Analysis of the Morley element for the Cahn-Hilliard equation and the Hele-Shaw flow
AANALYSIS OF THE MORLEY ELEMENT FOR THECAHN-HILLIARD EQUATION AND THE HELE-SHAW FLOW ∗ SHUONAN WU † AND
YUKUN LI ‡ Abstract.
The paper analyzes the Morley element method for the Cahn-Hilliard equation.The objective is to derive the optimal error estimates and to prove the zero-level sets of the Cahn-Hilliard equation approximate the Hele-Shaw flow. If the piecewise L ∞ ( H ) error bound is derivedby choosing test function directly, we cannot obtain the optimal error order, and we cannot establishthe error bound which depends on (cid:15) polynomially either. To overcome this difficulty, this paperproves them by the following steps, and the result in each next step cannot be established withoutusing the result in its previous one. First, it proves some a priori estimates of the exact solution u ,and these regularity results are minimal to get the main results; Second, it establishes L ∞ ( L ) andpiecewise L ( H ) error bounds which depend on (cid:15) polynomially based on the piecewise L ∞ ( H − )and L ( H ) error bounds; Third, it establishes piecewise L ∞ ( H ) optimal error bound which dependson (cid:15) polynomially based on the piecewise L ∞ ( L ) and L ( H ) error bounds; Finally, it proves the L ∞ ( L ∞ ) error bound and the approximation to the Hele-Shaw flow based on the piecewise L ∞ ( H )error bound. The nonstandard techniques are used in these steps such as the generalized coercivityresult, integration by part in space, summation by part in time, and special properties of the Morleyelements. If one of these techniques is lacked, either we can only obtain the sub-optimal piecewise L ∞ ( H ) error order, or we can merely obtain the error bounds which are exponentially dependenton (cid:15) . The approach used in this paper provides a way to bound the errors in higher norm fromthe errors in lower norm step by step, which has a profound meaning in methodology. Numericalresults are presented to validate the optimal L ∞ ( H ) error order and the asymptotic behavior of thesolutions of the Cahn-Hilliard equation. Key words.
Morley element, Cahn-Hilliard equation, generalized coercivity result, (cid:15) polyno-mial dependence, Hele-Shaw flow AMS subject classifications.
1. Introduction.
Consider the following Cahn-Hilliard equation with Neumannboundary conditions: u t + ∆( (cid:15) ∆ u − (cid:15) f ( u )) = 0 in Ω T := Ω × (0 , T ] , (1.1) ∂u∂n = ∂∂n ( (cid:15) ∆ u − (cid:15) f ( u )) = 0 on ∂ Ω T := ∂ Ω × (0 , T ] , (1.2) u = u in Ω × { t = 0 } , (1.3)where Ω ⊆ R is a bounded domain, f ( u ) = u − u is the derivative of a double wellpotential F ( u ) which is defined by(1.4) F ( u ) = 14 ( u − . The Allen-Cahn equation [3, 6, 12, 20, 17, 16, 19, 24] and the Cahn-Hilliard equation[2, 12, 25, 29] are two basic phase field models to describe the phase transition process.They are also proved to be related to geometric flow. For example, the zero-level setsof the Allen-Cahn equation approximate the mean curvature [15, 24] and the zero-levelsets of the Cahn-Hilliard equation approximate the Hele-Shaw flow [28, 2]. The Cahn-Hilliard equation was introduced by J. Cahn and J. Hilliard in [11] to describe the ∗ The work of Shuonan Wu is partially supported by the startup grant from Peking Unversity. † School of Mathematical Sciences, Peking University, China, 100871 ( [email protected] ) ‡ Department of Mathematics, The Ohio State University, Columbus, U.S.A. ( [email protected] )1 a r X i v : . [ m a t h . NA ] A ug S. WU AND Y. LI process of phase separation, by which the two components of a binary fluid separateand form domains pure in each component. It can be interpreted as the H − gradientflow [2] of the Cahn-Hilliard energy functional J (cid:15) ( v ) := (cid:90) Ω (cid:16) (cid:15) |∇ v | + 1 (cid:15) F ( v ) (cid:17) d x. (1.5)There are a few papers [4, 30, 13, 14] discussing the error bounds, which depend on theexponential power of (cid:15) , of the numerical methods for Cahn-Hilliard equation. Such anestimate is clearly not useful for small (cid:15) , in particular, in addressing the issue whetherthe computed numerical interfaces converge to the original sharp interface of the Hele-Shaw problem. Instead, the polynomial dependence in (cid:15) is proved in [21, 22] usingthe standard finite element method, and in [18, 26] using the discontinuous Galerkinmethod. Due to the high efficiency of the Morley elements, compared with mixedfinite element methods or C -conforming finite element methods, the Morley finiteelement method is used to derive the error bound which depends on (cid:15) polynomiallyin this paper.The highlights of this paper are fourfold. First, it establishes the piecewise L ∞ ( L ) and L ( H ) error bounds which depend on (cid:15) polynomially. If the standardtechnique is used, we can only prove that the error bounds depend on (cid:15) exponentially,which can not be used to prove our main theorem. To prove these bounds, specialproperties of the Morley elements are explored, i.e., Lemma 2.3 in [14], and piece-wise L ∞ ( H − ) and L ( H ) error bounds [27] are required. Second, by making useof the piecewise L ∞ ( L ) and L ( H ) error bounds above, it establishes the piecewise L ∞ ( H ) error bound which depends on (cid:15) polynomially. If the standard technique isused, we can only get the error bound in Remark 2, which does not have an optimalorder. The crux here is to employ the summation by part in time and integrationby part in space techniques simultaneously to handle the nonlinear term, togetherwith the special properties of the Morley elements. Third, the minimal regularity of u is used, i.e., (cid:107) u tt (cid:107) L ( L ) regularity instead of (cid:107) u tt (cid:107) L ∞ ( L ) regularity is used, andthe a priori estimate is derived in Theorem 2.2. Fourth, the L ∞ ( L ∞ ) error bound isestablished using the optimal piecewise L ∞ ( H ) error, by which the main result thatthe zero-level sets of the Cahn-Hilliard equation approximate the Hele-Shaw flow isproved in Section 5.The organization of this paper is as follows. In Section 2, the standard Sobolevspace notation is introduced, some useful lemmas are stated, and a new a priori esti-mate of the exact solution u is derived. In Section 3, the fully discrete approximationbased on the Morley finite element space is presented. In Section 4, first the polyno-mially dependent piecewise L ∞ ( L ) and L ( H ) error bounds are established basedon piecewise L ∞ ( H − ) and L ( H ) error bounds, then the polynomially dependentpiecewise L ∞ ( H ) error bound is established based on piecewise L ∞ ( L ) and L ( H )error bounds, by which the L ∞ ( L ∞ ) error bound is proved. In Section 5, the approx-imation of the zero-level sets of the Cahn-Hilliard equation of the Hele-Shaw flow isproved. In Section 6, numerical tests are presented to validate our theoretical results,including the optimal error orders and the approximation of the Hele-Shaw flow.
2. Preliminaries.
In this section, we present some results which will be used inthe following sections. Throughout this paper, C denotes a generic positive constantwhich is independent of interfacial length (cid:15) , spacial size h , and time step size k , and itmay have different values in different formulas. The standard Sobolev space notationbelow is used in this paper. ORLEY ELEMENT FOR THE CH EQUATION AND THE HS FLOW (cid:107) v (cid:107) ,p,A = (cid:18) (cid:90) A | v | p d x (cid:19) /p ≤ p < ∞ , (cid:107) v (cid:107) , ∞ ,A = ess sup A | v | , | v | m,p,A = (cid:18) (cid:88) | α | = m (cid:107) D α v (cid:107) p ,p,A (cid:19) /p ≤ p < ∞ , (cid:107) v (cid:107) m,p,A = (cid:18) m (cid:88) j =0 | v | pm,p,A (cid:19) /p . Here A denotes some domain, i.e., a single mesh element K or the whole domainΩ. When A = Ω, (cid:107) · (cid:107) H k , (cid:107) · (cid:107) L k are used to denote (cid:107) · (cid:107) H k (Ω) , (cid:107) · (cid:107) L k (Ω) respectively,and (cid:107) · (cid:107) , is also used to denote (cid:107) · (cid:107) L (Ω) . Let T h be a family of quasi-uniformtriangulations of domain Ω, and E h be a collection of edges, then the global meshdependent semi-norm, norm and inner product are defined below | v | j,p,h = (cid:18) (cid:88) K ∈T h | v | pj,p,K (cid:19) /p , (cid:107) v (cid:107) j,p,h = (cid:18) (cid:88) K ∈T h (cid:107) v (cid:107) pj,p,K (cid:19) /p , ( w, v ) h = (cid:88) K ∈T h (cid:90) K w ( x ) v ( x ) d x. Define L (Ω) as the mean zero functions in L (Ω). For Φ ∈ L (Ω), let u := − ∆ − Φ ∈ H (Ω) ∩ L (Ω) such that − ∆ u = Φ in Ω ,∂u∂n = 0 on ∂ Ω . Then we have − ( ∇ ∆ − Φ , ∇ v ) = (Φ , v ) in Ω ∀ v ∈ H (Ω) ∩ L (Ω) . (2.1)For v ∈ L (Ω) and Φ ∈ L (Ω), define the continuous H − inner product by(Φ , v ) H − := ( ∇ ∆ − Φ , ∇ ∆ − v ) = (Φ , − ∆ − v ) = ( v, − ∆ − Φ) . (2.2)As in [12, 18, 21, 22, 26, 27], we made the following assumptions on the initialcondition. These assumptions were used to derive the a priori estimates for thesolution of problem (1.1)–(1.4). General Assumption (GA)(1) Assume that m ∈ ( − ,
1) where m := 1 | Ω | (cid:90) Ω u ( x ) d x. (2) There exists a nonnegative constant σ such that J (cid:15) ( u ) ≤ C(cid:15) − σ . S. WU AND Y. LI (3) There exist nonnegative constants σ , σ and σ such that (cid:13)(cid:13) − (cid:15) ∆ u + (cid:15) − f ( u ) (cid:13)(cid:13) H (cid:96) ≤ C(cid:15) − σ (cid:96) (cid:96) = 0 , , . Under the above assumptions, the following a priori estimates of the solution wereproved in [18, 21, 22, 26].
