Analysis of a second order discontinuous Galerkin finite element method for the Allen-Cahn equation

Transcription

1 Graduate Teses and Dissertations Graduate College 06 Analysis of a second order discontinuous Galerin finite element metod for te Allen-Can equation Junzao Hu Iowa State University Follow tis and additional wors at: ttp://lib.dr.iastate.edu/etd Part of te Applied Matematics Commons Recommended Citation Hu, Junzao, "Analysis of a second order discontinuous Galerin finite element metod for te Allen-Can equation" 06. Graduate Teses and Dissertations ttp://lib.dr.iastate.edu/etd/4980 Tis Tesis is brougt to you for free and open access by te Graduate College at Iowa State University Digital Repository. It as been accepted for inclusion in Graduate Teses and Dissertations by an autorized administrator of Iowa State University Digital Repository. For more information, please contact digirep@iastate.edu.

2 Analysis of a second order discontinuous Galerin finite element metod for te Allen-Can equation by Junzao Hu A tesis submitted to te graduate faculty in partial fulfillment of te requirements for te degree of MASTER OF SCIENCE Major: Applied Matematics Program of Study Committee: Steven Hou, Major Professor Huaiqing Wu Zijun Wu Iowa State University Ames, Iowa 06 Copyrigt c Junzao Hu, 06. All rigts reserved.

3 ii TABLE OF CONTENTS ACKNOWLEDGEMENTS ABSTRACT iii iv CHAPTER. INTRODUCTION CHAPTER. PRELIMINARIES CHAPTER 3. FULLY DISCRETE IP-DG APPROXIMATIONS Discretized DG sceme Stability of te DG sceme Well-posedness of te DG sceme Error estimates analysis CHAPTER 4. CONVERGENCE OF THE NUMERICAL INTERFACE TO THE MEAN CURVATURE FLOW CHAPTER 5. NUMERICAL EXPERIMENTS BIBLIOGRAPHY

4 iii ACKNOWLEDGEMENTS I would lie to tae tis opportunity to express my tans to tose wo elped me wit various aspects of conducting researc and te writing of tis tesis. First and foremost, my adviser, Professor Steven Hou, for is guidance, patience and support trougout tis wor and te writing of tis tesis. I would also lie to tan my committee members Professors Huaiqing Wu and Zijun Wu for teir efforts.

5 iv ABSTRACT Te paper proposes and analyzes an efficient second-order in time numerical approximation for te Allen-Can equation wic is a nonlinear singular perturbation of te reaction-diffusion model arising from pase separation in alloys. We firstly present a fully discrete, nonlinear interior penalty discontinuous Galerin finite elementipdgfe metod, wic is based on te modified Cran-Nicolson sceme and a mid-point approximation of te potential term fu. We ten derive te stability analysis and error estimates for te proposed IPDGFE metod under some regularity assumptions on te initial function u 0. Tere are two ey wors in our analysis, one is to establis unconditionally energy-stable sceme for te discrete solutions. Te oter is to use a discrete spectrum estimate to andle te midpoint of te discrete solutions u m and u m+ in te nonlinear term, instead of using te standard Gronwall inequality tecnique. We obtain tat all our error bounds depend on reciprocal of te perturbation parameter ɛ only in some lower polynomial order, instead of exponential order.

6 CHAPTER. INTRODUCTION Let Ω R d d =, 3 be a bounded polygonal or polyedral domain. Consider te following nonlinear singular perturbation model of te reaction-diffusion equation u t u + ɛ fu = 0, in Ω T := Ω 0, T.. In tis paper, we consider te following omogenous Neumann boundary condition and initial condition u n = 0, in Ω T := Ω 0, T,. u = u 0, in Ω {t = 0},.3 were, n denotes te unit outward normal vector to te boundary Ω, and te boundary condition.3 means tat no mass loss occurs troug te boundary walls. Equation., wic is called te Allen-Can equation, was originally introduced by Allen and Can in [] to describe an interface evolving in time in te pase separation process of te crystalline solids. Herein, ɛ > 0 is a parameter related to te interface ticness, wic is small compared to te caracteristic lengt of te laboratory scale. u denotes te concentration of one of te two metallic species of te alloy, and fu = F u wit F u being some given energy potential. Several coices of F u ave been presented in te literature [-6]. In tis paper we focus on te following Ginzburg-Landau double-well potential F u = 4 u and fu = F u = u u..4 Altoug te potential term.4 as been widely used, its quartic growt at infinity leads to a variety of tecnical difficulties in te numerical approximation for te Allen-Can equation. For example, in order to assure tat our numerical sceme is second-order in time, we ave to

7 employ te modified Cran-Nicolson sceme and a second order in time approximation of te potential term fusee 3.4 in section 3..

