaces to be used in the solution approximation and error estimation. In § a posteriori and a priori error estimation are proposed. In §
5, an application ofthe a priori error estimation to the eigenvalue problem of the Stokes operator isprovided. § To make the argument concise, we only consider the equation over a 3D domain,while the 2D case can be regarded as a special case and can be processed in ananalogous way. Generally, a bounded Lipschitz domain will be preferred in numer-ical analysis. However, to have the domain completely partitioned by tetrahedra,the domain is assumed to a polyhedron in solving practical problems.Let L p ( Ω ) ( p >
0) and H k ( Ω ), H k ( Ω ) ( k = 1 , , · · · ) be the standard Sobolevspaces over Ω . The inner product in L ( Ω ) is denoted by ( · , · ) Ω or ( · , · ); the L normof a function in L ( Ω ) is denoted by k · k . The space L ( Ω )( ⊂ L ( Ω )) has functionwith degenerated average over the domain, that is, L ( Ω ) := { v ∈ L ( Ω ) | ( v, Ω =0 } . Let us introduce the divergence-free space V by V = { v ∈ ( H ( Ω )) | div v = 0 } , (1)where div v denotes the divergence of vector v . The inner product and the normof V are defined by( u , v ) V := Z Ω ∇ u · ∇ v d Ω, k u k V := p ( u , u ) V . Here, ∇ u denotes the gradient of function u . The H (div) space is defined by H (div ; Ω ) := { p ∈ L ( Ω ) | div p ∈ L ( Ω ) } . We further introduce the space ( H (div; Ω )) , the member function p = ( p , p , p )of which has the divergence as div p := (div p , div p , div p ) ∈ ( L ( Ω )) .In this paper, we consider the Stokes equation in a weak formulation: Given f ∈ ( L ( Ω )) , let u ∈ V be the exact solution such that( ∇ u , ∇ v ) = ( f , v ) , ∀ v ∈ V . (2)The solution existence and uniqueness of the above equation can be easily con-firmed by applying the Lax-Milgram theorem. The saddle point formulation uti-lizing test function space L ( Ω ) is given by (see, e.g., [4, § u ∈ ( H ( Ω )) and ρ ∈ L ( Ω ) such that( ∇ u , ∇ v ) + (div v , ρ ) + (div u , η ) = ( f , v ) , ∀ v ∈ ( H ( Ω )) , η ∈ L ( Ω ) . (3)Such a formulation will be used in the FEM approximation in §
3. As the objectiveof this paper, we will consider an conforming FEM approximation to (3) andprovide explicit error estimation.Below, let us introduce an extended version of the hypercircle method, whichwill help to construct an explicit error estimation for the Stokes equation.
Xuefeng LIU et al.
Lemma 1 (Extended Prager-Synge’s theorem)
Given f ∈ ( L ( Ω )) , let u be thesolution to (2) corresponding to f . Suppose that p ∈ H ( div ; Ω ) satisfies, div p + ∇ φ + f = 0 , for certain φ ∈ H ( Ω ) . (4) Then for any v ∈ V , the following Pythagoras equation holds, k∇ u − ∇ v k + k∇ u − p k = k p − ∇ v k . (5) Proof
The holding of the equality can be confirmed by the expansion of k p −∇ v k = k ( ∇ u − p ) −∇ ( u − v ) k , which has the cross term as zero due to the divergence-freecondition and the boundary condition of u and v :( ∇ u − p , ∇ ( u − v )) = ( f + div p , ( u − v )) = ( −∇ φ, ( u − v )) = ( φ, div ( u − v )) = 0 . Remark 1