Theorem
The solution u of problem (1.1) – (1.4) satisfies the following energyestimate: ess sup t ∈ [0 ,T ] (cid:16) (cid:15) (cid:107)∇ u (cid:107) L + 1 (cid:15) (cid:107) F ( u ) (cid:107) L (cid:17) + (cid:90) T (cid:107) u t ( s ) (cid:107) H − d s ≤ J (cid:15) ( u ) . (2.3) Moreover, suppose that GA (1)–(3) hold, u ∈ H (Ω) and ∂ Ω ∈ C , , then u satisfiesthe additional estimates: | Ω | (cid:90) Ω u ( x, t ) d x = m ∀ t ≥ , (2.4) ess sup t ∈ [0 ,T ] (cid:107) ∆ u (cid:107) L ≤ C(cid:15) − max { σ + ,σ +1 } , (2.5) ess sup t ∈ [0 ,T ] (cid:107)∇ ∆ u (cid:107) L ≤ C(cid:15) − max { σ + ,σ +1 } , (2.6) (cid:15) (cid:90) T (cid:107) ∆ u t (cid:107) L d s + ess sup t ∈ [0 ,T ] (cid:107) u t (cid:107) L ≤ C(cid:15) − max { σ + , σ + , σ +4 , σ } . (2.7) Furthermore, if there exists σ > such that (2.8) lim s → + (cid:107)∇ u t ( s ) (cid:107) L ≤ C(cid:15) − σ , then there hold ess sup t ∈ [0 ,T ] (cid:107)∇ u t (cid:107) L + (cid:15) (cid:90) T (cid:107)∇ ∆ u t (cid:107) L d s ≤ Cρ ( (cid:15) ) , (2.9) (cid:90) T (cid:107) u tt (cid:107) H − d s ≤ Cρ ( (cid:15) ) , (2.10) where ρ ( (cid:15) ) := (cid:15) − max { σ +5 , σ +2 }− max { σ + , σ + , σ +4 } + (cid:15) − σ + (cid:15) − max { σ +7 , σ +4 } ,ρ ( (cid:15) ) := (cid:15)ρ ( (cid:15) ) . Besides, an extra a priori estimates of solution u is needed in this paper. Theorem
Under the assumptions of Theorem 2.1 and if there exists σ > such that (cid:107) ∆ u t (0) (cid:107) L ≤ C(cid:15) − σ , (2.11) then there hold ess sup t ∈ [0 ,T ] (cid:107) ∆ u t (cid:107) L + (cid:15) (cid:90) T (cid:107) ∆ u t (cid:107) L d s ≤ Cρ ( (cid:15) ) , (2.12) ess sup t ∈ [0 ,T ] (cid:15) (cid:107) ∆ u t (cid:107) L + (cid:90) T (cid:107) u tt (cid:107) L d s ≤ Cρ ( (cid:15) ) , (2.13) ORLEY ELEMENT FOR THE CH EQUATION AND THE HS FLOW where ρ ( (cid:15) ) := (cid:15) − max { σ + , σ + , σ +4 , σ }− max { σ +5 , σ +2 }− + (cid:15) − max { σ + ,σ +1 }− ρ ( (cid:15) ) + (cid:15) − σ ,ρ ( (cid:15) ) := (cid:15)ρ ( (cid:15) ) . Proof.
Using the Gagliardo-Nirenberg inequalities [1] in two-dimensional space,we have (cid:107)∇ u (cid:107) L ∞ ≤ C (cid:18) (cid:107)∇ ∆ u (cid:107) L (cid:107) u (cid:107) L ∞ + (cid:107) u (cid:107) L ∞ (cid:19) ≤ C(cid:15) − max { σ + ,σ +1 } . (2.14)Since f (cid:48) ( u ) = 3 u −
1, using Sobolev embedding theorem [1], (2.3), (2.5), (2.6), (2.7)and (2.9), we have (cid:90) T (cid:107) ∆( f (cid:48) ( u ) u t ) (cid:107) L d s (2.15) = (cid:90) T (cid:107) uu t ∆ u + 12 u ∇ u · ∇ u t + 6 u t ∇ u · ∇ u + (3 u − u t (cid:107) L d s ≤ C (cid:90) T (cid:107) ∆ u (cid:107) L (cid:107) u t (cid:107) L ∞ d s + C (cid:90) T (cid:107)∇ u (cid:107) L ∞ (cid:107)∇ u t (cid:107) L d s + C (cid:90) T (cid:107)∇ u (cid:107) L ∞ (cid:107) u t (cid:107) L d s + C (cid:90) T (cid:107) ∆ u t (cid:107) L d s ≤ C (cid:107) ∆ u (cid:107) L ∞ ( L ) (cid:90) T (cid:107) u t (cid:107) H d s + C (cid:107)∇ u t (cid:107) L ∞ ( L ) (cid:107)∇ u (cid:107) L ∞ ( L ∞ ) + C (cid:107)∇ u (cid:107) L ∞ ( L ∞ ) (cid:107) u t (cid:107) L ∞ ( L ) + C (cid:90) T (cid:107) ∆ u t (cid:107) L d s ≤ C(cid:15) − max { σ + , σ + , σ +4 , σ }− max { σ +5 , σ +2 }− + C(cid:15) − max { σ + ,σ +1 } ρ ( (cid:15) )+ C(cid:15) − max { σ + , σ + , σ +4 , σ }− max { σ +5 , σ +2 } + C(cid:15) − max { σ + , σ + , σ +4 , σ }− ≤ C(cid:15) − max { σ + , σ + , σ +4 , σ }− max { σ +5 , σ +2 } + C(cid:15) − max { σ + ,σ +1 } ρ ( (cid:15) ) . Taking the derivative with respect to t on both sides of (1.1), we get u tt + (cid:15) ∆ u t − (cid:15) ∆( f (cid:48) ( u ) u t ) = 0 . (2.16)Testing (2.16) with ∆ u t , and taking the integral over (0 , T ), we obtain12 (cid:107) ∆ u t ( T ) (cid:107) L + (cid:15) (cid:90) T (cid:107) ∆ u t (cid:107) L d s (2.17) = 1 (cid:15) (cid:90) T (∆( f (cid:48) ( u ) u t ) , ∆ u t ) d s + 12 (cid:107) ∆ u t (0) (cid:107) L ≤ C(cid:15) (cid:90) T (cid:107) ∆( f (cid:48) ( u ) u t ) (cid:107) L d s + (cid:15) (cid:90) T (cid:107) ∆ u t (cid:107) L d s + C(cid:15) − σ . S. WU AND Y. LI
Then (2.12) is obtained by (2.15).Next we bound (2.13). Testing (2.16) with u tt , taking the integral over (0 , T ),and using (2.17), we obtain (cid:90) T (cid:107) u tt (cid:107) L d s + (cid:15) (cid:107) ∆ u t ( T ) (cid:107) L (2.18) ≤ (cid:15) (cid:107) ∆ u t (0) (cid:107) L + C(cid:15) (cid:90) T (cid:107) ∆( f (cid:48) ( u ) u t ) (cid:107) L d s + 12 (cid:90) T (cid:107) u tt (cid:107) L d s. Then (2.13) is obtained by (2.15).The next lemma gives an (cid:15) -independent lower bound for the principal eigenvalueof the linearized Cahn-Hilliard operator L CH defined below. The proof of this lemmacan be found in [12]. Lemma
Suppose that GA (1)–(3) hold. Given a smooth initial curve/surface Γ , let u be a smooth function satisfying Γ = { x ∈ Ω; u ( x ) = 0 } and some profiledescribed in [12]. Let u be the solution to problem (1.1) – (1.4) . Define L CH as L CH := ∆ (cid:18) (cid:15) ∆ − (cid:15) f (cid:48) ( u ) I (cid:19) . Then there exists < (cid:15) (cid:28) and a positive constant C such that the principleeigenvalue of the linearized Cahn-Hilliard operator L CH satisfies λ CH := inf (cid:54) = ψ ∈ H (Ω)∆ w = ψ (cid:15) (cid:107)∇ ψ (cid:107) L + (cid:15) ( f (cid:48) ( u ) ψ, ψ ) (cid:107)∇ w (cid:107) L ≥ − C for t ∈ [0 , T ] and (cid:15) ∈ (0 , (cid:15) ) .
3. Fully Discrete Approximation.
In this section, the backward Euler is usedfor time stepping, and the Morley finite element discretization is used for space dis-cretization.