8 3 CHAPTER. PRELIMINARIES Let T be a quasi-uniform triangulation of Ω suc tat Ω = K T K. Let K denote te diameter of K T and := max{ K ; K T }. We recall tat te standard broen Sobolev space H s T and DG finite element space V are defined as H s T := H s K, V := K T K T P r K, were P r K denotes te set of all polynomials wose degrees do not exceed a given positive integer r. Let E I denote te set of all interior faces/edges of T, E B denote te set of all boundary faces/edges of T, and E := E I EB. Te L -inner product for piecewise functions over te mes T is naturally defined by u, v T := K T K uv dx, and for any set S E, te L -inner product over S is defined by u, v := uv ds. S e S Let K, K T and e = K K and assume global labeling number of K is smaller tan tat of K. We coose n e := n K e = n K e as te unit normal on e and define te following standard jump and average notations across te face/edge e: [v] := v K v K on e E I, [v] := v on e EB, {v} := v K + v K e on e E I, {v} := v on e EB for v V. Let M be a large positive integer. Define τ := T/M and t m := mτ for m = 0,,,, M be a uniform partition of [0, T ]. For a sequence of functions {v m } M m=0, we define te bacward difference operator d t u m := um u m, m =,,, M.

9 4 First, we introduce te DG elliptic projection operator P r : H s T V by a v P r v, w + v P r v, w T = 0 w V. for any v H s T. We start wit a well-nown fact [8, 7] tat te Allen-Can equation. can be interpreted as te L -gradient flow for te following Can-Hilliard energy functional J ɛ v := v + ɛ F v dx. Ω Te following assumptions on te initial datum u 0 are made as in [6, 9] to derive a priori solution estimates. General Assumption GA Tere exists a nonnegative constant σ suc tat J ɛ u 0 Cɛ σ..3 Tere exists a nonnegative constant σ suc tat u 0 ɛ fu 0 L Ω Cɛ σ..4 3 Tere exists nonnegative constant σ 3 suc tat Te following solution estimates can be found in [6,9]. lim u ts L s 0 + Ω Cɛ σ 3..5 Proposition Suppose tat.3 and.4 old. Ten te solution u of problem..4 satisfies te following estimates: ess sup ut L Ω,.6 t [0, ess sup t [0, T 0 ess sup t [0, 0 J ɛ u + 0 u t s L Ω ds Cɛ σ,.7 us ds Cɛ σ +,.8 u t L Ω + u H Ω + u tt s H Ω + u ts H Ω 0 u t s L Ω ds Cɛ max{σ +,σ },.9 ds Cɛ max{σ +,σ }..0

10 5 In addition to.3 and.4, suppose tat.5 olds, ten u also satisfies ess sup u t L Ω + u tt s L ds Cɛ max{σ +,σ 3 },. t [0, 0 0 u t s L Ω ds Cɛ max{σ +,σ 3 }.. Next, we quote te following well nown error estimate results from [, ]. Lemma Let v W s, T, ten tere old v P r v L T + v P r v L T C min{r+,s} u H s T,.3 ln r v P r v L T + u P r u L T C min{r+,s} u W s, T..4 were r := min{, r} min{, r }. Let C = max ξ f ξ..5 and P r, corresponding to P r, denote te elliptic projection operator on te finite element space S := V C 0 Ω, tere olds te following estimate from []: u P r u L C d u H..6 We now state our discrete spectrum estimate for te DG approximation. Proposition Suppose tere exists a positive number γ > 0 suc tat te solution u of problem..4 satisfies ess sup ut W r+, Ω Cɛ γ..7 t [0,T ] Ten tere exists an ɛ-independent and -independent constant c 0 > 0 suc tat for ɛ 0, and a.e. t [0, T ] λ DG t := inf ψ V ψ 0 a ψ, ψ + ɛ f P r ut ψ, ψ ψ L T T c 0,.8 provided tat satisfies te constraint d C0 C C ɛ max{σ +3,σ +},.9 min{r+,s} ln r C 0 C C ɛ γ+,.0