The selection of p and v in (5) is not unique. It is easy to see that, for afixed solution u , to minimize k p − ∇ v k is equivalent to minimize both k∇ u − ∇ v k and k∇ u − p k independently. In this section, we introduce the FEM spaces to be used in the solution approxi-mation and error estimation.Let T h be a regular tetrahedron subdivision for domain Ω . Further requirementto the mesh for the purpose of a stable computation of the Stokes equation willbe explained afterward. On each element K ∈ T h , denote by P m ( K ) the set ofpolynomials with degree up to m . We choose the Scott-Vogelius type finite elementmethod to construct divergence-free FEM spaces to approximate the space V , Discontinuous space X h of degree d Let X ( d ) h be the set of piecewise polynomials ofdegree up to d . Let X ( d ) h := ( X ( d ) h ) . Let X ( d ) h, := L ( Ω ) ∩ X ( d ) h . Conforming FEM space U h ( ⊂ (cid:0) H ( Ω ) (cid:1) ) and V h ( ⊂ V ) of degree k . – Let U ( k ) h be the space consisted of piecewise polynomials of degree up to k ,which also belong to H ( Ω ). That is, U ( k ) h := H ( Ω ) ∩ X ( k ) ( Ω ). Define U ( k ) h :=( U ( k ) h ) . – Let U ( k ) h, := { u h ∈ U ( k ) h | u h = 0 on ∂Ω } , U ( k ) h, := ( U ( k ) h, ) . – Let V ( k ) h be the subspace of U ( k ) h, with divergence-free member function. Thatis, V ( k ) h = { u h ∈ U ( k ) h, | div u h = 0 } = U ( k ) h ∩ V . The Raviart-Thomas FEM space RT h of degree m Define RT ( m ) h by RT ( m ) h := { p h ∈ H (div; Ω ) | p h | K = a K + b K x , ∀ K ∈ T h } . Here, a K ∈ ( P m ( K )) , b K ∈ P m ( K ). Define the tensor space RT ( m ) h := ( RT ( m ) h ) .For the FEM spaces defined here, the following properties hold.div ( RT ( m ) h ) = X ( m ) h , ∇ ( U ( k ) h ) ⊂ X ( k − h . (6)We may omit the superscript of degree in the notation for FEM spaces to have,for example, X h , V h , RT h . xplicit error estimation for the finite element solution of Stokes equations 5 Construction of V h Generally, it is difficult to construct V h directly due to thedivergence-free condition. We turn to utilize the Scott-Vogelius type FEM space[3], which handles the divergence-free condition implicitly by utilizing the testfunctions. V ( k ) h = { v ∈ U ( k ) h, | (div v , η h ) = 0 ∀ η h ∈ X k − h } . The approximation to the Stokes equation with V h reads: Find u h ∈ V h such that( ∇ u h , ∇ v h ) = ( f , v h ) ∀ v h ∈ V h . (7)The saddle point formulation is given by: Find u h ∈ U ( k ) h, , η h ∈ X ( k − h, , s.t.,( ∇ u h , ∇ v h ) + (div v h , η h ) + (div u h , ρ h ) = ( f , v h ) ∀ v h ∈ U ( k ) h, , ρ h ∈ X ( k − h, . (8)To have the inf-sup condition hold for the above saddle point problem, weapply the method of S. Zhang [17] to create the tetrahedra division of domains(see detailed description in §
6) and select the degree of FEM spaces as below; d = m = k − , k ≥ . (9)Let P h be the projection P h : V → V h such that, for any v ∈ V ( ∇ ( v − P h v ) , ∇ v h ) = 0 , ∀ v h ∈ V h . (10)Thus, the solution u h of (8) is just u h = P h u . In this section, we consider the a posteriori and the a priori error estimation forthe finite element method solution to the Stokes equation.As a preparation, let us introduce the constant C ,h , which is used in the errorestimation of the L -projection π h : L ( Ω ) → X h : for any u ∈ V , k u − π h u k ≤ C ,h k∇ u k ( C ,h = O ( h )) . (11)It is easy to see that the constant b C ,h in the following inequality provides anupper bound for C ,h . k u − π h u k ≤ b C ,h k∇ u k ∀ u ∈ H ( Ω ) . (12)Here, by using the same notation as in (11), π h denotes the projection π h : L ( Ω ) → X h . The explicit bounds of b C ,h and C ,h are given in § f h := π h f ∈ X h with an auxillaryboundary value problem: Find u ∈ V such that( ∇ u , ∇ v ) = ( f h , v ) ∀ v ∈ V . (13)The estimate of k∇ ( u − u ) k can be obtained by applying the estimation (11) of π h to (2) − (13):( ∇ ( u − u ) , ∇ v ) = ( f − f h , v ) = ( f − f h , ( I − π h ) v ) ≤ C ,h k f − f h kk∇ v k . Xuefeng LIU et al.
