Error analysis of a decoupled finite element method for quad-curl problems
EERROR ANALYSIS OF A DECOUPLED FINITE ELEMENTMETHOD FOR QUAD-CURL PROBLEMS ∗ SHUHAO CAO † , LONG CHEN ‡ , AND
XUEHAI HUANG §¶ Abstract.
Finite element approximation to a decoupled formulation for the quad–curl problemis studied in this paper. The difficulty of constructing elements with certain conformity to thequad–curl problems has been greatly reduced. For convex domains, where the regularity assumptionholds for Stokes equation, the approximation to the curl of the true solution has quadratic orderof convergence and first order for the energy norm. If the solution shows singularity, a posteriorerror estimator is developed and a separate marking adaptive finite element procedure is proposed,together with its convergence proved. Both a priori and a posteriori error analysis are supported bythe numerical examples.
Key words. quad-curl problem, decoupled formulation, a posteriori error estimator, adaptivefinite element methods
AMS subject classifications.
1. Introduction.
Quad-curl problem arises from multiphysics simulation suchas modeling a magnetized plasma in magnetohydrodynamics (MHD). In both limitingregimes, resistive MHD ([4, 40]) and electron MHD ([29, 11, 38]), discretizing the quad-curl operator is one of the keys to simulate these models. In the mean time, quad-curloperator also plays an important role in approximating the Maxwell transmissioneigenvalue problem [32, 10]. Recently, the designing of the approximations for quad-curl problems gain quite a few attentions from the finite element community ([26, 24,39, 42, 43, 23, 44, 45]). The structures of the quad-curl problem are unique as theoperator has a much bigger kernel than the one in the curl-curl problem. Second,to construct the conforming finite element approximations, the stringent continuitycondition drives the local polynomial space’s dimension to be much bigger than that ofthe curl-curl problem, which renders them unpractical especially in three dimensions.The nonconforming elements [45] greatly simply the local structure of the space andis more preferable in approximating the quad-curl problem.In [13, 26], a novel way of further simplifying the structure of the quad-curlproblem is proposed. The quad-curl problem is decoupled into three sub-problems,two curl-curl equations, and one Stokes equation, all of which have mature finiteelement approximation theories (e.g., [17, 33, 28, 27]). In this paper, we have analyzedthis decoupled FEM for the quad-curl problem, and due to the decoupling mechanism,one of the major advantages is that the curl of the primal variable can be approximatedan order higher than the aforementioned conforming or nonconforming FEMs. § CORRESPONDING AUTHOR. ∗ Submitted to the editors on DATE.
Funding:
The first author was supported in part by the National Science Foundation undergrant DMS-1913080. The second author was supported in part by the National Science Foundationunder grants DMS-1913080 and DMS-2012465. The third author was supported by the NationalNatural Science Foundation of China Project 11771338, and the Fundamental Research Funds forthe Central Universities 2019110066. † Department of Mathematics and Statistics, Washington University, St. Louis, MO 63130, USA([email protected]). ‡ Department of Mathematics, University of California at Irvine, Irvine, CA 92697, USA ([email protected]). ¶ School of Mathematics, Shanghai University of Finance and Economics, Shanghai 200433, China([email protected]). 1 a r X i v : . [ m a t h . NA ] F e b S. CAO, L. CHEN, AND X. HUANG
Meanwhile, due to the nature of quad-curl operator [34], on a polyhedral domain,the singularities of solution may manifest themselves as either corner singularities ofthe Stokes system with Dirichlet boundary conditions, corner/edge singularities ofthe Maxwell problem, or both. To cope with such solutions with the presence ofsingularities, adaptive finite element method (AFEM) is favored over the finite ele-ment method performed on a uniformly refine mesh. The computational resources areadaptively allocated throughout different locations of the mesh based on the local es-timated approximation error. Thus the AFEM can achieve the same overall accuracywhile using fewer degrees of freedom than the one with uniform mesh.Opting for a decoupled system using existing and mature elements for each offersgreat facilitation to the AFEM pipeline. Now there are three major pieces to thepuzzle: the a posteriori error estimation for the conforming approximation to theMaxwell problem (e.g., [3, 9, 36, 16]), that for a nonconforming discretization to theStokes problem (e.g., [18, 41, 19]), and the design of a convergent AFEM algorithm([22, 48, 47]). In this paper, combining the ingredients from both conforming andnonconforming methods, we are able to show that the AFEM algorithm based on thea posteriori error estimation is convergent under common assumptions.This paper is organized as follows: Section 2 introduces the decoupled formulationas well as its well-posedness. Section 3 proves the a priori error estimation in both theenergy norm and the L -norm. Section 4 gives the a posteriori error analysis, Section5 shows the quasi-orthogonality of the solution, and a convergence proof is given in6. In Section 7 comparison of the rates of convergence of the AFEMs using variousmarking strategies is presented.
2. A quad- curl problem and a decoupled formulation.
Let Ω ⊂ R be apolyhedron homomorphism to a ball, and f ∈ H (div , Ω) with div f = 0. Considerthe quad-curl problem(2.1) (curl) u = f in Ω , div u = 0 in Ω , u × n = (curl u ) × n = on ∂ Ω . The primal formulation of the quad-curl problem (2.1) is to find u ∈ H (curl , Ω) suchthat(2.2) (curl curl u , curl curl v ) = ( f , v ) ∀ v ∈ H (curl , Ω) , where(2.3) H (curl , Ω) := { v ∈ L (Ω , R ) : curl v , curl curl v ∈ L (Ω , R ) , div v = 0 , and v × n = (curl v ) × n = 0 on Γ := ∂ Ω } . Conforming finite element spaces for H (curl , Ω) has been recently constructedin [42, 24] for two dimensional dimensions and [43, 23] in three dimensions. Noncon-forming and low order finite element spaces can be found in [45, 26].A natural mixed method is to mimic the biharmonic equation by introducing w = ∇×∇× u and write as a system for which standard edge elements can be used; see [39].The main drawback of this decoupling is the loss of the order of convergence due tothe fact that boundary condition (curl u ) × n = is imposed weakly. Indeed a naturalspace for w is H − (curl curl , Ω) := { v ∈ L (Ω; R ) : curl curl v ∈ H − (div , Ω) } . Here H − (div , Ω) := { v ∈ H − (Ω; R ) : div v ∈ H − (Ω) } is the dual space of H (curl , Ω) ECOUPLED FEM FOR QUAD-CURL PROBLEMS K c := { φ ∈ H (curl , Ω) : div φ = 0 } = H (curl , Ω) / grad H (Ω)equipped with norm (cid:107) · (cid:107) H (curl) . Due to the following commutative diagram H (Ω; R ) ∆ (cid:47) (cid:47) H − (Ω; R ) L (Ω) grad (cid:47) (cid:47) H − (curl , Ω) curl (cid:47) (cid:47) (cid:83) ( K c ) (cid:48) (cid:47) (cid:47) H (div , Ω) I (cid:79) (cid:79) K c (cid:79) (cid:79) curl (cid:111) (cid:111) , the primal formulation (2.2) of the quad-curl problem can be decoupled into thefollowing three systems [13, Section 3.4] (see also [44]):Step 1. Given f ∈ L (Ω), find w ∈ H (curl , Ω), σ ∈ H (Ω) s.t.(curl w , curl v ) + ( v , ∇ σ ) = ( f , v ) ∀ v ∈ H (curl , Ω) , (2.5) ( w , ∇ τ ) = 0 ∀ τ ∈ H (Ω) . (2.6)Step 2. Given w computed in Step 1, find φ ∈ H (Ω; R ), p ∈ L (Ω) s.t.( ∇ φ , ∇ ψ ) + (div ψ , p ) = (curl w , ψ ) ∀ ψ ∈ H (Ω; R ) , (2.7) (div φ , q ) = 0 ∀ q ∈ L (Ω) . (2.8)Step 3. Given φ computed in Step 2, find u ∈ H (curl , Ω) and ξ ∈ H (Ω) s.t.(curl u , curl χ ) + ( χ , ∇ ξ ) = ( φ , curl χ ) ∀ χ ∈ H (curl , Ω) , (2.9) ( u , ∇ ζ ) = 0 ∀ ζ ∈ H (Ω) . (2.10)That is the primal formulation (2.2) of the quad-curl problem (2.1) can be decoupledinto two Maxwell equations and one Stokes equation.Each system is well-posed and the solution ( w , σ, φ , p, u , ξ ) to (2.5)-(2.10) existsand is unique. Now we show briefly, without resorting to the abstract frameworkin [13], the equivalence of the decoupled formulation (2.5)-(2.10) and the primaryformulation (2.2).By taking χ = grad ξ in (2.9), we conclude the Lagrange multiplier ξ = 0. There-fore (2.9) becomes φ = curl u . Notice that the boundary condition curl u × n = 0implies that the tangential trace φ × n is zero, while u × n = 0 on boundary impliesthe normal trace φ · n = rot Γ u = 0. Together with curl φ = curl curl u ∈ L (Ω) anddiv φ = 0, by the embedding H (curl , Ω) ∩ H (div , Ω) (cid:44) → H (Ω) [35], we concludethat φ = curl u ∈ H (Ω). S. CAO, L. CHEN, AND X. HUANG
Furthermore by the identity( ∇ φ , ∇ ψ ) = (curl φ , curl ψ ) + (div φ , div ψ ) ∀ φ , ψ ∈ H (Ω; R ) , and div φ = 0, we can rewrite (2.7) as(2.11) (curl φ , curl ψ ) + (div ψ , p ) = (curl w , ψ ) ∀ ψ ∈ H (Ω; R ) . Noticing the fact div f = 0, by choosing v = ∇ σ , we get from (2.5) that theLagrange multipliers σ is also zero. Now choosing ψ = curl v in (2.11) for a v ∈ H (curl , Ω), we get(curl curl u , curl curl v ) = (curl φ , curl ψ ) = (curl w , ψ ) = ( f, v ) , which verifies that the solution u to (2.9)-(2.10) is also the solution to (2.2) and viceversa. Remark φ = curl u , but w (cid:54) =curl curl u . Equation (2.5) can be equivalently written as curl curl w = f but now w ∈ H (curl , Ω) while curl curl u ∈ H (div , Ω) may not satisfy the tangential boundarycondition.