11 6 were C arises from te following inequality: u P r u L 0,T ;L Ω C min{r+,s} ln r ɛ γ,. u P r u L 0,T ;L Ω C d ɛ max{σ +,σ }.. Lemma Let {S l } l be a positive nondecreasing sequence and {b l } l and { l } l be nonnegative sequences, and p > be a constant. If S p S l+ S l b l S l + l S p l for l,.3 + p l s= s a p s+ > 0 for l,.4 ten were S l a l { l + p S p a l := l s= s= s a p s+ } p for l,.5 + b s for l..6

12 7 CHAPTER 3. FULLY DISCRETE IP-DG APPROXIMATIONS 3. Discretized DG sceme We are now ready to introduce our fully discrete DG finite element metods for problem..4. Tey are defined by seeing u m V for m = 0,,,, M suc tat dt u m+, v + a T u m+, v + f m+ ɛ, v = 0 v T V, 3. were a u, v := u, v T { n u}, [v ] E I 3. + λ [u], { n v } E I + j u, v, j u, v := σ e [u], [v ] e, 3.3 e e E I were u m+ f m+ := 4[ u m+ 3 + u m+ u m + u m+ u m + u m 3] u m+ 3.4 = F um+ F u m u m+ u m. = um+ +u m, λ = 0, ± and σ e is a positive piecewise constant function on E I, wic will be cosen later see Lemma 3. In addition, we need to supply u 0 time-stepping, wose coice will be clear and will be specified below. to start te Lemma 3 Tere exist constants σ 0, α > 0 suc tat for σ e > σ 0 for all e E tere olds Φ v α v,dg v V, were v,dg := v L T + j v, v.

13 8 Now we introduce tree mes-dependent energy functionals wic can be regarded as DG counterparts of te continuous Can-Hilliard energy J ɛ defined in.. Φ v := v L T { n v}, [v] + E I j v, v v H T, 3.5 Jɛ v := Φ v + F v, ɛ T v H T, 3.6 Iɛ v := Φ v + F + ɛ c v, T v H T, 3.7 It is easy to cec tat Φ and I ɛ are convex functionals but J ɛ is not because F is not convex. Moreover, we ave: Lemma 4 Let λ = in 3., ten tere olds for all v, w V δφ v Φ v + sw Φ v, w := lim = a v, w, 3.8 δv T s 0 s δj ɛ v Jɛ v + sw Jɛ v, w : = lim 3.9 δv T s 0 s = a v, w + ɛ F v, w T, δi ɛ v Iɛ v + sw Iɛ v, w : = lim δv T s 0 s 3.0 = a v, w + ɛ F + c v, w T. 3. Stability of te DG sceme Teorem Te sceme is unconditionally stable for all, > 0. Proof: We ave te DG sceme as below: Let v = d t u m+, and we will get: dt u m+, v + a u m+ + u m dt u m+, d t u n+ + a u m+ + u m Rearrange it to get:, v + ɛ f m+, v = 0. 3., d t u m+ + ɛ F u m+ F u m u m+ u m, d t u m+ = d t u m+ L + d t[a u m+, u m+ ] + ɛ d tf u m+ = 0, 3.3 And te proof is complete. d t [ a u m+, u m+ + ɛ F um+, ]