By further taking v := u − u , we have k∇ ( u − u ) k ≤ C ,h k f − f h k . (14)Due to the properties in (6), for any φ h ∈ U h and f h ∈ X h , we can find p h ∈ RT h such that the following equation holds.div p h + ∇ φ h + f h = 0 . (15)4.1 A posteriori error estimationLet us consider the a posteriori error estimation based on the hypercircle in (5).
Theorem 1 (A posteriori error estimation)
For f ∈ L ( Ω ) , let u be the exactsolution to the Stokes equation corresponding to f . Let p h ∈ RT h be an approximationto ∇ u satisfying the condition in (15). Then we have an a posteriori error estimationfor both u h and p h k∇ ( u − u h ) k , k∇ u − p h k ≤ k p h − ∇ u h k + C ,h k f − f h k . Proof
Let u be the solution of (13). Replace f with f h in Lemma 1, then we have k∇ ( u − u h ) k ≤ k p h − ∇ u h k , k∇ u − p h k ≤ k p h − ∇ u h k . (16)By applying the triangle inequality, we have k∇ ( u − u h ) k ≤ k∇ ( u − u ) k + k∇ ( u − u h ) k . (17)With the estimation in (14) and the first inequality of (16), we have, k∇ ( u − u h ) k ≤ k p h − ∇ u h k + C ,h k f − f h k . Similarly, the estimation for k∇ u − p h k is obtained by noticing k∇ u − p h k ≤ k∇ ( u − u ) k + k∇ u − p h k . A priori error estimationLet us introduce a quantity κ h , which will play an important role in the a priori error estimation. κ h = max f h ∈ X h min p h ∈ RT h , v h ∈ V h k p h − ∇ v h kk f h k , (18)where the minimization with respect to p h is subject to the condition (15).By utilizing the quantity κ h , we obtain the a priori error estimation for FEMsolution. xplicit error estimation for the finite element solution of Stokes equations 7 Theorem 2
Given f ∈ L ( Ω ) , let u be the exact solution to the Stokes equation and u h = P h u . Then, we have k u − u h k ≤ C h k∇ u − ∇ u h k , k∇ u − ∇ u h k ≤ C h k f k . (19) Here, C h := q C ,h + κ h . Proof
Take f h := π h f ∈ X h and let u be the exact solution to the Stokes equationcorresponding to f h and u h = P h u . From the definition of projection P h , thehypercircle (5) and the definition of κ h , we have k∇ ( u − u h ) k = min v h ∈ V h k∇ ( u − v h ) k ≤ min v h ∈ V h min p h ∈ RT h k∇ v h − p h k ≤ κ h k f h k (20)where the minimization w.r.t. p h is subject to the condition (15).By applying the minimization principle to u h = P h u and the triangle inequal-ity, we have k∇ ( u − u h ) k ≤ k∇ ( u − u h ) k ≤ k∇ ( u − u ) k + k∇ ( u − u h ) k . Then, we can draw the conclusion from the estimation in (14) and (20). k∇ ( u − u h ) k ≤ C ,h k f − f h k + κ h k f h k ≤ q C ,h + κ h k f k . The estimation of k u − u h k can be obtained by applying the standard the Aubin-Nitsche duality method.4.3 Computation of κ h In this subsection, we explain how to calculate the quantity κ h . Given f h ∈ X h ,let u be the exact solution to the Stokes equation corresponding to f h , i.e.,( ∇ u , ∇ v ) = ( f h , v ) , ∀ ∈ V . Then, from the hypercircle k∇ u − ∇ v h k + k∇ u − p h k = k∇ v h − p h k , we know that to minimize k∇ v h − p h k is equivalent to solve the following twoproblems. min v h ∈ V h k∇ u − ∇ v h k , min p h ∈ RT h subject to (15) k∇ u − p h k . The minimizer u h and p h can be obtained by solving the following weak problems. Xuefeng LIU et al.