Define the Morley finite element spaces S h below [8, 10, 14]: S h := { v h ∈ L ∞ (Ω) : v h ∈ P ( K ) , v h is continuous at the vertices of all triangles, ∂v h ∂n is continuous at the midpoints of interelement edges of triangles } . We use the following notation H jE (Ω) := { v ∈ H j (Ω) : ∂v∂n = 0 on ∂ Ω } j = 1 , , . Corresponding to H jE (Ω), define S hE as a subspace of S h below: S hE := { v h ∈ S h : ∂v h ∂n = 0 at the midpoints of the edges on ∂ Ω } . We also define ˚ H jE (Ω) = H jE (Ω) ∩ L (Ω) , j = 1 , ,
3, and ˚ S hE = S hE ∩ L (Ω), where L (Ω) denotes the set of mean zero functions. ORLEY ELEMENT FOR THE CH EQUATION AND THE HS FLOW (cid:101) E h is restated [7, 8, 10]. Let (cid:101) S hE be the Hsieh-Clough-Tocher macro element space, which is an enriched space of the Morley finite elementspace S hE . Let p and m be the internal vertices and midpoints of triangles T h . Define (cid:101) E h : S hE → (cid:101) S hE by ( (cid:101) E h v )( p ) = v ( p ) ,∂ ( (cid:101) E h v ) ∂n ( m ) = ∂v∂n ( m ) , ( ∂ β ( (cid:101) E h v ))( p ) = average of ( ∂ β v i )( p ) | β | = 1 , where v i = v | T i and triangle T i contains p as a vertex.Define the interpolation operator I h : H E (Ω) → S hE such that( I h v )( p ) = v ( p ) ,∂ ( I h v ) ∂n ( m ) = 1 | e | (cid:90) e ∂v∂n d S, where p ranges over the internal vertices of all the triangles T , and m ranges over themidpoints of all the edges e . It can be proved that [7, 8, 10, 14] | v − I h v | j,p,K ≤ Ch − j | v | ,p,K ∀ K ∈ T h , ∀ v ∈ H ( K ) , j = 0 , , , (3.1) (cid:107) (cid:101) E h v − v (cid:107) j, ,h ≤ Ch − j | v | , ,h ∀ v ∈ S hE , j = 0 , , . (3.2)Notice that (cid:101) E h and I h cannot preserve the mean zero functions. Let ˚ (cid:101) S hE := (cid:101) S hE ∩ L (Ω). Define ˚ (cid:101) E h : ˚ S hE (cid:55)→ ˚ (cid:101) S hE such that˚ (cid:101) E h v = (cid:101) E h v − | Ω | (cid:90) Ω (cid:101) E h v d x. (3.3)Using (3.2), we have (cid:90) Ω (cid:101) E h v d x = ( (cid:101) E h v − v, ≤ | Ω | / (cid:107) (cid:101) E h v − v (cid:107) , ≤ Ch | v | , ,h ∀ v ∈ ˚ S hE . Then (cid:107) ˚ (cid:101) E h v − v (cid:107) j, ,h ≤ Ch − j | v | , ,h ∀ v ∈ ˚ S hE , j = 0 , , . (3.4)Finally the following spaces are needed H ,h (Ω) = S h ⊕ H (Ω) , H ,hE (Ω) = S hE ⊕ H E (Ω) ,H ,h (Ω) = S h ⊕ H (Ω) , H ,hE (Ω) = S hE ⊕ H E (Ω) ,H ,h (Ω) = S h ⊕ H (Ω) , H ,hE (Ω) = S hE ⊕ H E (Ω) , where, for instance, S hE ⊕ H E (Ω) := { u + v : u ∈ S hE and v ∈ H E (Ω) } . S. WU AND Y. LI
The weak form of (1.1)–(1.4) is to seek u ( · , t ) ∈ H E (Ω) suchthat ( u t , v ) + (cid:15)a ( u, v ) + 1 (cid:15) ( ∇ f ( u ) , ∇ v ) = 0 ∀ v ∈ H E (Ω) , (3.5) u ( · ,
0) = u ∈ H E (Ω) , (3.6)where the bilinear form a ( · , · ) is defined as a ( u, v ) := (cid:90) Ω ∆ u ∆ v + (cid:0) ∂ u∂x∂y ∂ v∂x∂y − ∂ u∂x ∂ v∂y − ∂ u∂y ∂ v∂x (cid:1) d x d y (3.7)with Poisson’s ratio .Next define the discrete bilinear form a h ( u, v ) := (cid:88) K ∈T h (cid:90) K ∆ u ∆ v + (cid:0) ∂ u∂x∂y ∂ v∂x∂y − ∂ u∂x ∂ v∂y − ∂ u∂y ∂ v∂x (cid:1) d x d y. (3.8)Based on the bilinear form (3.8), a fully discrete Galerkin method is to seek u nh ∈ S hE such that( d t u nh , v h ) + (cid:15)a h ( u nh , v h ) + 1 (cid:15) ( ∇ f ( u nh ) , ∇ v h ) h = 0 ∀ v h ∈ S hE , (3.9) u h = u h ∈ S hE , (3.10)where the difference operator d t u nh := u nh − u n − h k and u h := P h u ( t ), where the operator P h is defined below. P h . We define R := (cid:8) v ∈ H E (Ω) : ∆ v ∈ H E (Ω) (cid:9) . Then ∀ v ∈ R , define the elliptic operator P h (cf. [14]) by seeking P h v ∈ S hE such that˜ b h ( P h v, w ) := ( (cid:15) ∆ v − (cid:15) ∇ · ( f (cid:48) ( u ) ∇ v ) + αv, w ) ∀ w ∈ S hE , (3.11)where ˜ b h ( v, w ) := (cid:15)a h ( v, w ) + 1 (cid:15) ( f (cid:48) ( u ) ∇ v, ∇ w ) h + α ( v, w ) , (3.12)and α should be chosen as α = α (cid:15) − to guarantee the coercivity of ˜ b h ( · , · ). Moreprecisely, first we cite some lemmas in [14], which will be used in this paper. Lemma
Let w, z ∈ H ,hE (Ω) , then (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (cid:88) K ∈T h (cid:90) ∂K ∂w∂n z d S (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ Ch ( h (cid:107) w (cid:107) , ,h (cid:107) z (cid:107) , ,h + (cid:107) w (cid:107) , ,h (cid:107) z (cid:107) , ,h + (cid:107) w (cid:107) , ,h (cid:107) z (cid:107) , ,h ) . Lemma
Let z ∈ H ,h (Ω) and w ∈ H E (Ω) ∩ H (Ω) , anddefine B h ( w, z ) by B h ( w, z ) = (cid:88) K ∈T h (cid:90) ∂K (cid:18) ∆ w ∂z∂n + 12 ∂ w∂n∂s − ∂ w∂s ∂z∂n (cid:19) d S, then we have (3.13) | B h ( w, z ) | ≤ Ch | w | , ,h | z | , ,h . ORLEY ELEMENT FOR THE CH EQUATION AND THE HS FLOW w ∈ S hE , using Lemma 3.1 and the inverse inequality, we have | w | , ,h ≤ | w | , ,h (cid:107) w (cid:107) , + (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (cid:88) K ∈T h (cid:90) ∂K ∂w∂n z d S (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ C (cid:107) w (cid:107) , ,h (cid:107) w (cid:107) , ≤ C ( | w | , ,h (cid:107) w (cid:107) , + | w | , ,h (cid:107) w (cid:107) , + (cid:107) w (cid:107) , ) . The kick-back argument gives | w | , ,h ≤ C ( | w | , ,h (cid:107) w (cid:107) , + (cid:107) w (cid:107) , ) . Hence, ˜ b h ( w, w ) = (cid:15)a h ( w, w ) + 1 (cid:15) ( f (cid:48) ( u ) ∇ w, ∇ w ) + α (cid:15) ( w, w )(3.14) ≥ (cid:15) (cid:18) (cid:15) | w | , ,h − C(cid:15) | w | , ,h + α (cid:107) w (cid:107) , (cid:19) ≥ (cid:15) (cid:18) (cid:15) | w | , ,h + ( α − C ) (cid:107) w (cid:107) , (cid:19) , which implies the coercivity of ˜ b h ( · , · ) when α is large enough but independent of (cid:15) .Next we give the properties of P h . Define b h ( · , · ) := (cid:15) ˜ b h ( · , · ) and a norm ||| v ||| , ,h := (cid:15) | v | , ,h + (cid:15) | v | , ,h + (cid:107) v (cid:107) , , Lemma
Consider the following problems: b h ( v, η ) = F h ( η ) ∀ η ∈ H E (Ω) , (3.15) b h ( v h , χ ) = (cid:101) F h ( χ ) ∀ χ ∈ S hE . (3.16) Then we have ||| v − v h ||| , ,h (3.17) ≤ Ch (cid:40) ( (cid:15) + h ) | v | , + | v | , + sup χ ∈ S hE F h ( (cid:101) E h χ ) − (cid:101) F h ( χ ) + α ( v, χ − (cid:101) E h χ ) ||| χ ||| , ,h (cid:41) . Proof.
Using (3.14) and the Strang Lemma, we have ||| v − v h ||| , ,h ≤ C (cid:32) inf ψ ∈ S hE ||| v − ψ ||| , ,h + sup χ ∈ S hE b h ( v, χ ) − (cid:101) F h ( χ ) ||| χ ||| , ,h (cid:33) ≤ C (cid:32) inf ψ ∈ S hE ||| v − ψ ||| , ,h + sup χ ∈ S hE b h ( v, χ − (cid:101) E h χ ) + b h ( v, (cid:101) E h χ ) − (cid:101) F h ( χ ) ||| χ ||| , ,h (cid:33) ≤ C (cid:32) inf ψ ∈ S hE ||| v − ψ ||| , ,h + sup χ ∈ S hE b h ( v, χ − (cid:101) E h χ ) + F h ( (cid:101) E h χ ) − (cid:101) F h ( χ ) ||| χ ||| , ,h (cid:33) . Using Lemma 3.2 and (3.2), we have b h ( v, χ − (cid:101) E h χ ) = (cid:15) a h ( v, χ − (cid:101) E h χ ) + (cid:15) ( f (cid:48) ( u ) ∇ v, ∇ ( χ − (cid:101) E h χ )) + ( α v, χ − (cid:101) E h χ ) ≤ Ch (cid:0) (cid:15) | v | , | χ | , ,h + (cid:15) | v | , | χ | , ,h (cid:1) + ( α v, χ − (cid:101) E h χ ) ≤ Ch (cid:0) (cid:15) | v | , + | v | , (cid:1) ||| χ ||| , ,h + ( α v, χ − (cid:101) E h χ )0 S. WU AND Y. LI
Then we obtain the desired bound (3.17) by the approximation properties of Morleyinterpolation operator (3.1).
Theorem
Suppose u solves the Cahn-Hilliard equation (1.1) – (1.3) , thenwe have (cid:15) | u − P h u | , ,h + (cid:15) | u − P h u | , ,h + (cid:107) u − P h u (cid:107) , (3.18) ≤ Ch (cid:0) ( (cid:15) + h ) | u | , + | u | , + (cid:15)h (cid:107) u t (cid:107) , (cid:1) ,(cid:15) | u t − ( P h u ) t | , ,h + (cid:15) | u t − ( P h u ) t | , ,h + (cid:107) u t − ( P h u ) t (cid:107) , (3.19) ≤ Ch (cid:110) ( (cid:15) + h ) | u t | , + | u t | , + (cid:15)h (cid:107) u tt (cid:107) , + (cid:107) u t ∇ u (cid:107) , + (cid:15) − | ln h | / (cid:107) u t (cid:107) , (( (cid:15) + h ) | u | , + | u | , + (cid:15)h (cid:107) u t (cid:107) , ) (cid:111) . Proof.