3. Discrete Methods and A Priori Error Analysis.
We consider a conform-ing mixed finite element method of the Maxwell equations (2.5)-(2.6), and (2.9)-(2.10)but a nonconforming method for Stokes equation (2.7)-(2.8). We refer to [44] for aconforming mixed finite element method.Denote the p -th order Lagrange element space by V ph := { v h ∈ H (Ω) : v h | K ∈ P p ( K ) for each K ∈ T h } , and the lowest-order N´ed´elec edge element space [33] by V ch := { v h ∈ H (curl , Ω) : v h | K ∈ P ( K ; R ) ⊕ x ∧ P ( K ; R ) for each K ∈ T h } . We use V h − V ch to discretize the Maxwell equations (2.5)-(2.6). Find w h ∈ V ch , σ h ∈ V h s.t.(curl w h , curl v h ) + ( v h , ∇ σ h ) = ( f , v h ) ∀ v h ∈ V ch , (3.1) ( w h , ∇ τ h ) = 0 ∀ τ h ∈ V h . (3.2)We then use the nonconforming P - P element [17] to discretize the Stokes prob-lem (2.7)-(2.8). To this end, let V CR h := { ψ h ∈ L (Ω; R ) : ψ h | K ∈ P ( K ; R ) for each K ∈ T h , and ( (cid:74) ψ h (cid:75) , F = 0 for each F ∈ F h } , where (cid:74) v (cid:75) ( x ) := lim (cid:15) → + ( v | K ( x − (cid:15) n K ) − v | K ( x + (cid:15) n K )) is defined as the jump onface F the jump, for x ∈ F and n K being the outer unit normal to K on face F .Denote the piecewise constant space as Q h := { q h ∈ L (Ω) : q h | K ∈ P ( K ) for each K ∈ T h } . Given w h computed from (3.1)-(3.2), find φ h ∈ V CR h , p h ∈ Q h s.t.( ∇ h φ h , ∇ h ψ h ) + (div h ψ h , p h ) = (curl w h , ψ h ) ∀ ψ h ∈ V CR h , (3.3) (div h φ h , q h ) = 0 ∀ q h ∈ Q h . (3.4) ECOUPLED FEM FOR QUAD-CURL PROBLEMS φ h is a second-order approximation to curl u when the data is smooth. Finally, when one needs to seek a better approximation to u under L -norm, u h ∈ V c h and ξ h ∈ V h are sought such that they satisfy(curl u h , curl χ h ) + ( χ h , ∇ ξ h ) = ( φ h , curl χ h ) ∀ χ h ∈ V c h , (3.5) ( u h , ∇ ζ h ) = 0 ∀ ζ h ∈ V h . (3.6)Here V c h is the linear second family of N´ed´elec element: V c h := { v h ∈ H (curl , Ω) : v h | K ∈ P ( K ; R ) for each K ∈ T h } . and V h is the quadratic Lagrange element.The finite element pair ( V CR h , Q h ) is stable for the Stokes equation [5], i.e., wehave for any (cid:101) φ h ∈ V CR h and (cid:101) p h ∈ Q h that(3.7) | (cid:101) φ h | ,h + (cid:107) (cid:101) p h (cid:107) (cid:46) sup ψ h ∈ V CR h q h ∈Q h ( ∇ h (cid:101) φ h , ∇ h ψ h ) + (div h ψ h , (cid:101) p h ) + (div h (cid:101) φ h , q h ) | ψ h | ,h + (cid:107) q h (cid:107) . Next we focus on the a priori error analysis forthe decoupled mixed finite element method (3.1)-(3.6). First of all, since div f = 0and ∇ V h ⊂ V ch , we get from (3.1) and (3.5) that σ h = 0 and ξ h = 0. Lemma
Let ( w , ∈ H (curl , Ω) × H (Ω) be thesolution of the Maxwell equation (2.5) - (2.6) , and ( w h , ∈ V ch × V h be the solutionof the mixed method (3.1) - (3.2) . Then (3.8) (curl( w − w h ) , curl v h ) = 0 ∀ v h ∈ V ch . Proof.
As the Lagrange multiplier σ = 0 and its approximation σ h = 0, subtract-ing (3.1) from (2.5), we get the desired orthogonality.The error analysis of the mixed finite element method (3.1)-(3.2) is first studied byF. Kikuchi in [27, 28]. We recall it for completeness. Lemma
Let ( w , ∈ H (curl , Ω) × H (Ω) be the solution of the Maxwellequation (2.5) - (2.6) , and ( w h , ∈ V ch × V h the solution of the mixed method (3.1) - (3.2) . Assume curl w ∈ H (Ω; R ) , then we have (3.9) (cid:107) curl( w − w h ) (cid:107) (cid:46) h | curl w | . Proof.
The orthgonality (3.8) implies the best approximation (cid:107) curl( w − w h ) (cid:107) ≤ inf v h ∈ V ch (cid:107) curl( w − v h ) (cid:107) . This gives (3.9) by an interpolation error estimate (see e.g., [31]).According to the Poincar´e-Friedrichs inequality for piecewise H functions [7],the following inequality holds(3.10) (cid:107) ψ h (cid:107) (cid:46) | ψ h | ,h ∀ ψ h ∈ V CR h + H (Ω; R ) . Denote I sh as the nodal interpolation operator from H (Ω; R ) to V CR h , then(3.11) ( ∇ ( ψ − I sh ψ ) , τ ) K = 0 ∀ ψ ∈ H (Ω; R ) , τ ∈ P ( K ; M ) , K ∈ T h , S. CAO, L. CHEN, AND X. HUANG where P ( K ; M ) stands for the space of constant 3 × K , and for j = 1 , (cid:107) ψ − I sh ψ (cid:107) ,K + h K | ψ − I sh ψ | ,K (cid:46) h jK | ψ | j,K ∀ ψ ∈ H j (Ω; R ) , K ∈ T h . The error analysis for nonconforming P - P element approximation (3.3)–(3.4)of Stokes equation is standard [17]. Using the decoupled system to approximate thequad–curl problem, the subtlety is the perturbation of data. We shall present astability result for using curl w h to approximate curl w . To this end, we introducethe space Z := H (Ω) ∩ ker(div) . Subsequently (2.7) and the continuous problem using the perturbed data can bewritten as follows: − ∆ φ = curl w in Z (cid:48) and − ∆ ˜ φ = curl w h in Z (cid:48) , respectively. The second problem above is equivalent to( ∇ ˜ φ , ∇ ψ ) + (div ψ , ˜ p ) = (curl w h , ψ ) ∀ ψ ∈ H (Ω; R ) , (3.13) (div ˜ φ , q ) = 0 ∀ q ∈ L (Ω) . (3.14)The analysis is performed for this problem with the perturbed data. Lemma
Let ( φ , p ) , ( ˜ φ , ˜ p ) be the solutions to (2.7) – (2.8) and (3.13) – (3.14) ,respectively, where w and w h satisfy the orthogonality (3.8) . Then (cid:107) ˜ φ − φ (cid:107) (cid:46) h (cid:107) curl( w − w h ) (cid:107) . Proof.