1) Find u h ∈ V h s.t. ( ∇ u h , ∇ v h ) = ( f h , v h ) , ∀ v h ∈ V h . Since it is difficult to construct V h explicitly, we solve the problem by using U h and X h [17]: Find u h ∈ U h, , η h ∈ X h , c ∈ R s.t.( ∇ u h , ∇ v h ) + (div u h , ρ h ) + (div v h , η h ) + ( c, ρ h ) + ( d, η h ) = ( f h , v h ) , (21)for any v h ∈ U h, , ρ h ∈ X h , d ∈ R .2) Find p h ∈ RT h , η h ∈ X h , φ h ∈ U h such that( p h , q h ) + ( η h , div q h + ∇ ψ h ) + (div p h + ∇ φ h , ρ h ) = ( − f h , ρ h ) , (22)for any q h ∈ RT h , ρ h ∈ X h , and ψ h ∈ U h .By taking v h := u h in (21) and q h := p h in (22), we have the followingequalities for the minimizer u h and p h .( ∇ u h , ∇ u h ) = ( f h , u h ) , ( p h , p h ) = ( − η h , div p h + ∇ φ h ) = ( f h , η h ) . Thus, the error term k∇ u h − p h k can be presented by u h and η h . k∇ u h − p h k = ( ∇ u h − p h , ∇ u h − p h )= ( ∇ u h , ∇ u h ) − ∇ u h , p h ) + ( p h , p h )= ( f h , u h ) + 2( u h , − f h − ∇ φ h ) + ( f h , η h )= ( f h , η h − u h ) . Let K , K and K be the linear operators that map f h to u h , p h , η h , respec-tively. Then, the quantity κ h can be calculated by solving the following problems. κ h = max f h ∈ X h k∇ ( K f h ) − K f h k k f h k = max f h ∈ X h ( f h , ( K − K ) f h ) k f h k . For detailed computation of κ h , we can refer to [9], where a similar κ h for thePoisson’s equation is discussed and the value of κ h is solved by solving a matrixeigenvalue problem. Remark 2
An efficient computation of the approximate value of κ h is possible byapplying iteration methods to solve the matrix eigenvalue problem. In the itera-tion process, for an approximate eigenvector f h , one can solve the sub problem 1)and 2) with standard linear solvers for sparse matrices. However, the complexityto have a guaranteed estimation of quantity κ h is much higher than an approxi-mate estimation. Generally, to give an upper bound of the maximum eigenvalue ofeigenvalue problem Ax = λBx , the basic idea is to apply Sylvester’s law of inertiato show that b λB − A is positive definite for a candidate upper bound b λ . In thisprocess, the explicit forms of A and B are required. Thus, one has to calculatethe inverse of sparse matrices to create the matrices corresponding to the linearoperator K and K , which is quite time-consuming and requires huge computermemory. xplicit error estimation for the finite element solution of Stokes equations 9 In this section, we apply the a priori error estimation to the eigenvalue estimationproblem for the Stokes operator: Find u ∈ V and λ ∈ R such that( ∇ u , ∇ v ) = λ ( u , v ) ∀ v ∈ V . (23)Denote the eigenvalues of the above problem by λ ≤ λ ≤ · · · . The FEM approachfor the Stokes eigenvalue problem is as follows. Find u h ∈ V h and λ h ∈ R suchthat ( ∇ u h , ∇ v h ) = λ h ( u h , v h ) ∀ v h ∈ V h . (24)Denote the approximate eigenvalues by λ h, ≤ λ h, ≤ · · · ≤ λ h,n ( n = dim( V h )).Since V h ⊂ V , the min-max principle assures that the FEM approximation λ h,k gives upper bound for the exact eigenvalue λ k .Next theorem is a direct result by applying Theorem 2.1 of Liu [10] to theStokes eigenvalue problem (23). Theorem 3 (Application of Theorem 2.1 of [10])
Let P h be the project definedin (10). Then, the eigenvalue λ k of (23) has a lower bound as λ k ≥ λ h,k C h λ h,k (=: λ h,k ) , k = 1 , , · · · , dim ( V h ) . (25) Remark 3