Taking v = u and v h = P h u in Lemma 3.3, and noticing that F h ( ψ ) = ˜ F h ( ψ ) = ( (cid:15) ∆ u − (cid:15) ∆ f ( u ) + α u, ψ ) = ( (cid:15) u t + α u, ψ ) , we obtain the bound (3.18) from (3.2) and (3.17).Taking v = u t and v h = ( P h u ) t , we have F h ( ψ ) = ( (cid:15) ∆ u t − (cid:15) ∆ f ( u ) t + α u t , ψ ) − ( (cid:15) f (cid:48)(cid:48) ( u ) u t ∇ u, ∇ ψ ) h , (cid:101) F h ( ψ ) = ( (cid:15) ∆ u t − (cid:15) ∆ f ( u ) t + α u t , ψ ) − ( (cid:15) f (cid:48)(cid:48) ( u ) u t ∇ P h u, ∇ ψ ) h . Then we get F h ( (cid:101) E h χ ) − (cid:101) F ( χ ) + α ( u t , χ − (cid:101) E h χ )= ( (cid:15) ∆ u t − (cid:15) ∆ f ( u ) t , (cid:101) E h χ − χ ) − ( (cid:15) f (cid:48)(cid:48) ( u ) u t ∇ u, ∇ (cid:101) E h χ − ∇ χ ) − ( (cid:15) f (cid:48)(cid:48) ( u ) u t ∇ ( u − P h u ) , ∇ χ ) ≤ (cid:15) h (cid:107) u tt (cid:107) , | χ | , ,h + C(cid:15) h (cid:107) u t ∇ u (cid:107) , | χ | , ,h + C(cid:15) (cid:107) u t (cid:107) , (cid:107)∇ χ (cid:107) , ∞ | u − P h u | , ,h ≤ Ch (cid:110) (cid:15)h (cid:107) u tt (cid:107) , + (cid:107) u t ∇ u (cid:107) , + (cid:15) − | ln h | / (cid:107) u t (cid:107) , (( (cid:15) + h ) | u | , + | u | , + (cid:15)h (cid:107) u t (cid:107) , ) (cid:111) ||| χ ||| , ,h , where we use the discrete Sobolev inequality and the fact that ∇ χ belongs to theCrouzeix-Raviar finite element space [9]. This implies the bound (3.19).Combining with the a priori estimates of the bounds given in Section 2, we havethe following theorem. Theorem
Assume h ≤ C(cid:15) , then there hold (cid:15) | u − P h u | , ,h + (cid:15) | u − P h u | , ,h + (cid:107) u − P h u (cid:107) , ≤ Ch ρ ( (cid:15) ) , (3.20) (cid:90) T (cid:15) | u t − ( P h u ) t | , ,h + (cid:15) | u t − ( P h u ) t | , ,h + (cid:107) u t − ( P h u ) t (cid:107) , d s (3.21) ≤ Ch (cid:15) ρ ( (cid:15) ) + Ch | ln h | ρ ( (cid:15) ) , where ρ ( (cid:15) ) := (cid:15) − max { σ + , σ + , σ +4 , σ } +4 ,ρ ( (cid:15) ) := (cid:15) − { σ + , σ + , σ +4 , σ } +2 . ORLEY ELEMENT FOR THE CH EQUATION AND THE HS FLOW Proof.
Using (2.3), (2.6) and (2.7), we have( (cid:15) + h ) | u | , + | u | , + (cid:15) h (cid:107) u t (cid:107) , (3.22) ≤ C(cid:15) − max { σ +5 , σ +2 } +4 + C(cid:15) − σ − + C(cid:15) − max { σ + , σ + , σ +4 , σ } +4 ≤ Cρ ( (cid:15) ) , which implies the bound (3.20) by (3.18).Using (2.7), (2.13), (2.9) and (2.14), we obtain (cid:90) T ( (cid:15) + h ) | u t | , + | u t | , + (cid:15) h (cid:107) u tt (cid:107) , + (cid:107) u t ∇ u (cid:107) , d s ≤ C (cid:90) T (cid:15) | u t | , + | u t | , + (cid:15) (cid:107) u tt (cid:107) , + (cid:107) u t (cid:107) , (cid:107)∇ u (cid:107) , ∞ d s ≤ C(cid:15) ρ ( (cid:15) ) + Cρ ( (cid:15) ) + C(cid:15) ρ ( (cid:15) )+ C(cid:15) − max { σ + ,σ +1 }− max { σ + , σ + , σ +4 , σ } ≤ C(cid:15) ρ ( (cid:15) ) . Further, using (2.7) and (3.22), we obtain (cid:90) T (cid:15) − (cid:107) u t (cid:107) , (( (cid:15) + h ) | u | , + | u | , + (cid:15)h (cid:107) u t (cid:107) , ) d s ≤ Cρ ( (cid:15) ) . This implies the bound (3.21).
Corollary
Under the condition that (3.23) h ≤ C(cid:15) ρ − ( (cid:15) ) , h ≤ Cρ − ( (cid:15) ) , h | ln h | ≤ C(cid:15) ρ − ( (cid:15) ) , there hold | P h u | j, ,h ≤ C (1 + | u | j, ,h ) j = 0 , , , (3.24) (cid:90) T | P h u | j, ,h d s ≤ C (1 + (cid:90) T | u | j, ,h ) j = 0 , , , (cid:107) P h u (cid:107) , ∞ ≤ C. Proof.
By the Sobolev embedding and (3.20), we have (cid:107) P h u (cid:107) , ∞ ≤ (cid:107) u (cid:107) , ∞ + (cid:107) u − P h u (cid:107) , ,h ≤ C + Ch(cid:15) − ρ / ( (cid:15) ) ≤ C. The first two bounds are the direct consequences of Theorem 3.5.
4. Error Estimates.
In this section, first we derive the piecewise L ∞ ( L ) and L ( H ) error bounds which depend on (cid:15) polynomially based on the generalized co-ercivity result in Theorem 4.3, and piecewise L ∞ ( H − ) and L ( H ) error bounds.Then we prove the piecewise L ∞ ( H ) error bound based on the piecewise L ∞ ( L )and L ( H ) error bounds. Finally, the L ∞ ( L ∞ ) error bound is established.Decompose the error u − u nh = ( u − P h u ) + ( P h u − u nh ) := ρ n + θ n . (4.1)The following two lemmas will be used in this section.2 S. WU AND Y. LI
Lemma
Suppose { a n } (cid:96)n =0 and { b n } (cid:96)n =0 are two se-quences, then (cid:96) (cid:88) n =1 ( a n − a n − , b n ) = ( a (cid:96) , b (cid:96) ) − ( a , b ) − (cid:96) (cid:88) n =1 ( a n − , b n − b n − ) . Lemma
Suppose u ( t n ) to be the solution of (1.1) – (1.4) , and u nh to be thesolution of (3.9) – (3.10) , then ρ n ∈ ˚ S hE , θ n ∈ ˚ S hE . Proof.
Testing (1.1) with constant 1, and then taking the integration over (0 , t ),we can obtain for any t ≥ (cid:90) Ω u ( t ) dx = (cid:90) Ω u (0) dx. Then choosing v = u ( t ) , w = 1 in (3.11), we have for any t ≥ (cid:90) Ω P h u ( t ) d x = (cid:90) Ω u ( t ) d x. Choosing v h = 1 in (3.9), then (cid:90) Ω u nh d x = (cid:90) Ω u n − h d x = · · · = (cid:90) Ω u h d x. Therefore, if choosing u h = P h u (0), then (cid:90) Ω u nh d x = (cid:90) Ω u h d x = (cid:90) Ω P h u (0) d x = (cid:90) Ω u (0) d x = (cid:90) Ω u ( t n ) d x = (cid:90) Ω P h u ( t n ) d x. Hence, P h u ( t n ) − u nh ∈ ˚ S hE . L ∞ ( H − ) and L ( H ) errorestimates. We first cite the generalized coercivity result, piecewise L ∞ ( H − ) and L ( H ) error estimates established in [27]. Theorem
Suppose there exists a positive number γ > such that the solution u of problem (1.1) – (1.4) and elliptic operator P h satisfy (4.2) (cid:107) u − P h u (cid:107) L ∞ ((0 ,T ); L ∞ ) ≤ C h(cid:15) − γ . Then there exists an (cid:15) -independent and h -independent constant C > such that for (cid:15) ∈ (0 , (cid:15) ) , a.e. t ∈ [0 , T ] , and for any ψ ∈ ˚ S hE , ( (cid:15) − (cid:15) )( ∇ ψ, ∇ ψ ) h + 1 (cid:15) ( f (cid:48) ( P h u ( t )) ψ, ψ ) h ≥ − C (cid:107)∇ ∆ − ψ (cid:107) L − C(cid:15) − γ − h , provided that h satisfies the constraint h ≤ ( C C ) − (cid:15) γ +3 , (4.3) where γ = 2 γ + σ + 6 and C is determined by C := max | ξ |≤(cid:107) u (cid:107) L ∞ ((0 ,T ); L ∞ ) | f (cid:48)(cid:48) ( ξ ) | . ORLEY ELEMENT FOR THE CH EQUATION AND THE HS FLOW Remark Thanks to the Sobolev embedding theorem and (3.20) , we have (4.4) (cid:107) u − P h u (cid:107) , ∞ ≤ (cid:107) u − P h u (cid:107) , ,h ≤ Ch(cid:15) − ρ ( (cid:15) ) , which gives the explicit formulation of γ in (4.2) . Theorem L ∞ ( H − ) and L ( H ) error estimates). Assume u isthe solution of (1.1) – (1.4) , u nh is the numerical solution of scheme (3.9) – (3.10) . Underthe mesh constraints in Theorem 3.15 in [27], we have the following error estimate (cid:107)∇ (cid:101) ∆ − h θ (cid:96) (cid:107) , ,h + k (cid:96) (cid:88) n =1 (cid:107)∇ (cid:101) ∆ − h d t θ n (cid:107) , ,h + (cid:15) k (cid:96) (cid:88) n =1 ( ∇ θ n , ∇ θ n ) h + k(cid:15) (cid:96) (cid:88) n =1 (cid:107) θ n (cid:107) , ,h ≤ C (˜ ρ ( (cid:15) ) | ln h | h + ˜ ρ ( (cid:15) ) k ) , where ˜ ρ ( (cid:15) ) and ˜ ρ ( (cid:15) ) are polynomial (cid:15) -dependent functions and (cid:101) ∆ − h is a discreteinverse Laplace operator defined in [27]. L ∞ ( L ) and piecewise L ( H ) error estimates. Based on Theorem 4.4,the L ∞ ( L ) and piecewise L ( H ) error estimates which depend on (cid:15) polynomially,instead of exponentially, are derived below. Notice that the Theorem 4.4 is used tocircumvent the use of interpolation of (cid:107) · (cid:107) , ,h between (cid:107) · (cid:107) , ,h and (cid:107) · (cid:107) , ,h , by whichonly the exponential dependence can be derived. Theorem
Assume u is the solution of (1.1) – (1.4) , u nh is the numerical so-lution of scheme (3.9) – (3.10) . Under the mesh constraints in Theorem 3.15 in [27]and (3.23) , the following L ∞ ( L ) and piecewise L ( H ) error estimates hold (cid:107) θ (cid:96) (cid:107) , , Ω + k (cid:96) (cid:88) n =1 (cid:107) d t θ n (cid:107) , , Ω + (cid:15)k (cid:96) (cid:88) n =1 a h ( θ n , θ n )(4.5) ≤ C ˜ ρ ( (cid:15) ) | ln h | h + C ˜ ρ ( (cid:15) ) | ln h | k , where ˜ ρ ( (cid:15) ) := (cid:15) ρ ( (cid:15) ) + (cid:15) − σ − ρ ( (cid:15) ) + ρ ( (cid:15) ) + (cid:15) − ˜ ρ ( (cid:15) ) + (cid:15) − γ − γ − ˜ ρ ( (cid:15) ) , ˜ ρ ( (cid:15) ) := ρ ( (cid:15) ) + (cid:15) − ˜ ρ ( (cid:15) ) + (cid:15) − γ − γ − ˜ ρ ( (cid:15) ) . Proof.