14 9 3.3 Well-posedness of te DG sceme We want to get a second order approximation of fu m+, u m, wic leads to unconditionally energy stable scemes. We split te function F v = 4 v into te difference of two convex parts and get te convex decomposition F v = F c + v Fc v,were F c + v := 4 v4 + and Fc v := v. Now we want to construct a second-order energy-stable sceme to approximate te two convex functions F + c u and F c u. f + u m+, u m = F + c u m+ F + c u m u m+ u m, f u m+, u m = F c u m+ F c u m u m+ u m. Teorem Under te constraint < ɛ, tere exists a unique solution of te sceme Proof: Define te following functional: Ju m+ = 4 a u m+, u m+ + ɛ T F + u m+, u m ɛ um+ L T + a u m, u m+ + T ɛ um u m+. Tae te derivative of te functional Ju m+, and will get: δju m+ δu m+, v = T a u m+, v + ɛ f + u m+, u m 3.6 T + 4ɛ um+, v T + a u m, v + ɛ um, v T. Rearrange it, and we will get: δju m+ δu m+, v = d t u m+, v + a T T u m+, v + f m+ ɛ, v = T Also we can see from 3.0 te first two terms of Ju m+ are convex, also since te last two terms are linear wit respect to u m+, so tey are also convex, so if we restrict te coefficient of tird term to be positive, tat is, if we restrict < ɛ, ten te Ju m+ will be a convex functional, and te uniqueness of te solution to tis sceme is approved.

15 0 3.4 Error estimates analysis Te main result of tis subsection is te following error estimate teorem. Teorem 3 suppose σ e > max{σ 0, σ 0 }. Let u and {um }M denote respectively te solutions of problems..4 and Assume u H 0, T ; L Ω L 0, T ; W s, Ω and suppose GA and.7 old. Ten, under te following mes and initial value constraints: d C0 C C ɛ max{σ +3,σ +}, min{r+,s} ln r C 0 C C ɛ γ+, < Aɛ, u 0 S suc tat u 0 u 0 L T C min{r+,s}, tere old max ut m u m L 0 m M T C + min{r+,s} ɛ σ M ut m u m H T C + min{r+,s} ɛ σ+3, 3.9 max 0 m M ut m u m L T C min{r+,s} ln r ɛ γ C d + min{r+,s} ɛ σ +. Step : Proof: Since te proof is long, we split te proof into four steps: We write: ut m u m = η m + ξ m, η m := ut m P r ut m, ξ m := P r ut m u m. Multiply v on bot sides of te Allen-Can equation in. at te point ut m+ ut t m+, v T + a ut m+, v + futm+ ɛ, v = 0, 3. T

16 for all v V, were t m+ = t m++t m. Subtract 3. from 3., we get te following equation: ut t m+ um+ u m, v T + a utm+ + futm+ ɛ f m+, v = 0. T um+ + u m, v 3. From Taylor expansion: ut m+ = u t m+ + t m + u t t m+ + t m t m+ t m + R m, were R m = u ttξ t m+ t m. ut m = u t m+ + t m u t t m+ + t m t m+ t m + R m, were R m = u ttξ t m+ t m. And we will get: ut m+ = ut m+ + ut m u t t m+ = ut m+ ut m Use 3.3 and 3.4 into 3., we will get: ξ m+ ξ m + a ξ m+ + ξ m + ηm+ η m + ηm+ + η m + futm+ ɛ f m+, v = 0. T R m + R m, 3.3 R m R m. 3.4 R m R m, v 3.5 T R m + R m, v dt ξ m+, v + a T ξm+ + ξ m, v futm+ ɛ f m+, v T = R m Rm, v d T t η m+, v T a ηm+ + η m R m, v + a + R m, v = R m Rm, v d T t η m+, v T + ηm+ + η m R m, v T + a + R m, v.

17 Let v = ξm+ +ξ m, for te first term on te left and side: dt ξ m+, ξm+ + ξ m T = d t ξ m+ L T. 3.7 We split te tird term on te left and side in 3.6 into two parts and deal wit tem separately: futm+ ɛ f m+, ξm+ + ξ m 3.8 T = futm+ ɛ f ut m+ + ut m, ξm+ + ξ m T + ut m+ + ut m f ɛ f m+, ξm+ + ξ m. T Let ût m+ = ut m++ut m, te we ave te following: fut m+ fût m+ 3.9 = f ût m+ 8 u ξ + u ξ fût m+ = f ξ 8 u ξ + u ξ C. Since f and u bot are bounded, we will get te following inequality by Caucy-Scwarz inequality: futm+ ɛ fût m+ ɛ C, ξm+ + ξ m T ɛ 4 C4 ξ m+ L T., ξm+ + ξ m 3.30 T For te last term of te rigt and side in 3.6: R m a + Rm R m a + Rm ɛ, ξm+ + ξ m, Rm + Rm ɛ C 4 ɛ + ɛ a ξ m+, ξ m+. R = m + Rm a, ɛξm+ + ξ m ɛ ɛξ m+ + ξ m + a, ɛξm+ + ξ m 3.3