The difference between the two pairs satisfies the Stokes equation − ∆( φ − ˜ φ ) + ∇ ( p − ˜ p ) = curl( w − w h ) in ( H (Ω)) (cid:48) , div( φ − ˜ φ ) = 0 in L (Ω) . Applying the definition of the duality pair testing against φ − ˜ φ , we get | φ − ˜ φ | = (curl( w − w h ) , φ − ˜ φ ) . Moreover, since div( φ − ˜ φ ) = 0, by [21, Chapter 1 Theorem 3.4] there exists v ∈ H (Ω) such that φ − ˜ φ = curl v , (cid:107) v (cid:107) (cid:46) (cid:107) φ − ˜ φ (cid:107) . Then it follows from (3.8) that | φ − ˜ φ | = (curl( w − w h ) , curl v ) = (curl( w − w h ) , curl( v − v h )) , ∀ v h ∈ V ch . Thus | φ − ˜ φ | ≤ (cid:107) curl( w − w h ) (cid:107) inf v h ∈ V ch (cid:107) curl( v − v h ) (cid:107) (cid:46) h (cid:107) curl( w − w h ) (cid:107) | φ − ˜ φ | , which implies the desired result.In the next step, we treat φ h as the approximation of ˜ φ and use the standard erroranalysis to obtain the following estimate. Here the H -regularity of Stokes equationis assumed to hold (e.g., see [30, Section 11.5]) for smooth or convex Ω, thus thestandard a priori estimate for the stable nonconforming P - P pair holds. ECOUPLED FEM FOR QUAD-CURL PROBLEMS Theorem
Let ( φ , p ) ∈ H (Ω; R ) × L (Ω) be the solution of the Stokesequation (2.7) - (2.8) , and ( φ h , p h ) ∈ V CR h × Q h the solution of the mixed method (3.3) - (3.4) . Assume the H -regularity of Stokes equation holds, then (3.15) | φ − φ h | ,h (cid:46) h (cid:107) curl w (cid:107) . Proof.
First by a standard estimate [17], and the elliptic regularity estimate ofthe approximation φ h for the Stokes problem with curl w h as data, we have | (cid:101) φ − φ h | ,h (cid:46) h | (cid:101) φ | (cid:46) h (cid:107) curl w h (cid:107) . Furthermore, as w h is the projection of w to the discrete space in the energy norm,from the orthogonality (3.8), we have (cid:107) curl w h (cid:107) ≤ (cid:107) curl w (cid:107) . Consequently, thetheorem follows from combining the estimate with the ones in Lemma 3.3. Remark u · n = div Γ ( u × n ). Consequently, the presence ofnonhomogeneous u × n and/or curl u × n leads to the necessity of impose compatibleDirichlet boundary conditions with the divergence free condition for problems (2.7)–(2.8), (3.13)–(3.14). Let φ I be the standard nodal interpolation in Crouzeix-Raviartelement of a sufficiently smooth φ , by a standard decomposition argument we can seethat aside from the terms on the right hand side of (3.15), for the nonhomogeneousboundary condition, the estimate should include: | φ − φ I | / ,h,∂ Ω := (cid:32) (cid:88) F ∈F h | φ − φ I | / ,F (cid:33) / (cid:46) h | φ | . Next we present the L -error estimate for the Stokes equation. Lemma
Let ( w h , , φ h , p h ) ∈ V ch × V h × V CR h × Q h be the solutions of themixed method (3.1) - (3.4) on triangulations T h . Assume H -regularity of Stokes equa-tion holds, then (3.16) (cid:107) φ − φ h (cid:107) (cid:46) h | φ − φ h | ,h + h (cid:107) curl( w − w h ) (cid:107) + h (cid:107) curl w (cid:107) . Furthermore if curl w ∈ H (Ω) , then we have the second order estimate (3.17) (cid:107) φ − φ h (cid:107) (cid:46) h (cid:107) curl w (cid:107) . Proof.
Consider the following dual problem: seek ( ˆ φ , ˆ p ) ∈ H (Ω) × L (Ω) suchthat (cid:40) − ∆ ˆ φ + ∇ ˆ p = φ − φ h , div ˆ φ = 0 . The H -regularity to the problem above (e.g., see [30, Section 11.5]) reads(3.18) (cid:107) ˆ φ (cid:107) + (cid:107) ˆ p (cid:107) (cid:46) (cid:107) φ − φ h (cid:107) . Since div h ( φ − φ h ) = 0, it follows (cid:107) φ − φ h (cid:107) = ( φ − φ h , − ∆ ˆ φ + ∇ ˆ p )= ( ∇ h ( φ − φ h ) , ∇ ˆ φ ) + (cid:88) K ∈T h ( φ − φ h , ˆ p n − ∂ n ˆ φ ) ∂K . (3.19) S. CAO, L. CHEN, AND X. HUANG
Employing (3.11) and the fact div h I sh ˆ φ = 0, we obtain( ∇ h ( φ − φ h ) , ∇ ˆ φ ) = ( ∇ φ , ∇ ˆ φ ) − ( ∇ h φ h , ∇ h I sh ˆ φ )= (curl w , ˆ φ ) − (curl w h , I sh ˆ φ )= (curl w − curl w h , ˆ φ ) + (curl w h , ˆ φ − I sh ˆ φ ) . Applying the same argument in Lemma 3.3 by treating ˆ φ as a stream function andinserting a curl of its interpolation, we achieve(curl w − curl w h , ˆ φ ) (cid:46) h (cid:107) curl( w − w h ) (cid:107) | ˆ φ | . Besides from (3.12), we have(curl w h , ˆ φ − I sh ˆ φ ) (cid:46) h (cid:107) curl w (cid:107) | ˆ φ | . Hence(3.20) ( ∇ h ( φ − φ h ) , ∇ ˆ φ ) (cid:46) h (cid:107) curl( w − w h ) (cid:107) | ˆ φ | + h (cid:107) curl w (cid:107) | ˆ φ | . Due to the continuity condition of Crouzeix-Raviart element, by a standard techniqueof inserting a constant on each face (e.g., see [8, Chapter 10.3]) we get(3.21) (cid:88) K ∈T h ( φ − φ h , ˆ p n − ∂ n ˆ φ ) ∂K (cid:46) h | φ − φ h | ,h ( (cid:107) ˆ φ (cid:107) + (cid:107) ˆ p (cid:107) ) . Combining (3.19)-(3.21) and (3.18) yields (cid:107) φ − φ h (cid:107) (cid:46) h | φ − φ h | ,h + h (cid:107) curl( w − w h ) (cid:107) + h (cid:107) curl w (cid:107) , which is (3.16).We now consider the approximation (3.5)–(3.6) of the last Maxwell equation.Due to the inexactness of the data, the orthogonality is lost but the perturbation ismeasured in L -norm of the difference φ − φ h , which is controllable. Lemma
Let ( u , ∈ H (curl , Ω) × H (Ω) be the solution of the Maxwellequation (2.9) - (2.10) , and ( u h , ∈ V c h × V h the solution of the mixed method (3.5) - (3.6) , then (3.22) (cid:107) curl( u − u h ) (cid:107) (cid:46) (cid:107) φ − φ h (cid:107) + inf v h ∈ V ch (cid:107) curl( u − v h ) (cid:107) , and when curl w , curl u ∈ H (Ω) , (3.23) (cid:107) curl( u − u h ) (cid:107) (cid:46) h | curl w | + h | curl u | . Proof.