It follows from (3.9), (3.11), and (3.12) that for any v h ∈ S hE ,( d t θ n , v h ) + (cid:15)a h ( θ n , v h )(4.6) = [( d t P h u, v h ) + (cid:15)a h ( P h u, v h )] − [( d t u nh , v h ) + (cid:15)a h ( u nh , v h )]= − ( d t ρ n , v h ) + ( u t + (cid:15) ∆ u − (cid:15) ∆ f ( u ) + αu, v h ) + ( R n ( u tt ) , v h ) − (cid:15) ( f (cid:48) ( u ) ∇ P h u, ∇ v h ) h − α ( P h u, v h ) + 1 (cid:15) ( ∇ f ( u nh ) , ∇ v h ) h = ( − d t ρ n + αρ n , v h ) − (cid:15) ( f (cid:48) ( u ) ∇ P h u − ∇ f ( u nh ) , ∇ v h ) h + ( R n ( u tt ) , v h ) , where the remainder(4.7) R n ( u tt ) := u ( t n ) − u ( t n − ) k − u t ( t n ) = − k (cid:90) t n t n − ( s − t n − ) u tt ( s ) d s. S. WU AND Y. LI
Choosing v h = θ n , taking summation over n from 1 to (cid:96) , multiplying k on both sidesof (4.6), we have12 (cid:107) θ (cid:96) (cid:107) , + k (cid:96) (cid:88) n =1 (cid:107) d t θ n (cid:107) , + (cid:15)k (cid:96) (cid:88) n =1 a h ( θ n , θ n )(4.8) = k (cid:96) (cid:88) n =1 ( − d t ρ n + αρ n , θ n ) − k(cid:15) (cid:96) (cid:88) n =1 ( f (cid:48) ( u ) ∇ P h u − ∇ f ( u nh ) , ∇ θ n ) h + k (cid:96) (cid:88) n =1 ( R n ( u tt ) , θ n ) := I + I + I . Estimate of I : The first term on the right hand side of (4.6) can be bounded by I = k (cid:96) (cid:88) n =1 ( − d t ρ n + αρ n , θ n )(4.9) ≤ Ck (cid:96) (cid:88) n =1 (cid:107) d t ρ n (cid:107) , + Ck (cid:96) (cid:88) n =1 α (cid:107) ρ n (cid:107) , + Ck (cid:96) (cid:88) n =1 (cid:107) θ n (cid:107) , ≤ C ( (cid:15) ρ ( (cid:15) ) + (cid:15) − ρ ( (cid:15) )) h + Cρ ( (cid:15) ) | ln h | h + Ck (cid:96) (cid:88) n =1 (cid:107) θ n (cid:107) , , where by (3.20) and (3.21) k (cid:96) (cid:88) n =1 (cid:107) d t ρ n (cid:107) , = 1 k (cid:96) (cid:88) n =1 (cid:107) (cid:90) t n t n − ρ t d s (cid:107) , ≤ (cid:96) (cid:88) n =1 (cid:90) t n t n − (cid:107) ρ t (cid:107) , d s (4.10) ≤ (cid:90) T (cid:107) ρ t (cid:107) , d s ≤ C(cid:15) ρ ( (cid:15) ) h + Cρ ( (cid:15) ) | ln h | h ,k (cid:96) (cid:88) n =1 α (cid:107) ρ n (cid:107) , ≤ C(cid:15) − sup ≤ n ≤ (cid:96) (cid:107) ρ n (cid:107) , ≤ C(cid:15) − ρ ( (cid:15) ) h . (4.11)Estimate of I : The second term on the right hand side of (4.8) can be written as − k(cid:15) (cid:96) (cid:88) n =1 ( f (cid:48) ( u ) ∇ P h u − ∇ f ( u nh ) , ∇ θ n ) h (4.12) = − k(cid:15) (cid:96) (cid:88) n =1 ( f (cid:48) ( u ) ∇ P h u − f (cid:48) ( P h u ) ∇ P h u, ∇ θ n ) h − k(cid:15) (cid:96) (cid:88) n =1 ( ∇ f ( P h u ) − f (cid:48) ( P h u ) ∇ u nh , ∇ θ n ) h − k(cid:15) (cid:96) (cid:88) n =1 ( f (cid:48) ( P h u ) ∇ u nh − ∇ f ( u nh ) , ∇ θ n ) h := J + J + J . By (2.3), (3.20) and mesh condition (3.23), we have (cid:107)∇ P h u (cid:107) , ≤ (cid:107)∇ u (cid:107) , + C ≤ (cid:15) − σ − . ORLEY ELEMENT FOR THE CH EQUATION AND THE HS FLOW L ( H ) error estimate given in Theorem 4.4, thefirst term on the right-hand side of (4.12) can be bounded below J = − k(cid:15) (cid:96) (cid:88) n =1 ( ρ n ( u + P h u ) ∇ P h u, ∇ θ n ) h (4.13) ≤ Ck(cid:15) (cid:96) (cid:88) n =1 (cid:107) u + P h u (cid:107) , ∞ (cid:107) ρ n (cid:107) , ∞ (cid:107)∇ P h u (cid:107) , + Ck(cid:15) (cid:96) (cid:88) n =1 ( ∇ θ n , ∇ θ n ) h ≤ C(cid:15) − σ − ρ ( (cid:15) ) h + C(cid:15) − ˜ ρ ( (cid:15) ) | ln h | h + C(cid:15) − ˜ ρ ( (cid:15) ) k . Again, thanks to the piecewise L ( H ) error estimate given in Theorem 4.4, thesecond term on the right-hand side of (4.12) can be written as J = − k(cid:15) (cid:96) (cid:88) n =1 ( f (cid:48) ( P h u ) ∇ θ n , ∇ θ n ) h ≤ Ck(cid:15) (cid:96) (cid:88) n =1 ( ∇ θ n , ∇ θ n ) h (4.14) ≤ C(cid:15) − ˜ ρ ( (cid:15) ) | ln h | h + C(cid:15) − ˜ ρ ( (cid:15) ) k . By the discrete Sobolev inequality and Theorem 3.14 in [27], we have for any n , (cid:107) u nh (cid:107) , ∞ ,h ≤ C | ln h | (cid:107) u nh (cid:107) , ,h ≤ C(cid:15) − γ | ln h | . (4.15)Then, the third term on the right-hand side of (4.12) can be bounded by J = − k(cid:15) (cid:96) (cid:88) n =1 ( θ n ( P h u + u h ) ∇ u nh , ∇ θ n )(4.16) ≤ Ck (cid:96) (cid:88) n =1 (cid:107) θ n (cid:107) , + Ck(cid:15) (cid:96) (cid:88) n =1 (cid:107) P h u + u nh (cid:107) , ∞ (cid:107) u nh (cid:107) , ∞ ,h (cid:107)∇ θ n (cid:107) , ≤ Ck (cid:96) (cid:88) n =1 (cid:107) θ n (cid:107) , + C(cid:15) − γ − γ − | ln h | k (cid:96) (cid:88) n =1 (cid:107)∇ θ n (cid:107) , ≤ Ck (cid:96) (cid:88) n =1 (cid:107) θ n (cid:107) , + C(cid:15) − γ − γ − (˜ ρ ( (cid:15) ) | ln h | h + ˜ ρ ( (cid:15) ) | ln h | k ) . Estimate of I : The third term on the right hand side of (4.6) can be bounded by I = k (cid:96) (cid:88) n =1 ( R n ( u tt ) , θ n ) ≤ Ck (cid:96) (cid:88) n =1 (cid:107) R n ( u tt ) (cid:107) , + Ck (cid:96) (cid:88) n =1 (cid:107) θ n (cid:107) , (4.17) ≤ Cρ ( (cid:15) ) k + Ck (cid:96) (cid:88) n =1 (cid:107) θ n (cid:107) , , where by (2.13) and (4.7), k (cid:96) (cid:88) n =1 (cid:107) R n ( u tt ) (cid:107) , ≤ k (cid:96) (cid:88) n =1 (cid:16)(cid:90) t n t n − ( s − t n − ) d s (cid:17)(cid:16)(cid:90) t n t n − (cid:107) u tt ( s ) (cid:107) , d s (cid:17) (4.18) ≤ Cρ ( (cid:15) ) k . S. WU AND Y. LI L ∞ ( L ) and piecewise L ( H ) error estimates: Taking (4.9), (4.13), (4.14), (4.16),(4.16) into (4.8), we have12 (cid:107) θ (cid:96) (cid:107) , + k (cid:96) (cid:88) n =1 (cid:107) d t θ n (cid:107) , + (cid:15)k (cid:96) (cid:88) n =1 a h ( θ n , θ n )(4.19) ≤ Ck (cid:96) (cid:88) n =1 (cid:107) θ n (cid:107) , + C ( (cid:15) ρ ( (cid:15) ) + (cid:15) − σ − ρ ( (cid:15) )) h + C ( ρ ( (cid:15) ) + (cid:15) − ˜ ρ ( (cid:15) )) | ln h | h + (cid:15) − γ − γ − ˜ ρ ( (cid:15) ) | ln h | h + C ( ρ ( (cid:15) ) + (cid:15) − ˜ ρ ( (cid:15) )) k + C(cid:15) − γ − γ − ˜ ρ ( (cid:15) ) | ln h | k . The desired result (4.5) is therefore obtained by the Gronwall’s inequality. L ∞ ( H ) and L ∞ ( L ∞ ) error estimates. In this subsection, wegive the (cid:107) θ (cid:96) (cid:107) , ,h estimate by taking the summation by parts in time and integrationby parts in space, and using the special properties of the Morley element. The (cid:107) θ (cid:96) (cid:107) , ,h estimate below is “almost” optimal with respect to time and space. Theorem