18 3 Substitute 3.7,3.30 and 3.3 into 3.6, and we will get: d t ξ m+ L T + a ξm+ + ξ m, ξm+ + ξ m fûtm+ ɛ f m+, ξm+ + ξ m T = R m Rm, v d T t η m+, v T + ηm+ + η m, v T + a Rm + Rm, v. R m Rm L T + d tη m+ L T + ηm+ + η m ξ m+ + ξ m L T L T + C 4 [ɛ 4 + ɛ ] + ɛ a ξ m+, ξ m+ + ξ m+ L T. Using te integral form of Taylor formula, we can get: Rm Rm = u ttξ u tt ξ 4 = u tttξ ξ ξ C. 4 Hence Rm Rm L T C Summing in m from to l, using 3.3,3.3 and 3.33, and we will get te following inequality: ξ l L T + + a ξm + ξ m fûtm ɛ f m, ξm + ξ m T, ξm + ξ m 3.34 ξ 0 L T + C min{r+,s} u H 0,T ;H s Ω + C 4 [ɛ 4 + ɛ + ] + ɛ a ξ m, ξ m + 4 ξ m L T. Step : We want to bound te term fût m+ f m+, ξm+ +ξ m T 3.34: on te left and side of fût m+ f m+ = [fût m+ fp r ût m+ ] + [fpr ût m+ f m+ ]. 3.35

19 4 For te first part on te rigt and side of 3.35,we get: fu m+ fp r ût m+ = f ξ ûtm+ P r ût m+ For te second part on te rigt and side of 3.35,we get: η m+ + η m C fpr ût m+ f m+ = Pr ut m+ + Pr ut m 3 P r ut m+ + Pr ut m 3.37 [ 4 [um+ 3 + u m+ u m + u m+ u m + u m 3 ] um+ + u m ] P = r ut m+ + Pr ut m [um+ 3 + u m+ u m + u m+ u m + u m 3 ] [ Pr ut m+ + Pr ut m u m+ + u m ] P = r ut m+ + Pr ut m [P r ut m+ ξ m+ 3 + Pr ut m+ ξ m+ Pr ut m ξ m + Pr ut m+ ξ m+ Pr ut m ξ m + Pr ut m ξ m 3 ] ξm+ + ξ m. We split te above into four terms: constant term wit resect to ξ m+ and ξ m, linear, quadratic and cubic in terms of ξ m+ and ξ m. For constant term, we ave 8 P r ut m+ Pr ut m Pr ut m+ + Pr ut m, ξm+ + ξ m 3.38 T C 4 +, ξm+ + ξ m T C C ξ m+ L T. By te boundness of P r u m and P r ut m+ P r ut m +. For te linear term, we ave te following: l = 4{ ξ m+ [3P r ut m+ + P r ut m+ P r ut m + P r ut m ] ξ m [3P r ut m + P r ut m+ P r ut m + P r ut m+ ] } ξm+ + ξ m = 4 ξm+ + ξ m P r ut m+ + P r ut m + [ξm+ Pr ut m+ + ξ m Pr ut m ] ξm+ + ξ m.

20 5 And we ave: 4 ξm+ + ξ m Pr ut m+ + Pr ut m [ξm+ Pr ut m+ + ξ m Pr ut m ], ξm+ + ξ m T = P r ut m+ + Pr ut m, ξm+ + ξ m T + [ξm+ Pr ut m+ + ξ m Pr ut m ], ξm+ + ξ m. T By using te Scwarz Inequality and P r ut m+ P r ut m C +, we get te following inequalities for te first and second terms of te rigt and side of 3.40: P r ut m+ + Pr ut m, ξm+ + ξ m 3.4 T Pr ut m, ξm+ + ξ m C + ξ m+ T L T [ξm+ Pr ut m+ + ξ m Pr ut m ], ξm+ + ξ m 3.4 T Pr ut m, ξm+ + ξ m C + ξ m+ T L T. ξ m+ + ξ m l, T Pr ut m, ξm+ + ξ m C + ξ m+ T L T = f Pr ut m, ξm+ + ξ m C + ξ m+ T L T. For te quadratic term, we get te inequality below; q = 3ξ m+ P r ut m+ + ξ m+ P r ut m + ξ m+ ξ m P r ut m ξ m P r ut m+ + ξ m+ ξ m P r ut m + 3ξ m P r ut m C [ξ m+ + ξ m ].