Subtracting (3.5) from (2.9), we get(3.24) (curl( u − u h ) , curl χ h ) = ( φ − φ h , curl χ h ) ∀ χ h ∈ V c h . Taking χ h = v h − u h with v h ∈ V c h , we acquire (cid:107) curl( u − u h ) (cid:107) = (curl( u − u h ) , curl( u − u h ))= (curl( u − u h ) , curl( u − v h )) + ( φ − φ h , curl( v h − u h )) ≤ (cid:107) curl( u − u h ) (cid:107) (cid:107) curl( u − v h ) (cid:107) + (cid:107) φ − φ h (cid:107) ( (cid:107) curl( u − v h ) (cid:107) + (cid:107) curl( u − u h ) (cid:107) ) , which indicates (cid:107) curl( u − u h ) (cid:107) (cid:46) (cid:107) φ − φ h (cid:107) + inf v h ∈ V c h (cid:107) curl( u − v h ) (cid:107) . ECOUPLED FEM FOR QUAD-CURL PROBLEMS φ = curl u and (cid:107) φ − φ h (cid:107) is at least first order h . Therefore if stillmerely the lowest order edge element is used in (3.5)–(3.6), no approximation to curl u better than φ h could be obtained. By the duality argument for Stokes equation, theerror (cid:107) φ − φ h (cid:107) can be of second order h if the H -regularity result holds. As a resultin the last Maxwell equation, we opt to use the second family N´ed´elec element toimprove the L approximation of u to second order. Theorem
Let ( u , ∈ H (curl , Ω) × H (Ω) be the solution of the Maxwellequation (2.9) - (2.10) , and ( u h , ∈ V c h × V h the solution of the mixed method (3.5) - (3.6) . Assume Ω is convex, then (3.25) (cid:107) u − u h (cid:107) (cid:46) inf v h ∈ V c h (cid:110) (cid:107) u − v h (cid:107) + h (cid:107) curl( u − v h ) (cid:107) (cid:111) + h (cid:107) curl( u − u h ) (cid:107) + (cid:107) φ − φ h (cid:107) , and when curl w ∈ H (Ω) and u ∈ H (Ω) , (3.26) (cid:107) u − u h (cid:107) (cid:46) h ( | curl w | + (cid:107) curl u (cid:107) + | u | ) . Proof.
The proof is adapted from a similar argument in [49] without the dataperturbation. Denote e h := u − u h , then by (2.10) and (3.6), we have ( e h , ∇ ζ h ) = 0for ζ h ∈ V h , thus for any fixed v h ∈ V c h (cid:107) e h (cid:107) = ( e h , u − v h ) + ( e h , s h + ∇ q h ) = ( e h , u − v h ) + ( e h , s h ) , where a discrete Helmholtz decomposition(3.27) v h − u h = s h + ∇ q h , and ( s h , ∇ r h ) = 0 , ∀ r h ∈ V h is applied such that s h ∈ V c h . As a result,(3.28) (cid:107) e h (cid:107) (cid:46) (cid:107) u − v h (cid:107) + (cid:107) s h (cid:107) . An H (curl)-lifting s ∈ H (curl , Ω) (see [31, Lemma 7.6, Remark 3.52]) of s h is soughtsuch that curl s = curl s h , div s = 0 , and (cid:107) s − s h (cid:107) (cid:46) h (cid:107) curl s h (cid:107) . By the triangle inequality and (3.27), (cid:107) s h (cid:107) ≤ (cid:107) s (cid:107) + (cid:107) s − s h (cid:107) (cid:46) (cid:107) s (cid:107) + h (cid:107) curl s h (cid:107) = (cid:107) s (cid:107) + h (cid:107) curl( u h − v h ) (cid:107) , hence it suffices to bound (cid:107) s (cid:107) . Consequently, the Aubin-Nitsche argument is appliedon s , where we seek an ( r , ξ ) ∈ H (curl , Ω) × H (Ω) s.t.(curl r , curl χ ) + ( χ , ∇ ξ ) = ( s , χ ) ∀ χ ∈ H (curl , Ω) , (3.29) ( r , ∇ ζ ) = 0 ∀ ζ ∈ H (Ω) . (3.30)We have ξ = 0 since s is divergence free, and letting χ = s yields (cid:107) s (cid:107) = (curl r , curl s ) = (cid:0) curl r , curl( s + u h − v h ) (cid:1) + (cid:0) curl r , curl( − u h + v h ) (cid:1) = − (cid:0) curl r , curl( ∇ q h ) (cid:1) + (cid:0) curl r , curl( u − u h ) (cid:1) − (cid:0) curl r , curl( u − v h ) (cid:1) . S. CAO, L. CHEN, AND X. HUANG
By an embedding result (see [21, Chapter 1 Section 3.4]), the N´ed´elec nodal interpo-lation I c h r is well-defined, inserting which into the first above, letting χ = u − v h in(3.29)–(3.30), and by (3.24), we have (cid:107) s (cid:107) = (cid:0) curl e h , curl( r − I c h r ) (cid:1) + (curl e h , curl I c h r ) − ( s , u − v h )= (cid:0) curl e h , curl( r − I c h r ) (cid:1) + ( φ − φ h , curl I c h r ) − ( s , u − v h ) ≤ (cid:107) curl( r − I c h r ) (cid:107) (cid:107) curl e h (cid:107) + (cid:107) φ − φ h (cid:107) (cid:107) curl I c h r (cid:107) + (cid:107) s (cid:107) (cid:107) u − v h (cid:107) . By standard approximation and stability estimates for the nodal interpolation, as wellas a regularity estimate for problem (3.29)–(3.30), we have (cid:107) curl( r − I c h r ) (cid:107) (cid:46) h | curl r | (cid:46) h (cid:107) s (cid:107) and (cid:107) curl I c h r (cid:107) (cid:46) (cid:107) curl r (cid:107) (cid:46) (cid:107) s (cid:107) . As a result, we have (cid:107) s (cid:107) (cid:46) h (cid:107) curl e h (cid:107) + (cid:107) φ − φ h (cid:107) + (cid:107) u − v h (cid:107) Lastly, the desired estimate follows from combining the estimates for (cid:107) s h (cid:107) and (cid:107) s (cid:107) into (3.28). In this section, we verify the a priori convergenceresults shown in the previous subsection. The first example has a smooth solution u ( x, y, z ) = (cid:10) , , (sin x sin y ) sin z (cid:11) on Ω = (0 , π ) . Because the true solution is notdivergence free, problem (2.10) needs to be modified to ( u , ∇ ζ ) = ( g, ζ ) with g = div u is computed from the true solution. and the discretization changes accordingly. Ωis partitioned into a uniform tetrahedral mesh, and the convergence plot is in Figure3.1a. It can be seen that when u and curl u are smooth, the rates of convergence of | φ I − φ h | and (cid:107) φ − φ h (cid:107) are optimal, being O ( h ) and O ( h ), respectively. For thesolution u h obtained from the last Maxwell equation, (cid:107) curl( u − u h ) (cid:107) is still O ( h )and the L error (cid:107) u − u h (cid:107) is improved to O ( h ). (a) (b) Fig. 3.1: On a uniformly refined mesh: (a) The convergence of approximat-ing u ( x, y, z ) = (cid:10) , , (sin x sin y ) sin z (cid:11) . (b) The convergence of approximating u ( x, y, z ) = curl (cid:10) , , r / sin(2 θ/ (cid:11) . ECOUPLED FEM FOR QUAD-CURL PROBLEMS u which is present in (3.16)–(3.17)shall affect we choose a singular solution on an L-shaped domain (Figure 3.2). Thetrue solution is u = curl (cid:104) , , µ (cid:105) for a potential function µ = r / sin(2 θ/
3) in thecylindrical coordinate on Ω = (1 , × (0 , / \ ([0 , × [ − , × [0 , / µ is bi-harmonic so that f = , and curl u ∈ H / − (cid:15) (Ω). Theconvergence of the approximation φ = curl u in | · | ,h and (cid:107) · (cid:107) are both sub-optimal(Figure 3.1b) because φ = curl u (cid:54)∈ H (Ω) which is required to achieve the optimalrate of convergence (see Theorem 3.4 and Remark 3.5). While the approximationfor u is optimal as (3.22)’s dependence only on the L -error (cid:107) φ − φ h (cid:107) and theapproximation property of the linear N´ed´elec space for u ∈ H / − (cid:15) (Ω). -1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1-1-0.8-0.6-0.4-0.200.20.40.60.81 (a) (b) Fig. 3.2: The true solution vector field shown in (a) of the L-shaped domain exampleviewed from above on z = 1 / z -component.A coarse mesh ( h = 1 /
2) can be found in (b).