Assume u is the solution of (1.1) – (1.4) , u nh is the numerical so-lution of scheme (3.9) – (3.10) . Under the mesh constraints in Theorem 3.15 in [27]and (3.23) , the following piecewise L ∞ ( H ) error estimate holds k (cid:96) (cid:88) n =1 (cid:107) d t θ n (cid:107) L + (cid:15)k (cid:96) (cid:88) n =1 a h ( d t θ n , d t θ n ) + (cid:15) (cid:107) θ (cid:96) (cid:107) , ,h (4.20) ≤ C ˜ ρ ( (cid:15) ) | ln h | h + C ˜ ρ ( (cid:15) ) | ln h | k , where ˜ ρ ( (cid:15) ) = (cid:15) − σ − ρ ( (cid:15) ) + (cid:15) − ρ ( (cid:15) ) ρ ( (cid:15) ) + (cid:15) − σ − ρ ( (cid:15) )+ (cid:16) (cid:15) − γ − + (cid:15) − γ − + (cid:15) − max { σ +5 , σ +2 }− + (cid:15) γ − max { σ + , σ + , σ +4 , σ }− (cid:17) ˜ ρ ( (cid:15) ) , ˜ ρ ( (cid:15) ) = (cid:16) (cid:15) − γ − + (cid:15) − γ − + (cid:15) − max { σ +5 , σ +2 }− + (cid:15) γ − max { σ + , σ + , σ +4 , σ }− (cid:17) ˜ ρ ( (cid:15) ) . Proof.
Choosing v h = θ n − θ n − = kd t θ n in (4.6), taking summation over n from1 to (cid:96) , we get k (cid:96) (cid:88) n =1 (cid:107) d t θ n (cid:107) L + (cid:15) a h ( θ (cid:96) , θ (cid:96) ) + (cid:15)k (cid:96) (cid:88) n =1 a h ( d t θ n , d t θ n )(4.21) = k (cid:96) (cid:88) n =1 ( − d t ρ n + αρ n , d t θ n ) − k(cid:15) (cid:96) (cid:88) n =1 ( f (cid:48) ( u ) ∇ P h u − ∇ f ( u nh ) , ∇ ( d t θ n )) h + k (cid:96) (cid:88) n =1 ( R n ( u tt ) , d t θ n ) := I + I + I . ORLEY ELEMENT FOR THE CH EQUATION AND THE HS FLOW (cid:15)a h ( θ n , θ n − θ n − ) = (cid:15)k a h ( d t θ n , d t θ n ) + (cid:15) a h ( θ n , θ n ) − (cid:15) a h ( θ n − , θ n − ) . Estimates of I and I : Similar to (4.9), using (4.10) and (4.11), we have I ≤ Ck (cid:96) (cid:88) n =1 (cid:107) d t ρ n (cid:107) L + Ck (cid:96) (cid:88) n =1 α (cid:107) ρ n (cid:107) L + k (cid:96) (cid:88) n =1 (cid:107) d t θ n (cid:107) L (4.22) ≤ C ( (cid:15) ρ ( (cid:15) ) + (cid:15) − ρ ( (cid:15) )) h + Cρ ( (cid:15) ) | ln h | h + k (cid:96) (cid:88) n =1 (cid:107) d t θ n (cid:107) , ,h . From (4.17) and (4.18), we also obtain the estimate of I below I = k (cid:96) (cid:88) n =1 ( R n ( u tt ) , d t θ n ) ≤ Ck (cid:96) (cid:88) n =1 (cid:107) R n ( u tt ) (cid:107) L + k (cid:96) (cid:88) n =1 (cid:107) d t θ n (cid:107) , (4.23) ≤ Cρ ( (cid:15) ) k + k (cid:96) (cid:88) n =1 (cid:107) d t θ n (cid:107) , . Estimate of I : Next we bound the more complicated term I . Using integration byparts, we have I = − k(cid:15) (cid:96) (cid:88) n =1 ( f (cid:48) ( u ) ∇ P h u − ∇ f ( P h u ) , d t ∇ θ n ) h − k(cid:15) (cid:96) (cid:88) n =1 ( ∇ ( f ( P h u ) − f ( u nh )) , d t ∇ θ n ) h (4.24)= − k(cid:15) (cid:96) (cid:88) n =1 ( f (cid:48) ( u ) ∇ P h u − ∇ f ( P h u ) , d t ∇ θ n ) h + k(cid:15) (cid:96) (cid:88) n =1 ( f ( P h u ) − f ( u nh ) , d t ∆ θ n ) h − k(cid:15) (cid:96) (cid:88) n =1 (cid:88) E ∈E h ( { f ( P h u ) − f ( u nh ) } , d t (cid:74) ∇ θ n (cid:75) ) E − k(cid:15) (cid:96) (cid:88) n =1 (cid:88) E ∈E h ( (cid:74) f ( P h u ) − f ( u nh ) (cid:75) , {∇ d t θ n } ) E := J + J + J + J . Here we adopt the standard DG notation and the DG identity, see [5, Equ. (3.3)].Next we bound J to J respectively. • Estimate of J . Using summation by parts in Lemma 4.1, we have J = k(cid:15) (cid:96) (cid:88) n =1 ( d t ( ρ ( u + P h u ) ∇ P h u ) , ∇ θ n − ) h − (cid:15) ( ρ (cid:96) ( u (cid:96) + P h u (cid:96) ) ∇ P h u (cid:96) , ∇ θ (cid:96) ) h . (4.25)8 S. WU AND Y. LI
Thanks to (2.3), (2.7), (2.9), (3.20), (3.21), (3.24), and the piecewise L ( H ) estimatein Theorem 4.4, the first term on the right hand side of (4.25) can be bounded by k(cid:15) (cid:96) (cid:88) n =1 ( d t ( ρ ( u + P h u ) ∇ P h u ) , ∇ θ n − ) h (4.26) ≤ k (cid:96) (cid:88) n =1 (cid:107) (cid:90) t n t n − ( ρ ( u + P h u ) ∇ P h u ) t d s (cid:107) , + C(cid:15) − k (cid:96) (cid:88) n =1 | θ n − | , ,h ≤ ess sup t ∈ [0 ,T ] (cid:107)∇ P h u (cid:107) , (cid:90) T (cid:107) ρ t (cid:107) , ∞ d s + ess sup t ∈ [0 ,T ] (cid:107) ρ (cid:107) , ∞ (cid:90) T (cid:107)∇ ( P h u ) t (cid:107) , d s + ess sup t ∈ [0 ,T ] (cid:107) ρ (cid:107) , ∞ (cid:107)∇ P h u (cid:107) , (cid:90) T (cid:107) u t + ( P h u ) t (cid:107) , ∞ d s + C(cid:15) − k (cid:96) (cid:88) n =1 | θ n − | , ,h ≤ C(cid:15) − σ − ( ρ ( (cid:15) ) + (cid:15) − ρ ( (cid:15) ) | ln h | ) h + C(cid:15) − ρ ( (cid:15) ) ρ ( (cid:15) ) h + C(cid:15) − σ − − max { σ + , σ + , σ +4 , σ } ρ ( (cid:15) ) h + C(cid:15) − ˜ ρ ( (cid:15) ) | ln h | h + C(cid:15) − ˜ ρ ( (cid:15) ) k . Thanks to (2.3), (3.20) and the L ∞ ( L ) estimate in Theorem 4.5, the second term onthe right hand of (4.25) can be bounded by − (cid:15) ( ρ (cid:96) ( u (cid:96) + P h u (cid:96) ) ∇ P h u (cid:96) , ∇ θ (cid:96) ) h (4.27) ≤ C(cid:15) − (cid:107) ρ l (cid:107) , ∞ | P h u l | , ,h + C(cid:15) − (cid:107) θ (cid:107) , + (cid:15) a h ( θ l , θ l ) ≤ C(cid:15) − σ − ρ ( (cid:15) ) h + C(cid:15) − ˜ ρ ( (cid:15) ) | ln h | h + C(cid:15) − ˜ ρ ( (cid:15) ) | ln h | k + (cid:15) a h ( θ l , θ l ) . Combining (4.26) and (4.27), simplifying the coefficients according to the definitionof ρ i ( (cid:15) ) and ˜ ρ i ( (cid:15) ), we obtain the bound for J : J ≤ C ( (cid:15) − σ − ρ ( (cid:15) ) + (cid:15) − ρ ( (cid:15) ) ρ ( (cid:15) ) + (cid:15) − σ − ρ ( (cid:15) ) + (cid:15) − ˜ ρ ( (cid:15) )) | ln h | h (4.28) + C(cid:15) − ˜ ρ ( (cid:15) ) | ln h | k + (cid:15) a h ( θ l , θ l ) . • Estimate of J . Define f ( P h u ) − f ( u nh ) := M n θ n , where M n is given as M n := ( P h u ( t n )) + P h u ( t n ) u nh + ( u nh ) − . Using summation by parts in Lemma 4.1, we have J = − k(cid:15) (cid:96) (cid:88) n =1 ( d t ( M n θ n ) , ∆ θ n − ) h + 1 (cid:15) ( M l θ l , ∆ θ l ) h (4.29) ≤ Ck(cid:15) (cid:96) (cid:88) n =1 (cid:107) d t ( M n θ n ) (cid:107) , | θ | , ,h + C(cid:15) (cid:107) M l θ l (cid:107) , | θ l | , ,h . ORLEY ELEMENT FOR THE CH EQUATION AND THE HS FLOW d t u nh = d t ( P h u n ) − d t θ n , a direct calculation shows that d t ( M n θ n ) = θ n d t M n + M n − d t θ n = M n − d t θ n + θ n ( P h u n + P h u n − ) d t ( P h u n )+ θ n u nh d t ( P h u n ) + θ n P h u n − d t ( P h u n ) − θ n P h u n − d t θ n + θ n ( u nh + u n − h ) d t ( P h u n ) − θ n ( u nh + u n − h ) d t θ n = ( M n − − θ n P h u n − − θ n ( u nh + u n − h )) d t θ n + ( P h u n + 2 P h u n − + 