21 6 So we get: ξ m+ + ξ m q, 3.45 T C ξ m+ + ξ m, ξm+ + ξ m T C ξ m+ 3 L 3 T. For cubic term, we ave: c = [ ξ m+ 3 + ξ m+ ξ m + ξ m+ ξ m + ξ m 3] 4 = 4[ ξ m+ + ξ m ] ξ m+ + ξ m, 3.46 te we ave ξ m+ + ξ m c, Combine all above togeter, we will get: T = ξ m+ + ξ m, ξ m+ + ξ m T fp r ût m+ f m+, ξm+ + ξ m 3.48 T C η m+, ξ m+ T C C ξ m+ L T f Pr ut m, ξm+ + ξ m C + ξ m+ T L T C ξm+ 3 L 3 T + 4 ξ m+ ɛ + ξ m, ξ m+ + ξ m. T

22 7 Summing in m a from to l and we will get te following: ɛ fp r ût m+ f m+, ξm+ + ξ m 3.49 T C ɛ + ɛ C ɛ η m+ T ξ m+ T C ɛ C ɛ f Pr ut m, ξm+ + ξ m C T ɛ + ξ m+ 3 L 3 T + ξ m L T ξ m L T ξ m + ξ m+, ξ m + ξ m+, T C min{r+,s} ɛ 4 u L 0,T ;H s Ω C ɛ ɛ + ɛ f Pr ut m, ξm+ + ξ m T ξ m + ξ m+, ξ m + ξ m+ C T ɛ ξ m+ 3 L 3 T C ɛ + + ξ m L T ξ m L T. Substitute te inequality above into 3.34, and we get: ξ l L T + ɛ + ɛ + ξ m + ξ m, ξ m + ξ m 3.50 a ξ m, ξ m + ɛ f P r ut m, ξ m T f P r ut m, ξ m T ξ 0 L T + C min{r+,s} u H 0,T ;H s Ω + ɛ 4 u C L 0,T ;H s Ω + ɛ C 4 [ɛ 4 + ɛ + ] + C + ɛ ɛ + + ɛ ξ m L T + C ɛ ξ m+ 3 L 3 T. Step 3: In order to control te last two terms on te rigt-and side of 3.49 we use te following Gagliardo-Nirenberg inequality [3]: v 3 L 3 K v C d 6 d L K v L K + v 3 L K K T,

23 8 to get C ɛ ξ m 3 L 3 T ɛ α ɛ α ξ m L T + ɛ + Cɛ 4+d 4 d K T ξ m L T ξ m L T 3.5 ξ m 6 d 4 d + Cɛ 4+d 4 d ξ m 6 d 4 d L T. L K Finally, for te tird term on te left-and side of te above inequality, we utilize te discrete spectrum estimate.8 to bound it from below as follows: ɛ a ξ m, ξ m + ɛ f P r ut m, ξ m T + 4 f Pr ut m, ξ m T = ɛ a ξ m, ξ m + ɛ f Pr ut m, ξ m T + ɛ a ξ m, ξ m + 4 ɛ c 0 ξ m L T + 4ɛ α f P r ut m, ξ m T ξ m,dg C ξ m L T. 3.5 Step 4: Substitute 3.5 and 3.5 into 3.50, and we get te following: ξ l L T + 3ɛ α C + ɛ ɛ + + ɛ ξ m,dg + ɛ ξ m + ξ m, ξ m + ξ m 3.53 ξ m 4+d L T + Cɛ 4 d ξ m 6 d 4 d L T + ξ 0 L T + C min{r+,s} u H 0,T ;H s Ω + ɛ 4 u L 0,T ;H s Ω + C ɛ C 4 [ɛ 4 + ɛ + ]. Notice tat on te rigt and side, we need to coose te appropriate initial value u 0, so tat ξ 0 L T = O min{r+,s} to maintain te optimal rate of convergence in. Clearly, bot te