4. A posteriori error analysis.
In this section we will propose a reliable andefficient error estimator for the decoupled mixed finite element method (3.1)-(3.4).We aim to get an accurate approximation of u in the energy norm which can becontrolled by (cid:107) curl u − φ h (cid:107) = (cid:107) φ − φ h (cid:107) (cid:46) | φ − φ h | ,h . Therefore we do not includeproblem (3.5)-(3.6) into the adaptive procedure.To this end, we first recall a quasi-interpolation [36, 14, 20] and a decompositionof tensor-valued functions [18]. Lemma [36] There exists an operator Π h : H (curl , Ω) → V ch such that forany v ∈ H (curl , Ω) there exist τ ∈ H (Ω) and χ ∈ H (Ω; R ) satisfying v − Π h v = ∇ τ + χ , (4.1) (cid:88) K ∈T h ( h − K (cid:107) χ (cid:107) ,K + h − K (cid:107) χ (cid:107) ,∂K ) (cid:46) (cid:107) curl v (cid:107) . Lemma
Let τ be a tensor-valued function in L (Ω; M ) .There exist r ∈ H (Ω; R ) , q ∈ L (Ω) , s ∈ H (Ω; M ) and v ∈ H (Ω; R ) such that τ = ∇ r − q I + curl s , r = curl v , q = tr( curl s ) , (cid:107) r (cid:107) + (cid:107) s (cid:107) + (cid:107) q (cid:107) + (cid:107) v (cid:107) (cid:46) (cid:107) τ (cid:107) . S. CAO, L. CHEN, AND X. HUANG
For any subset M h ⊆ T h , define error estimators(4.2) η ( w h , f , M h ) := (cid:88) K ∈M h h K (cid:107) f (cid:107) ,K + (cid:88) F ∈F ih ( M h ) h F (cid:107) (cid:74) (curl w h ) × n F (cid:75) (cid:107) ,F , (4.3) η ( φ h , w h , M h ) := (cid:88) K ∈M h h K (cid:107) curl w h (cid:107) ,K + (cid:88) F ∈F h ( M h ) h F (cid:107) (cid:74) n F × ( ∇ h φ h ) (cid:75) (cid:107) ,F . Let Q K f be the L -projection of the data onto (cid:81) K ∈T h P ( K ; R ), then the dataoscillation is defined asosc ( f , M h ) := (cid:88) K ∈M h h K (cid:107) f − Q K f (cid:107) ,K . Let I SZh be the tensorial Scott-Zhang interpolation from H (Ω; M ) to the tensoriallinear Lagrange element space [37]. It holds(4.4) (cid:88) K ∈T h (cid:16) h − K (cid:107) v − I SZh v (cid:107) ,K + | v − I SZh v | ,K (cid:17) (cid:46) | v | ∀ v ∈ H (Ω; M ) . We first present a posterior analysis of error w − w h which is well-documented forthe saddle point formulation of Maxwell’s equation (see e.g., [3, 48, 46]). We includea proof here for the completness. Lemma
Let ( w , ∈ H (curl , Ω) × H (Ω) be the solution of the Maxwellequation (2.5) - (2.6) , and ( w h , ∈ V ch × V h the solution of the mixed method (3.1) - (3.2) . We have (4.5) (cid:107) curl( w − w h ) (cid:107) (cid:46) η ( w h , f , T h ) , (4.6) η ( w h , f , T h ) (cid:46) (cid:107) curl( w − w h ) (cid:107) + osc( f , T h ) . Proof.
Applying Lemma 4.1 to v = w − w h , we get from (3.8) and (2.5) that (cid:107) curl( w − w h ) (cid:107) = (curl( w − w h ) , curl( v − Π h v )) = (curl( w − w h ) , curl χ )= ( f , χ ) − (curl w h , curl χ )= ( f , χ ) − (cid:88) K ∈T h ((curl w h ) × n , χ ) ∂K = ( f , χ ) − (cid:88) F ∈F ih ( (cid:74) (curl w h ) × n F (cid:75) , χ ) F . Hence we have derived (4.5) by (4.1).The efficiency (4.6) follows from the standard bubble function techniques (seee.g., [3]).
Lemma
Let ( w , , φ , p ) ∈ H (curl , Ω) × H (Ω) × H (Ω; R ) × L (Ω) be thesolution of the variational formulation (2.5) - (2.8) , and ( w h , , φ h , p h ) ∈ V ch × V h × V CR h × Q h the solution of the mixed method (3.1) - (3.4) . We have (4.7) | φ − φ h | ,h + h (cid:107) curl( w − w h ) (cid:107) (cid:46) hη ( w h , f , T h ) + η ( φ h , w h , T h ) ,hη ( w h , f , T h ) + η ( φ h , w h , T h ) (cid:46) | φ − φ h | ,h + (cid:107) p − p h (cid:107) + h (cid:107) curl( w − w h ) (cid:107) + h osc( f , T h ) . (4.8) ECOUPLED FEM FOR QUAD-CURL PROBLEMS Proof.
Applying Lemma 4.2 to ∇ h ( φ − φ h ), there exist r ∈ H (Ω; R ), q ∈ L (Ω), s ∈ H (Ω; M ) and v ∈ H (Ω; R ) such that ∇ h ( φ − φ h ) = ∇ r − q I + curl s , r = curl v , (4.9) (cid:107) r (cid:107) + (cid:107) s (cid:107) + (cid:107) q (cid:107) + (cid:107) v (cid:107) (cid:46) (cid:107) ∇ h ( φ − φ h ) (cid:107) . Note that div h ( φ − φ h ) = 0 from (2.8) and (3.4). Since div r = 0, we get from (2.7)that | φ − φ h | ,h = ( ∇ h ( φ − φ h ) , ∇ r − q I + curl s ) = ( ∇ h ( φ − φ h ) , ∇ r + curl s )= ( ∇ h ( φ − φ h ) , ∇ r ) + (div r , p − p h ) + ( ∇ h ( φ − φ h ) , curl s )= (curl w , r ) − ( ∇ h φ h , ∇ r ) − (div r , p h ) − ( ∇ h φ h , curl s ) . (4.10)It follows from (3.11) and (3.3) that( ∇ h φ h , ∇ r ) + (div r , p h ) = ( ∇ h φ h , ∇ ( I sh r )) + (div( I sh r ) , p h ) = (curl w h , I sh r ) . Noticing that ( ∇ h φ h , curl ( I SZh s )) = 0, we obtain from the last three identities that | φ − φ h | ,h = (curl w , r ) − (curl w h , I sh r ) − ( ∇ h φ h , curl ( s − I SZh s ))= (curl( w − w h ) , curl v ) + (curl w h , r − I sh r ) − ( ∇ h φ h , curl ( s − I SZh s )) . (4.11)Next we estimate the right hand side of (4.11) term by term. Employing (3.8) and(4.5), it follows(curl( w − w h ) , curl v ) = inf v h ∈ V ch (curl( w − w h ) , curl( v − v h )) ≤ (cid:107) curl( w − w h ) (cid:107) inf v h ∈ V ch (cid:107) curl( v − v h ) (cid:107) (cid:46) hη ( w h , f , T h ) | curl v | = hη ( w h , f , T h ) | r | . According to (3.12), we have(curl w h , r − I sh r ) (cid:46) η ( φ h , w h , T h ) | r | . And we get from (4.4) that − ( ∇ h φ h , curl ( s − I SZh s )) = (cid:88) K ∈T h ( n × ( ∇ h φ h ) , s − I SZh s ) ∂K = (cid:88) F ∈F h ( (cid:74) n F × ( ∇ h φ h ) (cid:75) , s − I SZh s ) F (cid:46) η ( φ h , w h , T h ) | s | . Combining the last three inequalities and (4.11), we get from (4.9) that(4.12) | φ − φ h | ,h (cid:46) hη ( w h , f , T h ) + η ( φ h , w h , T h ) . On the other side, by applying the bubble function techniques, we get(4.13) (cid:88) K ∈T h h K (cid:107) curl w h (cid:107) ,K (cid:46) h (cid:107) curl( w − w h ) (cid:107) + | φ − φ h | ,h + (cid:107) p − p h (cid:107) , S. CAO, L. CHEN, AND X. HUANG (cid:88) F ∈F h h F (cid:107) (cid:74) n F × ( ∇ h φ h ) (cid:75) (cid:107) ,F (cid:46) | φ − φ h | ,h . Combining the last two inequalities shows η ( φ h , w h , T h ) (cid:46) h (cid:107) curl( w − w h ) (cid:107) + | φ − φ h | ,h + (cid:107) p − p h (cid:107) . Therefore we conclude (4.8) from (4.6).By the a priori L -estimate of the Stokes problem, when we assume the H s -regularity ( s ∈ (1 / , (cid:107) φ − φ h (cid:107) . Lemma
Let ( w , , φ , p ) ∈ H (curl , Ω) × H (Ω) × H (Ω; R ) × L (Ω) be thesolution of the variational formulation (2.5) - (2.8) , and ( w h , , φ h , p h ) ∈ V ch × V h × V CR h × Q h the solution of the mixed method (3.1) - (3.4) . We have (4.14) (cid:107) φ − φ h (cid:107) (cid:46) h s | φ − φ h | ,h + h (cid:107) curl( w − w h ) (cid:107) + h s (cid:107) curl w h (cid:107) .