2 u nh + u n − h ) θ n d t ( P h u n ) . Using the L ( H ) error estimate (4.5) and the assumption on the L ∞ bound of u nh ,we get Ck(cid:15) (cid:96) (cid:88) n =1 (cid:107) d t ( M n θ n ) (cid:107) , | θ n | , ,h (4.30) ≤ C(cid:15) − γ − k (cid:96) (cid:88) n =1 (cid:107) d t θ n (cid:107) , | θ n | , ,h + C(cid:15) − γ − k (cid:96) (cid:88) n =1 (cid:107) θ n d t ( P h u ) (cid:107) , | θ n | , ,h ≤ k (cid:96) (cid:88) n =1 (cid:107) d t θ n (cid:107) , + C(cid:15) − γ − k (cid:96) (cid:88) n =1 | θ | , ,h + C(cid:15) γ k (cid:96) (cid:88) n =1 (cid:107) θd t ( P h u ) (cid:107) , ≤ k (cid:96) (cid:88) n =1 (cid:107) d t θ n (cid:107) , + C(cid:15) − γ − (˜ ρ ( (cid:15) ) | ln h | h + ˜ ρ ( (cid:15) ) | ln h | k )+ C(cid:15) γ − max { σ + , σ + , σ , σ }− (˜ ρ ( (cid:15) ) | ln h | h + ˜ ρ ( (cid:15) ) | ln h | k ) , where by (2.7) and the L ∞ ( L ) error estimate (4.5), k (cid:96) (cid:88) n =1 (cid:107) θd t ( P h u ) (cid:107) , ≤ sup ≤ n ≤ (cid:96) (cid:107) θ n (cid:107) , k (cid:107) (cid:90) t n t n − ( P h u ) t d s (cid:107) , ∞ ≤ sup ≤ n ≤ (cid:96) (cid:107) θ n (cid:107) , (cid:90) T (cid:107) ( P h u ) t (cid:107) , ∞ d s ≤ C(cid:15) − max { σ + , σ + , σ , σ }− (˜ ρ ( (cid:15) ) | ln h | h + ˜ ρ ( (cid:15) ) | ln h | k ) . And the second term on the right hand side of (4.29) can be bounded by
C(cid:15) (cid:107) M l θ l (cid:107) , | θ l | , ,h ≤ C − γ − (cid:107) θ l (cid:107) , + (cid:15) a h ( θ l , θ l )(4.31) ≤ C(cid:15) − γ − (˜ ρ ( (cid:15) ) | ln h | h + ˜ ρ ( (cid:15) ) | ln h | k ) + (cid:15) a h ( θ l , θ l ) . Combining (4.30) and (4.31), we obtain the bound for J : J ≤ k (cid:96) (cid:88) n =1 (cid:107) d t θ n (cid:107) , + (cid:15) a h ( θ l , θ l ) + C(cid:15) − γ − (˜ ρ ( (cid:15) ) | ln h | h + ˜ ρ ( (cid:15) ) | ln h | k )(4.32) + C(cid:15) γ − max { σ + , σ + , σ +4 , σ }− (˜ ρ ( (cid:15) ) | ln h | h + ˜ ρ ( (cid:15) ) | ln h | k ) . S. WU AND Y. LI • Estimate of J . Notice that θ n ∈ S hE and (cid:90) E (cid:74) ∇ θ n (cid:75) d S = 0 ∀ E ∈ E h . Using summation by parts in Lemma 4.1, Lemma 2.2 in [14] and inverse inequality,we have J = k(cid:15) (cid:96) (cid:88) n =1 (cid:88) E ∈E h ( d t { M n θ n } , (cid:74) ∇ θ n − (cid:75) ) E − (cid:15) (cid:88) E ∈E h ( { M (cid:96) θ (cid:96) } , (cid:74) ∇ θ (cid:96) (cid:75) ) E ≤ Ck(cid:15) (cid:96) (cid:88) n =1 (cid:107) d t ( M n θ n ) (cid:107) , | θ | , ,h + C(cid:15) (cid:107) M (cid:96) θ (cid:96) (cid:107) , | θ l | , ,h . Hence, J has the same bound as J . • Estimate of J . Since P h u and u h are continuous at vertexes of T h , thanks toLemma 2.6 in [14], we have J ≤ Ck(cid:15) (cid:96) (cid:88) n =1 h | M n θ n | , ,h | d t θ n | , ,h (4.33) ≤ Ck(cid:15) (cid:96) (cid:88) n =1 | M n θ n | , ,h (cid:107) d t θ n (cid:107) , ≤ Ck(cid:15) (cid:96) (cid:88) n =1 | M n θ n | , ,h + k (cid:96) (cid:88) n =1 (cid:107) d t θ n (cid:107) , . Using the piecewise L ( H ) estimate given in Theorem 4.4, we have Ck(cid:15) (cid:96) (cid:88) n =1 | M n θ n | , ,h (4.34) ≤ Ck(cid:15) (cid:96) (cid:88) n =1 (cid:0) (cid:107) M n (cid:107) , ∞ | θ n | , ,h + | M n | , ,h | θ n | , ,h + | M n | , ,h (cid:107) θ n (cid:107) , ∞ (cid:1) ≤ C(cid:15) sup ≤ n ≤ (cid:96) (cid:107) M n (cid:107) , ,h k (cid:96) (cid:88) n =1 (cid:107) θ n (cid:107) , ,h ≤ C ( (cid:15) − γ − + (cid:15) − max { σ +5 , σ +2 }− )(˜ ρ ( (cid:15) ) | ln h | h + ˜ ρ ( (cid:15) ) | ln h | k ) , where by (2.6) and the fact that (cid:107) u nh (cid:107) , ,h ≤ C(cid:15) − γ (c.f. [27, Theorem 3.14]) (cid:107) M n (cid:107) , ,h ≤ C ( (cid:107) ( P h u n ) (cid:107) , ,h + (cid:107) u nh P h u n (cid:107) , ,h + (cid:107) ( u nh ) (cid:107) , ,h ) ≤ C ( (cid:107) P h u n (cid:107) , ,h + (cid:107) P h u n (cid:107) , ,h + (cid:107) u h (cid:107) , ∞ (cid:107) u nh (cid:107) , ,h + (cid:107) u nh (cid:107) , ,h + (cid:107) u nh (cid:107) , ,h + (cid:107) u nh (cid:107) , ∞ (cid:107) P h u n (cid:107) , ,h + (cid:107) u nh (cid:107) , ,h (cid:107) P h u n (cid:107) , ,h ) ≤ C ( (cid:15) − γ + (cid:15) − max { σ +5 , σ +2 } ) . Piecewise L ∞ ( H ) error estimate: Taking (4.22), (4.23), (4.28), (4.32) and (4.33) ORLEY ELEMENT FOR THE CH EQUATION AND THE HS FLOW k (cid:96) (cid:88) n =1 (cid:107) d t θ n (cid:107) L + (cid:15) a h ( θ (cid:96) , θ (cid:96) ) + (cid:15)k (cid:96) (cid:88) n =1 a h ( d t θ n , d t θ n )(4.35) ≤ C ( (cid:15) ρ ( (cid:15) ) + (cid:15) − ρ ( (cid:15) )) h + Cρ ( (cid:15) ) | ln h | h + Cρ ( (cid:15) ) k + C ( (cid:15) − σ − ρ ( (cid:15) ) + (cid:15) − ρ ( (cid:15) ) ρ ( (cid:15) ) + (cid:15) − σ − ρ ( (cid:15) ) + (cid:15) − ˜ ρ ( (cid:15) )) | ln h | h + C(cid:15) − ˜ ρ ( (cid:15) ) | ln h | k + C(cid:15) − γ − (˜ ρ ( (cid:15) ) | ln h | h + ˜ ρ ( (cid:15) ) | ln h | k )+ C(cid:15) − max { σ + , σ + , σ +4 , σ }− (˜ ρ ( (cid:15) ) | ln h | h + ˜ ρ ( (cid:15) ) | ln h | k )+ C ( (cid:15) − γ − + (cid:15) − max { σ +5 , σ +2 }− )(˜ ρ ( (cid:15) ) | ln h | h + ˜ ρ ( (cid:15) ) | ln h | k ) . Then the theorem can be proved by simplifying the coefficients according to thedefinitions of ρ i ( (cid:15) ) and ˜ ρ i ( (cid:15) ). Remark If the summation by part for time and integration by part for spacetechniques are not employed simultaneously, one can only obtain a coarse estimate (cid:107) θ (cid:96) (cid:107) , ,h + k (cid:96) (cid:88) n =1 (cid:107) d t θ n (cid:107) L + (cid:15)k (cid:96) (cid:88) n =1 a h ( d t θ n , d t θ n ) ≤ Ck − ( (cid:15) − γ | ln h | h + (cid:15) − γ | ln h | k ) , where γ , γ denote some positive constants. Finally, using (4.4), Theorem 4.6 and the Sobolev embedding theorem, we canprove the desired L ∞ ( L ∞ ) error estimate. Theorem
Assume u is the solution of (1.1) – (1.4) , u nh is the numerical so-lution of scheme (3.9) – (3.10) . Under the mesh constraints in Theorem 3.15 in [27]and (3.23) , we have the L ∞ ( L ∞ ) error estimate (cid:107) u ( t n ) − u nh (cid:107) L ∞ ≤ C | ln h | ((˜ ρ ( (cid:15) )) | ln h | h + (˜ ρ ( (cid:15) )) k ) ∀ ≤ n ≤ (cid:96). (4.36) Remark The mesh constraints in Theorem 3.15 in [27] and (3.23) can beachieved by h = C(cid:15) p and k = C(cid:15) p for certain positive p , p . Hence, the | ln h | k decreases asymptoticly as k when (cid:15) goes to zero.