24 9 L and te elliptic projection of u 0 wor. and in te latter case, we get ξ 0 = 0. It ten follows from.7,.9,. and 3.53 tat ξ l L T + 3ɛ α C + ɛ ɛ + + ɛ ξ m,dg + ɛ ξ m + ξ m, ξ m + ξ m 3.54 ξ m 4+d L T + Cɛ 4 d ξ m 6 d 4 d + C min{r+,s} ɛ σ + + C ɛ C 4 [ɛ 4 + ɛ + ]. L T since u l can be written as ten by.3 and 3., we get u l = d t u m + u 0, 3.55 u l L T d t u m L T + u 0 L T Cɛ σ By te boundedness of te projection, we ave ξ l L T Cɛ σ were Ten te above inequality is equivalent to te form below: ξ l L T + 3ɛ α ξ m,dg H + H, 3.58 H : = C + ɛ ɛ + + l ɛ ξ m L T 4+d + Cɛ 4 d ξ m 6 d 4 d 3.59 L T l + C min{r+,s} ɛ σ + + C ɛ C 4 [ɛ 4 + ɛ + ], H : = C + ɛ ɛ + + ɛ ξ l L T + 4+d Cɛ 4 d ξ l 6 d 4 d L T It is easy to cec tat H < ξl L T provided tat < Aɛ. 3.6

25 0 By 3.58 we ave ξ l L T + 3ɛ α ξ m,dg H 3.6 C + ɛ ɛ + + l ɛ ξ m L T 4+d + Cɛ 4 d ξ m 6 d 4 d l + C min{r+,s} ɛ σ + + C ɛ C 4 [ɛ 4 + ɛ + ] L T C + ɛ ɛ + + l ɛ ξ m L T 4+d + Cɛ 4 d ξ m 6 d 4 d l L T + C min{r+,s} ɛ σ + + C ɛ C 4 [ɛ 4 + ɛ + ]. Let d l 0 be te slac variable suc tat ξ l L T + 3ɛ α ξ m,dg + d l = C + ɛ ɛ + + l ɛ ξ m L T 4+d + Cɛ 4 d ξ m 6 d 4 d 3.63 L T l + C min{r+,s} ɛ σ + + C ɛ C 4 [ɛ 4 + ɛ + ]. and define for l S l+ : = ξ l L T + 3ɛ α ξ m,dg + d l, 3.64 S : = C min{r+,s} ɛ σ + + C ɛ C 4 [ɛ 4 + ɛ + ] ten we ave S l+ S l C + ɛ ɛ + + ɛ S l + Cɛ 4+d 6 d 4 d S 4 d l for l Applying Lemma to {S l } l defined above, we obtain for l S l a l { S 4 d C 4 d l ɛ 4+d 4 d a 4 d s+ s= } 4 d 3.67 provided tat C l 4 d S 4 d s= ɛ 4+d 4 d a 4 d s+ >

26 We note tat a s s l are all bounded as 0, terefore, 3.68 olds under te mes constraint stated in te teorem. It follows from 3.66 and 3.67 tat S l a l S C 4 ɛ σ + + C min{r+,s} ɛ σ Finally, using te above estimate and te properties of te operator P r we obtain 3.8 and 3.9. Te estimate 3.0 follows from 3.9 and te inverse inequality bounding te L -norm by te L -norm and.. Te proof is complete.