The original Theorem 2.1 in [10] also works for non-conforming FEMspace. In [15], the Crouzeix-Raviart non-conforming FEM is utilized to bound theeigenvalue of the Stokes operator on 2D domains. In case of 3D domains, let theapproximate eigenvalue obtained by the Crouzeix-Raviart FEM space be λ NCh,k ,then we have λ k ≥ λ NCh,k . h ) λ NCh,k (=: λ NCh,k ) , k = 1 , , · · · , dim( V h ) . (26)Here, the quantity 0 . a priori estimation and the one based on the non-conforming FEMspace are compared in the section of numerical computations. In this section, we solve the Stokes equation over several 3D domains, includingthe cube domain, the L-shape domain and the cube-minus-cube domain.To have a stable computing of the Stokes equation, we apply Zhang’s method[17] in the mesh generation process. First, each domain is subdivided into uniformsmall cubes. Then each cube is divided into 5 tetrahedra. Finally, by followingZhang’s method, each tetrahedron is partitioned into 4 sub-tetrahedra by using thebarycentric of the tetrahedron. Note that for 2D case, a mesh without degeneratepoint is required for a stable computation [3].Let h be the edge of length of small cubes in the subdivision. From the results in[8], we have an upper bound for the Poincar´e constant over the special tetrahedraresulted by Zhang’s subdivision method. C ,h ≤ b C ,h ≤ . h ( h : the largest edge length of sub-cubes) . (27) For the FEM spaces over the mesh, the degrees of FEM function spaces areselected such that d = m = k − ≥ k ≥ d = m = 1 , k = 2. Notice that it is difficult to obtain guaranteedestimation for κ h with k = 2.To have rigorous bounds for the estimation of κ h and C h , we apply the veri-fied computation method in the computation and use the INTLAB toolbox [12]for interval arithmetic. However, since the algorithm in verified computation ofeigenvalues involves inverse computation of sparse matrices, the verified computa-tion requires huge computer memory and the computation takes longer time thanapproximate computation. See the comparison of resource consuming in Table 3.For the approximate estimation of κ h , the involved matrix eigenvalue problem fora dense mesh is solved by using the SLEPc eigenvalue solver in the PETSc library[5, 1].In the last subsection, we also solve the eigenvalue problem (23) on a cubedomain to estimate the Poincar´e constant over the divergence-free space V .6.1 A priori error estimation over 3D domainsThis subsection displays the a priori error estimation results for several 3D do-mains. In the tables of the computation results, κ h . Cube domain
For the unit cube domain Ω = (0 , , the computation results with k = 2 and k = 3 are listed in Table 1 and 2, respectively. In this case, we confirmthe convergence rate of C h in both cases is about one. Table 1
The a priori error estimation (Cube domain, d = m = 1 , k = 2)N κ h C ,h C h Order1 20 886 1.34E-1 2.84E-1 3.14E-1 -2 160 6774 1.04E-1 1.42E-1 1.76E-1 0.844 1280 53050 5.54E-2 7.10E-2 9.01E-2 0.978 10240 420018 2.83E-2 3.55E-2 4.54E-2 0.99
L-shaped domain
We consider the L-shaped domain Ω := (( − , \ [ − , ) × (0 , a priori error estimationresults are displayed in Table 4 and 5. Here, N denotes the number of sub-cubesalong z direction in the subdivision process. Since the domain is non-convex, thesolution of Stokes equation may have singularity around the re-entrant boundary,which will cause a dropped convergence rate. xplicit error estimation for the finite element solution of Stokes equations 11 Table 2
The a priori error estimation (Cube domain, d = m = 2 , k = 3)N DOF κ h C ,h C h Order1 2284 1.22E-1 2.84E-1 3.09E-1 -2 17664 7.99E-2 1.42E-1 1.63E-1 0.993 139030 3.23E-2 7.10E-2 7.80E-2 1.004 1103370 1.61E-2 3.55E-2 3.90E-2 1.00Note: Underlined numbers are rigorous bounds by using verified computation. Approximateesitmation shows that κ h ≈ .