5. Quasi-orthogonality.
In this section we will develop the quasi-orthogonalityof the decoupled mixed finite element method.
First recall a nonconformingdiscretization of the following Stokes complex in three dimensions [26](5.1) 0 (cid:71)(cid:71)(cid:71)(cid:65) H (Ω) ∇ (cid:71)(cid:71)(cid:71)(cid:71)(cid:71)(cid:65) H (grad curl , Ω) curl (cid:71)(cid:71)(cid:71)(cid:71)(cid:71)(cid:71)(cid:71)(cid:71)(cid:65) H (Ω; R ) div (cid:71)(cid:71)(cid:71)(cid:71)(cid:71)(cid:71)(cid:71)(cid:65) L (Ω) (cid:71)(cid:71)(cid:71)(cid:65) , where H (grad curl , Ω) := { v ∈ H (curl , Ω) : curl v ∈ H (Ω; R ) } . Note that H (grad curl , Ω) = H (curl , Ω) in (2.3) (cf. [44])..The space of the shape functions of the H (grad curl) nonconforming elementproposed in [26] is P ( K ; R ) ⊕ x ∧ P ( K ; R ), and the local degrees of freedom aregiven by (cid:90) e v · t e d s on each e ∈ E ( K ) , (5.2) (cid:90) F (curl v ) · t F,i d s on each F ∈ F ( K ) with i = 1 , . (5.3)The global H (grad curl) nonconforming element space is then defined as W h := { v h ∈ L (Ω; R ) : v h | K ∈ P ( K ; R ) ⊕ x ∧ P ( K ; R ) for each K ∈ T h , all the degree of freedom (5.2)-(5.3) are single-valued , and all the degree of freedom (5.2)-(5.3) on ∂ Ω vanish } . Now the nonconforming discrete Stokes complex in [26] is presented as(5.4) 0 (cid:71)(cid:71)(cid:71)(cid:65) V h ∇ (cid:71)(cid:71)(cid:71)(cid:71)(cid:71)(cid:65) W h curl h (cid:71)(cid:71)(cid:71)(cid:71)(cid:71)(cid:71)(cid:71)(cid:71)(cid:71)(cid:65) V CR h div h (cid:71)(cid:71)(cid:71)(cid:71)(cid:71)(cid:71)(cid:71)(cid:71)(cid:65) Q h (cid:71)(cid:71)(cid:71)(cid:65) . We also need the help of the discrete de Rham complex. Recall the lowest-orderRaviart-Thomas element space [35, 33](5.5) V dh := { v h ∈ H (div , Ω) : v h | K ∈ P ( K ; R ) + x P ( K ) for each K ∈ T h } , ECOUPLED FEM FOR QUAD-CURL PROBLEMS (cid:71)(cid:71)(cid:71)(cid:65) V h ∇ (cid:71)(cid:71)(cid:71)(cid:71)(cid:71)(cid:65) V ch curl (cid:71)(cid:71)(cid:71)(cid:71)(cid:71)(cid:71)(cid:71)(cid:71)(cid:65) V dh div (cid:71)(cid:71)(cid:71)(cid:71)(cid:71)(cid:71)(cid:71)(cid:65) Q h (cid:71)(cid:71)(cid:71)(cid:65) . Let I ch be the nodal interpolation operator from Dom( I ch ) to V ch , and I dh the nodalinterpolation operator from Dom( I dh ) to V dh , where Dom( I ch ) and Dom( I dh ) are thedomains of the operators I ch and I dh respectively. It holds for any v h ∈ W h that(5.6) (cid:107) curl h ( v h − I ch v h ) (cid:107) ,K = (cid:107) curl h v h − I dh (curl h v h ) (cid:107) ,K (cid:46) h K | curl h v h | ,K . Hereforth, consider two nested conforming triangulations T h and T H , where T h isa refinement of T H . We have the commutative diagram property [1, 2](5.7) curl( I cH v h ) = I dH (curl v h ) ∀ v h ∈ V ch + W h . To derive the quasi-orthogonality, we need the following interpolation error estimationfor I sH [22](5.8) (cid:107) ψ h − I sH ψ h (cid:107) ,K (cid:46) h K (cid:107)∇ h ψ h (cid:107) ,K ∀ ψ h ∈ V CR h , K ∈ T H \T h . According to (3.1), since the triangulations T h and T H are nested, we get thefollowing Galerkin orthogonality(5.9) (curl( w h − w H ) , curl v H ) = 0 ∀ v H ∈ V cH . Lemma
It holds (5.10) (cid:88) K ∈T H h − K (cid:107) ψ h − I dH ψ h (cid:107) ,K (cid:46) | ψ h | ,h ∀ ψ h ∈ V CR h . Proof.
By the averaging technique [6, 25], there exists (cid:101) ψ h ∈ V h := V h ⊗ R suchthat(5.11) | (cid:101) ψ h | + (cid:88) K ∈T h (cid:16) h − K (cid:107) ψ h − (cid:101) ψ h (cid:107) ,K + h − K (cid:107) ψ h − (cid:101) ψ h (cid:107) ,∂K (cid:17) (cid:46) | ψ h | ,h . Applying the scaling argument, we get (cid:88) K ∈T H h − K (cid:107) I dH ( ψ h − (cid:101) ψ h ) (cid:107) ,K (cid:46) (cid:88) K ∈T H h − K (cid:107) ( ψ h − (cid:101) ψ h ) · n (cid:107) ,∂K ≤ (cid:88) K ∈T H h − K (cid:32) (cid:88) K (cid:48) ∈T h ( K ) (cid:107) ψ h − (cid:101) ψ h (cid:107) ,∂K (cid:48) (cid:33) ≤ (cid:88) K ∈T h h − K (cid:107) ψ h − (cid:101) ψ h (cid:107) ,∂K . Noting that (cid:101) ψ h ∈ H (Ω), it follows (cid:88) K ∈T H h − K (cid:107) (cid:101) ψ h − I dH (cid:101) ψ h (cid:107) ,K (cid:46) | (cid:101) ψ h | ,h . S. CAO, L. CHEN, AND X. HUANG
Combining the last two inequalities and (5.11) yields (cid:88) K ∈T H h − K (cid:16) (cid:107) I dH ( ψ h − (cid:101) ψ h ) (cid:107) ,K + (cid:107) ψ h − (cid:101) ψ h (cid:107) ,K + (cid:107) (cid:101) ψ h − I dH (cid:101) ψ h (cid:107) ,K (cid:17) (cid:46) | ψ h | ,h . On the other hand, we have (cid:107) ψ h − I dH ψ h (cid:107) ,K ≤ (cid:107) I dH ( ψ h − (cid:101) ψ h ) (cid:107) ,K + (cid:107) ψ h − (cid:101) ψ h (cid:107) ,K + (cid:107) (cid:101) ψ h − I dH (cid:101) ψ h (cid:107) ,K . Therefore we conclude (5.10) from the last two inequalities.
Lemma φ ). Let ( w h , , φ h , p h ) ∈ V ch × V h × V CR h ×Q h and ( w H , , φ H , p H ) ∈ V cH × V h × V CR h ×Q H the solutions of the mixed method (3.1) - (3.4) on triangulations T h and T H respectively. We have ( ∇ h ( φ − φ h ) , ∇ h ( φ h − φ H )) (cid:46) H | φ − φ h | ,h (cid:107) curl( w h − w H ) (cid:107) + | φ − φ h | ,h (cid:32) (cid:88) K ∈T H \T h h K (cid:107) curl w H (cid:107) ,K (cid:33) / . (5.12) Proof.