5. Convergence of the Numerical Interface.
In this section, we prove thatthe numerical interface defined as the zero level set of the Morley element interpolationof the solution U n converges to the moving interface of the Hele-Shaw problem underthe assumption that the Hele-Shaw problem has a unique global (in time) classicalsolution. We first cite the following convergence result established in [2]. Theorem
Let Ω be a given smooth domain and Γ be a smooth closed hy-persurface in Ω . Suppose that the Hele-Shaw problem starting from Γ has a uniquesmooth solution (cid:0) w, Γ := (cid:83) ≤ t ≤ T (Γ t × { t } ) (cid:1) in the time interval [0 , T ] such that Γ t ⊆ Ω for all t ∈ [0 , T ] . Then there exists a family of smooth functions { u (cid:15) } <(cid:15) ≤ which are uniformly bounded in (cid:15) ∈ (0 , and ( x, t ) ∈ Ω T , such that if u (cid:15) solves theCahn-Hilliard problem (1.1) – (1.3) , then (i) lim (cid:15) → u (cid:15) ( x, t ) = (cid:40) if ( x, t ) ∈ O− if ( x, t ) ∈ I uniformly on compact subsets, where I and O stand for the “inside” and “outside” of Γ ; S. WU AND Y. LI (ii) lim (cid:15) → (cid:0) (cid:15) − f ( u (cid:15) ) − (cid:15) ∆ u (cid:15) (cid:1) ( x, t ) = − w ( x, t ) uniformly on Ω T . We are now ready to state the first main theorem of this section.
Theorem
Let { Γ t } t ≥ denote the zero level set of the Hele-Shaw problemand U (cid:15),h,k ( x, t ) denotes the piecewise linear interpolation in time of the numericalsolution u nh , namely, U (cid:15),h,k ( x, t ) := t − t n − k u nh ( x ) + t n − tk u n − h ( x ) , (5.1) for t n − ≤ t ≤ t n and ≤ n ≤ M . Then, under the mesh and starting valueconstraints of Theorem 4.6 and k = O ( h q ) with < q < , we have (i) U (cid:15),h,k ( x, t ) (cid:15) (cid:38) −→ uniformly on compact subset of O , (ii) U (cid:15),h,k ( x, t ) (cid:15) (cid:38) −→ − uniformly on compact subset of I .Proof. For any compact set A ⊂ O and for any ( x, t ) ∈ A , we have | U (cid:15),h,k − | ≤ | U (cid:15),h,k − u (cid:15) ( x, t ) | + | u (cid:15) ( x, t ) − | (5.2) ≤ | U (cid:15),h,k − u (cid:15) ( x, t ) | L ∞ (Ω T ) + | u (cid:15) ( x, t ) − | . Theorem 4.7 infers that(5.3) | U (cid:15),h,k − u (cid:15) ( x, t ) | L ∞ (Ω T ) ≤ C (˜ ρ ( (cid:15) )) h q | ln h | . where ˜ ρ ( (cid:15) ) = max { ˜ ρ ( (cid:15) ) , ˜ ρ ( (cid:15) ) } . The first term on the right-hand side of (5.2) tends to 0 when (cid:15) (cid:38) h, k (cid:38)
0, too). The second term converges uniformly to 0 on the compact set A ,which is ensured by (i) of Theorem 5.1. Hence, the assertion (i) holds.To show (ii), we only need to replace O by I and 1 by − Theorem
Let Γ (cid:15),h,kt := { x ∈ Ω; U (cid:15),h,k ( x, t ) = 0 } be the zero level set of U (cid:15),h,k ( x, t ) , then under the assumptions of Theorem 5.2, we have sup x ∈ Γ (cid:15),h,kt dist( x, Γ t ) (cid:15) (cid:38) −→ uniformly on [0 , T ] . Proof.
For any η ∈ (0 , N η of width 2 η of Γ t (5.4) N η := { ( x, t ) ∈ Ω T ; dist( x, Γ t ) < η } . Let A and B denote the complements of the neighborhood N η in O and I , respectively, A = O \ N η and B = I \ N η . Note that A is a compact subset outside Γ t and B is a compact subset inside Γ t .By Theorem 5.2, there exists (cid:15) >
0, which only depends on η , such that for any (cid:15) ∈ (0 , (cid:15) ) | U (cid:15),h,k ( x, t ) − | ≤ η ∀ ( x, t ) ∈ A, (5.5) | U (cid:15),h,k ( x, t ) + 1 | ≤ η ∀ ( x, t ) ∈ B. (5.6) ORLEY ELEMENT FOR THE CH EQUATION AND THE HS FLOW t ∈ [0 , T ] and x ∈ Γ (cid:15),h,kt , from U (cid:15),h,k ( x, t ) = 0 we have | U (cid:15),h,k ( x, t ) − | = 1 ∀ ( x, t ) ∈ A, (5.7) | U (cid:15),h,k ( x, t ) + 1 | = 1 ∀ ( x, t ) ∈ B. (5.8)(5.5) and (5.7) imply that ( x, t ) is not in A , and (5.6) and (5.8) imply that ( x, t ) isnot in B , then ( x, t ) must lie in the tubular neighborhood N η . Therefore, for any (cid:15) ∈ (0 , (cid:15) ),(5.9) sup x ∈ Γ (cid:15),h,kt dist( x, Γ t ) ≤ η uniformly on [0 , T ] . The proof is complete.
6. Numerical experiments.
In this section, we present two two-dimensionalnumerical tests to gauge the performance of the proposed fully discrete Morley finiteelement method for Cahn-Hilliard equation. The square domain Ω = [ − , is usedin both tests. Test 1.
Consider the Cahn-Hilliard problem with an ellipse initial interface de-termined by Γ : x . + y . = 0. The initial condition is chosen to have the form u ( x, y ) = tanh( d ( x,y ) √ (cid:15) ), where d ( x, y ) denotes the signed distance from ( x, y ) to theinitial ellipse interface Γ and tanh( t ) = ( e t − e − t ) / ( e t + e − t ).Figure 1 displays four snapshots at four fixed time points of the numerical interfacewith four different (cid:15) ’s. Here time step size k = 1 × − and space size h = 0 .
01 areused. They clearly indicate that at each time point the numerical interface convergesto the sharp interface Γ t of the Hele-Shaw flow as (cid:15) tends to zero. Note that thisinitial condition may not satisfy the General Assumption (GA) due to the singularityof the signed distance function. We will adopt a smooth initial condition in the latertest. Test 2.
Consider the following initial condition, which is also adopted in [23], u ( x, y ) = tanh (cid:16) (( x − . + y − . ) /(cid:15) (cid:17) tanh (cid:16) (( x + 0 . + y − . ) /(cid:15) (cid:17) . Table 1 and 2 show the errors of spatial L , H and H semi-norms and the ratesof convergence at T = 0 . T = 0 . (cid:15) = 0 .
08 is used to generate the table. k = 1 × − is chosen so that the error in time is relatively small to the error in space.The L ∞ ( H ) norm error is in agreement with the convergence theorem, but L ∞ ( L )and L ∞ ( H ) norm errors are one order higher than our theoretical results. We notethat in [14], the second order convergence for both L ∞ ( L ) and L ∞ ( H ) norms areproved, whereas only (cid:15) -exponential dependence can be derived. L ∞ ( L ) error order L ∞ ( H ) error order L ∞ ( H ) error order h = 0 . √ h = 0 . √ h = 0 . √ h = 0 . √ h = 0 . √ Table 1
Spatial errors and convergence rates of Test 2: (cid:15) = 0 . , k = 1 × − , T = 0 . . Figure 2 displays six snapshots at six fixed time points of the numerical interfacewith four different (cid:15) . Again, they clearly indicate that at each time point the numericalinterface converges to the sharp interface Γ t of the Hele-haw flow as (cid:15) tends to zero.4 S. WU AND Y. LI t=0 -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 ǫ = 0.08 ǫ = 0.04 ǫ = 0.03 ǫ = 0.02 t=0.005 -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 ǫ = 0.08 ǫ = 0.04 ǫ = 0.03 ǫ = 0.02 t=0.015 -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 ǫ = 0.08 ǫ = 0.04 ǫ = 0.03 ǫ = 0.02 t=0.03 -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 ǫ = 0.08 ǫ = 0.04 ǫ = 0.03 ǫ = 0.02 Fig. 1 . Test 1: Snapshots of the zero-level sets of u (cid:15),k at t = 0 , . , . , . and (cid:15) =0 . , . , . , . . L ∞ ( L ) error order L ∞ ( H ) error order L ∞ ( H ) error order h = 0 . √ h = 0 . √ h = 0 . √ h = 0 . √ h = 0 . √ Table 2
Spatial errors and convergence rates of Test 2: (cid:15) = 0 . , k = 1 × − , T = 0 . . Acknowledgements.
The authors Shuonan Wu and Yukun Li highly thankProfessor Xiaobing Feng in the University of Tennessee at Knoxville for his motivationfor this paper.
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