121 for N = 1, κ h ≈ . N = 2. Table 3
Resource consuming comparison between approximate scheme and verified compu-tation Approximate scheme Verified computationComputer memory 19.7MB 205.8MBComputing time 0.5 second 54.8 seconds
Table 4
The a priori error estimation and eigenvalue error bounds (L-shaped domain, d = m = 1 , k = 2) N κ h C ,h C h Order1 60 2640 1.36E-1 2.84E-1 3.15E-1 -2 480 20126 1.27E-1 1.42E-1 1.91E-1 0.73 3840 158410 7.67E-2 7.10E-2 1.05E-1 0.94 30720 1257170 4.83E-2 3.55E-2 5.99E-2 0.8
Table 5
The a priori error estimation and eigenvalue error bounds (L-shaped domain, d = m = 2 , k = 3) N DOF κ h C ,h C h Order1 6746 1.38E-1 2.84E-1 3.16E-1 -2 52604 8.00E-2 1.42E-1 1.63E-1 0.964 415598 4.43E-2 7.10E-2 8.37E-1 0.968 3304250 2.92E-2 3.55E-2 4.59E-2 0.87Note: Approximate esitmation shows that κ h ≈ N = 1. Cube-minus-cube domain
Let us consider the Stokes equation over a cube-minus-cube domain Ω := ((0 , \ [1 , ), which is consisted of 7 unit cubes. In themesh generation process, denote by N the number of sub-cubes along z direction.Notice that the initial domain has N = 2 while the edge length of sub-cube is 1.The computational results are listed in Table 6 and 7. To have verified bound for κ h , it takes about 2.4 hours and about 9GB memory for the case N = 2. Moreover,due to the accumulation of rounding error in verified computing, the estimatedrigorous bound of κ h is about 1 . Table 6
Quantities in the a priori error estimation (Cube-minus-cube domain, d = m =1 , k = 2) N κ h C ,h C h Order2 140 5962 2.02E-1 2.84E-1 3.49E-1 -4 1120 46554 1.33E-1 1.42E-1 1.94E-1 0.858 8960 368050 8.19E-2 7.10E-2 1.08E-1 0.8416 8960 2927202 5.30E-2 3.55E-2 6.38E-2 0.77
Table 7
Quantities in the a priori error estimation (Cube-minus-cube domain, d = m =2 , k = 3) N DOF κ h C ,h C h Order2 15526 1.85E-1 2.84E-1 3.39E-1 -4 121926 8.52E-1 1.42E-1 1.66E-1 1.038 966538 4.64E-1 7.10E-2 8.48E-2 0.9716 7697298 3.19E-2 3.55E-2 4.77E-2 0.83Note: Approximate estimation shows that κ h ≈ N = 2. a priori to estimate the Poincar´e constant by solving the corre-sponding eigenvalue problem of the Stokes operator over the unit cube domain.The Poincar´e constant C p over V is the optimal quantity to make the followinginequality hold. k v k L ( Ω ) ≤ C p k∇ v k L ( Ω ) ∀ v ∈ V . (28)Th constant C p is determined by the first eigenvalue of the Stokes operator definedin (23). That is, C p = 1 / √ λ Recall the Poincar´e constant C ′ p over H ( Ω ) such that, k v k ≤ C ′ p k∇ v k ∀ v ∈ H ( Ω ) . (29)It is easy to see that C p ≤ C ′ p . The estimation of C ′ p has the optimal value as C ′ p = √ / (3 π ) ≈ . · · · for the unit cube domain. The equality in (29) holdsfor v = sin πx sin πy sin πz .By applying the lower bound estimation in (26) along with the explicit valueof projection error quantity C h in Table 2, we obtain the two-side bounds for both λ and C p . As a comparison, the lower bounds based on the Crouzeix-Raviartnonconforming method are also provided. The estimation results are displayed inTable 8. xplicit error estimation for the finite element solution of Stokes equations 13 Table 8
Estimation of eigenvalue and the Poincar´e constant over the unit cube domainh C h λ h, λ h, λ NCh, Estimation of C p ≤ C p ≤ ≤ C p ≤ ≤ C p ≤ ≤ C p ≤ C ′ p ≈ . C p . Conclusion
For the Stokes equation over general 3D domains, an explicit error estimationfor the finite element approximation is proposed. The difficulty in dealing withthe divergence-free condition is solved by introducing the extended hypercirclemethod and utilizing the conforming FEM space provided by the Scott-Vogeliusscheme and Zhang’s mesh. The resulted estimation plays an important role in thecomputer-assisted proof for the solution existence verification to the Navier–Stokesequation, which will be challenged in the next step of our research.
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