Let ψ h = I sh ( φ − φ h ), then we get from (3.11) and (3.12) that(5.13) div h ψ h = 0 , | ψ h | ,h (cid:46) | φ − φ h | ,h . It follows from (3.4), (3.11) and (3.3) that( ∇ h ( φ − φ h ) , ∇ h ( φ h − φ H )) = ( ∇ h ( φ − φ h ) , ∇ h φ h + p h I − ( ∇ H φ H + p H I ))= ( ∇ h φ h + p h I − ( ∇ H φ H + p H I ) , ∇ h ψ h )= (curl w h , ψ h ) − ( ∇ H φ H + p H I , ∇ h ψ h )= (curl w h , ψ h ) − ( ∇ H φ H + p H I , ∇ h ( I sH ψ h ))= (curl w h , ψ h ) − (curl w H , I sH ψ h ) . Thus we have( ∇ h ( φ − φ h ) , ∇ h ( φ h − φ H )) = (curl( w h − w H ) , ψ h )+ (curl w H , ψ h − I sH ψ h ) . (5.14)Next we estimate the right hand side of the last equation term by term.Due to the discrete Stokes complex (5.4) and the fact div h ψ h = 0, there exists v h ∈ W h such that ψ h = curl v h . Employing (5.6), it follows(curl( w h − w H ) , curl( v h − I ch v h )) ≤ h (cid:107) curl( w h − w H ) (cid:107) | ψ h | ,h . Due to (5.9) and (5.7), we have(curl( w h − w H ) , ψ h ) = (curl( w h − w H ) , curl v h )= (curl( w h − w H ) , curl( v h − I cH v h ))= (curl( w h − w H ) , curl v h − I dH curl v h )= (curl( w h − w H ) , ψ h − I dH ψ h ) . ECOUPLED FEM FOR QUAD-CURL PROBLEMS w h − w H ) , ψ h ) (cid:46) H (cid:107) curl( w h − w H ) (cid:107) | ψ h | ,h . Thanks to (5.8), we obtain(curl w H , ψ h − I sH ψ h ) = (cid:88) K ∈T H \T h (curl w H , ψ h − I sH ψ h ) K (cid:46) | ψ h | ,h (cid:32) (cid:88) K ∈T H \T h h K (cid:107) curl w H (cid:107) ,K (cid:33) / . Finally we conclude from (5.14), the last two inequalities and (5.13).
6. Convergence.
In this section, we propose an adaptive algoirithm (Algorithm6.1) based on the estimators in Section 4. Then its convergence is proved usingthe quasi-orthogonality in Section 5. The methodology in the convergence mainlyfollows that of [22]. There are two major modifications: one first needs to control theperturbation of data for the Stokes problem, then the convergence is proved only for (cid:107) curl( u − u h ) (cid:107) and | φ − φ h | ,h without the Lagrange multiplier variable.For the ease of the readers, the following short notations are adopted throughoutthe proof of the contraction in Theorem 6.3 and the lemmas needed. For V h definedon T k , we denote them by V ck and V CR k . Similarly, the approximations on T k aredenoted by w k , φ k , and u k respectively. Let h k := max K ∈T k h K , and denote the quantitiesinvolving two consecutive levels of meshes as follows: E k := | φ − φ k | ,k := (cid:32) (cid:88) K ∈T k (cid:107)∇ k ( φ − φ k ) (cid:107) ,K (cid:33) / ,R k +1 := | φ k − φ k +1 | ,k +1 := (cid:88) K ∈T k +1 (cid:107)∇ k +1 ( φ k − φ k +1 ) (cid:107) ,K / ,e k := (cid:107) curl( w − w k ) (cid:107) , r k +1 := (cid:107) curl( w k − w k +1 ) (cid:107) ,η ,k ( M ) := η ( w k , f , M ) ,η ,k ( M ) := η ( φ k , w k , M ) ,g k ( M ) := (cid:32) (cid:88) K ∈M h K (cid:107) curl w k (cid:107) ,K (cid:33) / , where ∇ k stands for the discrete gradient ∇ h defined piecewisely on all K ∈ T k , and M can be T k , T k +1 , or T k \T k +1 .The following two lemmas concern the contraction and the continuity of the es-timators on two nested meshes, the proofs are standard in the AFEM literature thusomitted, the reader can refer to, e.g., [22, 47]. Lemma
Let ( w k , φ k ) ∈ V ck × V CR k and ( w k +1 , φ k +1 ) ∈ V ck +1 × V CR k +1 be the solutions to (3.1) - (3.4) on triangulations T k and T k +1 obtained through Algorithm 6.1, ρ = 1 − − / and there exists positive constants β i ∈ (1 − ρθ i , for i = 1 , such that (6.1) η i,k ( T k +1 ) ≤ β i η i,k ( T k ) − (cid:2) β i − (1 − ρθ i ) (cid:3) η i,k ( T k ) . S. CAO, L. CHEN, AND X. HUANG
Algorithm 6.1
An adaptive nonconforming finite element method
Input: T , f , tol , θ , θ ∈ (0 , Output: T N , φ N , u N . η = tol , η = tol , k = 0 . while η (cid:62) tol and η (cid:62) tol do SOLVE : Solve (2.5)–(2.6) and (2.7)–(2.8) on T k to get ( w k , φ k ); ESTIMATE : Compute η ( w k , f , K ) and η ( φ k , w k , K ) for all K ∈ T k MARK : Seek a minimum
M ⊆ T k such that(M) η ( w k , f , M ) ≥ θ η ( w k , f , T k )and η ( w k , φ k , M ) ≥ θ η ( w k , φ k , T k ) REFINE : Bisect K ∈ M and their neighbors to form a conforming T k +1 ; η ← η ( w k , f , T k ) η ← η ( φ k , w k , T k ) k ← k + 1 end while N ← k Solve (2.7)–(2.8) and (2.9)–(2.10) on T N to get φ N and u N . Lemma
Under the same assumption withLemma 6.1 then, then given positive constants δ i ∈ (0 , for i = 1 , , (6.2) η ,k +1 ( T k +1 ) ≤ (1 + δ ) η ,k ( T k +1 ) + C δ r k +1 , and η ,k +1 ( T k +1 ) ≤ (1 + δ ) η ,k ( T k +1 ) + C δ (cid:0) h k +1 r k +1 + R k +1 (cid:1) , where C i depends on the shape-regularity of the mesh. Theorem
Let ( w , φ ) ∈ H (curl , Ω) × H (Ω; R ) be the so-lutions to (2.5) - (2.8) without the Lagrange multipliers, and ( w k +1 , φ k +1 ) ∈ V ck +1 × V CR k +1 and ( w k , φ k ) ∈ V ck × V CR k be their approximations in problems (3.1) - (3.4) on T k +1 and T k , respectively. If T k +1 is a conforming refinement from T k with h k +1 ≤ h k ,then there exist γ , γ , µ, β > , and < α < such that the AFEM in Algorithm 6.1satisfies (6.3) | φ − φ k +1 | ,k +1 + µh k +1 (cid:107) curl( w − w k +1 ) (cid:107) + γ h k +1 η ( w k +1 , f , T k +1 ) + γ ˜ η ( φ k +1 , w k +1 , T k +1 ) ≤ α (cid:16) | φ − φ k | ,k + µh k (cid:107) curl( w − w k ) (cid:107) + γ h k η ( w k , f , T k ) + γ ˜ η ( φ k , w k , T k ) (cid:17) where ˜ η ( w k , φ k , T k ) := (cid:18) η ( w k , φ k , T k ) + β (cid:88) K ∈T k h K (cid:107) curl w k (cid:107) ,K (cid:19) is the modified estimator.Proof. First by the Galerkin orthogonality (3.8), we have e k +1 = e k − r k +1 . As-suming the constant in the quasi-orthogonality (5.12) is (cid:112) C Q /
2, we have by Young’s
ECOUPLED FEM FOR QUAD-CURL PROBLEMS (cid:15) ∈ (0 , E k +1 = E k − R k +1 − (cid:0) ∇ k +1 ( φ − φ k +1 ) , ∇ k +1 ( φ k +1 − φ k ) (cid:1) ≤ E k − R k +1 + 2 (cid:112) C Q (cid:0) h k r k +1 + g k ( T k \T k +1 ) (cid:1) E k +1 ≤ E k − R k +1 + (cid:15)E k +1 + C Q (cid:15) (cid:16) h k r k +1 + g k ( T k \T k +1 ) (cid:17) . Now from Lemmas 6.1 and 6.2, and h k +1 ≤ h k , we choose δ i ( i = 1 ,
2) such that β i := (1 − ρθ i )(1 + δ i ) ∈ (0 , η ,k +1 ( T k +1 ) ≤ β η ,k ( T k ) + (cid:104) δ β − (1 + δ ) (cid:0) β − (1 − ρθ ) (cid:1)(cid:105) η ,k ( T k ) + C δ r k +1 , and(6.5) η ,k +1 ( T k +1 ) ≤ β η ,k ( T k ) + (cid:104) δ β − (1 + δ ) (cid:0) β − (1 − ρθ ) (cid:1)(cid:105) η ,k ( T k )+ C δ (cid:0) h k r k +1 + R k +1 (cid:1) . Next for the element residual term in η on each K we have: (cid:107) curl w k +1 (cid:107) ,K = (cid:107) curl w k (cid:107) ,K − (cid:107) curl( w k − w k +1 ) (cid:107) ,K − (cid:0) curl w k +1 , curl( w k − w k +1 ) (cid:1) K . By the Young’s inequality for a δ ∈ (0 , − δ ) (cid:107) curl w k +1 (cid:107) ,K ≤ (cid:107) curl w k (cid:107) ,K + 1 − δ δ (cid:107) curl( w k − w k +1 ) (cid:107) ,K , and consequently applying similar technique with Lemma 6.1 yields:(6.6) (1 − δ ) g k +1 ( T k +1 ) ≤ g k ( T k ) − ρg k ( T k \T k +1 ) + C δ h k +1 r k +1 . To prove the contraction result, we define G k +1 := (1 − (cid:15) ) E k +1 + µh k +1 e k +1 + γ h k +1 η ,k +1 ( T k +1 ) + γ η ,k +1 ( T k +1 ) + (1 − δ ) γ g k +1 ( T k +1 )and G k := E k + µh k e k + γ β h k η ,k ( T k ) + γ β η ,k ( T k ) + γ g k ( T k ) . Combining estimates (6.4), (6.5), and (6.6) above,(6.7) G k +1 ≤ G k − (cid:18) − γ C δ (cid:19) R k +1 − µ − C Q (cid:15) − (cid:88) j =1 γ j C j δ j h k r k +1 + γ h k (cid:104) δ β − (1 + δ ) (cid:0) β − (1 − ρθ ) (cid:1)(cid:105) η ,k ( T k )+ γ (cid:104) δ β − (1 + δ ) (cid:0) β − (1 − ρθ ) (cid:1)(cid:105) η ,k ( T k ) − (cid:18) γ ρ − C Q (cid:15) (cid:19) g k ( T k \T k +1 ) . S. CAO, L. CHEN, AND X. HUANG
The constants are formulated such that all terms on the right hand side except thefirst in the inequality above vanish for a fixed (cid:15) ∈ (0 , δ i ( i = 1 ,
2) above, we choose γ and γ to be(6.8) γ = δ C , and γ = C Q ρ(cid:15) , and γ is a free constant. Additionally, µ is free as well, and is to be chosen sufficientlylarge such that the following holds regardless of what values (cid:15) and δ take(6.9) µ − C Q (cid:15) − (cid:88) j =1 γ j C j δ j ≥ . Consequently, (6.7) becomes G k +1 ≤ G k . For an α ∈ (0 ,
1) we rewrite right handside of (6.7) as G k = α G k + R k where R k := (cid:0) − α (1 − (cid:15) ) (cid:1) E k + µ (1 − α ) h k e k + γ ( β − α ) h k η ,k ( T k ) + γ ( β − α ) η ,k ( T k ) + γ (cid:0) − α (1 − δ ) (cid:1) g k ( T k ) . It suffices to show that for the constants chosen and to be determined, there existsan α ∈ (0 ,
1) such that R k ≤
0. Now assuming that the constant in the reliabilityestimate (4.7) is √ C R , and the fact that g k ( T k ) ≤ η ,k ( T k ) we have(6.10) R k ≤ (cid:0) h k η ,k ( T k ) + η ,k ( T k ) (cid:1)(cid:104) C R (cid:0) − α (1 − (cid:15) ) + µ (1 − α ) (cid:1) + ( β − α ) γ + ( β − α ) γ + γ (cid:0) − α (1 − δ ) (cid:1)(cid:105) . It is straightforward to verify the second term on the right hand side above vanisheswhen letting(6.11) α := C R (1 + µ ) + γ β + γ β + γ C R (1 + µ − (cid:15) ) + γ + γ + γ (1 − δ ) > . To make α <
1, by (6.8), the following inequality is needed:(6.12) (cid:15)C R + δ C Q ρ(cid:15) < γ (1 − β ) + γ (1 − β ) . Choosing δ = ρ(cid:15) and γ = γ β − β , we have (6.12) holds with 0 < (cid:15) < min (cid:26) δ C ( C R + C Q ) , (cid:27) and thus α defined in (6.11) is in (0 , G k +1 ≤ α G k is shown, and finallythe theorem follows by acknowledging that g k ( T k ) is a part of η ,k ( T k ). ECOUPLED FEM FOR QUAD-CURL PROBLEMS
7. Numerical examples.
The numerical experiments in this section, as wellas in Section 3.2, are carried out using i FEM [12] which is publicly available athttps://github.com/lyc102/ifem. We mainly compare the performance of the adaptivealgorithm under the proposed separate marking strategy (M) in Algorithm 6.1 versustwo single marking strategies: A minimum
M ⊆ T k is sought such that(M1) η ( w h , φ h , f , M ) ≥ θη ( w h , φ h , f , T k ) , or(M2) (cid:101) η ( w h , φ h , f , M ) ≥ θ (cid:101) η ( w h , φ h , f , T k ) , where η is defined as(7.1) η ( w h , φ h , f , M ) := η ( w h , f , M h ) + η ( φ h , w h , M ) , and (cid:101) η ( w h , φ h , f , M ) is defined by the L -sum of the weighted η and η :(7.2) (cid:101) η ( w h , φ h , f , M ) := (cid:88) K ∈M h K (cid:107) f (cid:107) ,K + (cid:88) F ∈F ih ( M ) h F (cid:107) (cid:74) (curl w h ) × n F (cid:75) (cid:107) ,F + (cid:88) K ∈M h K (cid:107) curl w h (cid:107) ,K + (cid:88) F ∈F h ( M ) h F (cid:107) (cid:74) n F × ( ∇ h φ h ) (cid:75) (cid:107) ,F . On a uniform mesh, (cid:101) η can be viewed as approximately h η + η .To demonstrate the reason why we opt for the proposed separate marking strategy(M), and not single markings such as (M1) and (M2), we construct a toy exampleusing u = curl (cid:104) , , µ (cid:105) for a potential function µ = r / sin(2 θ/
3) in the cylindricalcoordinate as in Section 3.2 example 2 on an L-shaped domain. In this case we havea regular φ = curl u ∈ H / − (cid:15) (Ω) while w ∈ H / − (cid:15) (Ω) has a mild singularity nearthe nonconvex corner.The results for the convergence of Algorithm 6.1 using marking strategies (M),(M1), and (M2) can be found in Figure 7.1a, 7.1b, and 7.1c, respectively.Using the proposed marking (M), we obtain the desired optimal convergencefor | φ h − curl u | ,h being optimal in that the convergence is at the rate of “linear” ≈ − / . Additionally, (cid:107) φ h − curl u (cid:107) and (cid:107) u − u h (cid:107) converge “quadratically”,i.e., in the order of approximately − / even though the error estimator’sreliability and efficiency are not directly measured in those norms. In this experiment, θ = 0 . θ = 0 . θ = 0 .
3, if η is unweighted in the η in (7.1), due to w being singular while φ being regular, thanks to η being locally efficient, the markedelements are dominantly concentrated on which η are large. As a result, marking(M1) drives the AFEM algorithm favoring reducing the error for w while the errors inapproximating φ and u barely change (Figure 7.1b). However, due to the regularitylifting effect from w h being the data for the problem of φ h (c.f. Lemma 3.3), toachieve the optimal rate of convergence, w h does not have to be approximated to thesame precision with φ h .If marking (M2) with θ = 0 . η is locally weighted by the meshsize h K , the optimal rates of convergence is restored. However, one does benefit fromchoose different marking parameters for approximating w and φ due to the regularitydifference (see e.g., [15]). Moreover, one does not have the Galerkin orthogonality for (cid:107) curl( w − w h ) (cid:107) to exploit in the proof of the contraction in Theorem 6.3, because2 S. CAO, L. CHEN, AND X. HUANG the consecutive difference is measured under a norm weighted by the mesh size. Asa result, it needs new tools that are not available in any of the current literature toprove a similar contraction result when the local error indicator is further weightedby the local mesh size h K . (a) (b) (c) Fig. 7.1: The convergence results for AFEM 6.1 using different marking strategies:(a) separate marking for η and η . (b) single marking for η . (c) single marking for (cid:101) η . REFERENCES[1]
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