27 CHAPTER 4. CONVERGENCE OF THE NUMERICAL INTERFACE TO THE MEAN CURVATURE FLOW In tis section, we prove te rate of convergence of te numerical interface to its limit geometric interface of te Allen-Can equation. Tis convergence teory is based on te maximum norm error estimates, wic is proven above. Te rate of convergence can be proven by te sarper error estimates, wic is te negative polynomial function of te interaction lengt ɛ. It can t be proven if te coarse error estimate, wic is te exponential function of ɛ, is used. For all te DG problem, te te zero-level set of u n may not be well defined since te zero-level set may not be continuous. Terefore, we introduce te finite element approximation û m of te DG solution um It is defined by using te averaged degrees of freedom of un as te degrees of freedom for determining û m cf. [4]. We get te following results [4]. Teorem 4 Let T be a conforming mes consisting of triangles wen d =, and tetraedra wen d = 3. For v V, let v be te finite element approximation of v as defined above. Ten for any v V and i = 0, tere olds v v H i K C K T e E I i e [v ] L e, 4. were C > 0 is a constant independent of and v but may depend on r and te minimal angle θ 0 of te triangles in T. Using te above approximation result we can sow tat te error estimates of Teorem 3 also old for û n. Teorem 5 Let u m denote te solution of te DG sceme and ûm denote its finite element approximation as defined above. Ten under te assumptions of Teorem 3 te error

28 3 estimates for u m given in Teorem 3 are still valid for ûm, in particular, tere olds max ut m û m L 0 m M T C min{r+,s} ln r ɛ γ 4. + C d + min{r+,s} ɛ σ +. Proof: We only give a proof for 4. because oter estimates can be proved liewise. By te triangle inequality we ave ut m û m L T ut m u m L T + u m ûm L T. 4.3 Hence, it suffices to sow tat te second term on te rigt-and side is an equal or iger order term compared to te first one. Let u I t denote te finite element interpolation of ut into S. It follows from 4. and te trace inequality tat u m ûm L T C e E I = C e E I e [u m ] L e 4.4 e [u m ui t m ] L e C K T e K um ui t m L K C u m ut m L T + ut m u I t m L T. Substituting 4.4 into 4.3 after using te inverse inequality yields ut m û m L T ut m u m L T + C d u m û m L T ut m u m L T + C d u m ut m L T + ut m u I t m L T, wic togeter wit 3.8 implies te desired estimate 4.. Te proof is complete. We are now ready to state te main teorem of tis section.

29 4 Teorem 6 Let {Γ t } denote te generalized mean curvature flow defined in [5], tat is, Γ t is te zero-level set of te solution w of te following initial value problem: w t = w D wdw Dw Dw in R d 0,, 4.5 w, 0 = w 0 in R d. 4.6 Let u ɛ,, denote te piecewise linear interpolation in time of te numerical solution {û m } defined by u ɛ,, x, t := t t m û m+ x + t m+ t û m x, t m t t m+ 4.7 for 0 m M. Let {Γ ɛ,, t } denote te zero-level set of u ɛ,,, namely, Γ ɛ,, t = {x Ω; u ɛ,, x, t = 0}. 4.8 Suppose Γ 0 = {x Ω; u 0 x = 0} is a smoot ypersurface compactly contained in Ω, and = O. Let t be te first time at wic te mean curvature flow develops a singularity, ten tere exists a constant ɛ > 0 suc tat for all ɛ 0, ɛ and 0 < t < t tere olds sup {distx, Γ t } Cɛ ln ɛ. x Γ ɛ,, t Proof: We note tat since u ɛ,, x, t is continuous in bot t and x, ten Γ ɛ,, t is well defined. Let I t and O t denote te inside and te outside of Γ t defined by I t := {x R d ; wx, t > 0}, O t := {x R d ; wx, t < 0}. 4.9 Let dx, t denote te signed distance function to Γ t wic is positive in I t and negative in O t. By Teorem 6. of [6], tere exist ɛ > 0 and Ĉ > 0 suc tat for all t 0 and ɛ 0, ɛ tere old u ɛ x, t ɛ x {x Ω; dx, t Ĉɛ ln ɛ }, 4.0 u ɛ x, t + ɛ x {x Ω; dx, t Ĉɛ ln ɛ }. 4. Since for any fixed x Γ ɛ,, t, u ɛ,, x, t = 0, by 4. wit = O, we ave u ɛ x, t = u ɛ x, t u ɛ,, x, t C min{r+,s} ln r ɛ γ + d + min{r+,s} ɛ σ +.

30 5 Ten tere exists ɛ > 0 suc tat for ɛ 0, ɛ tere olds u ɛ x, t < ɛ. 4. Terefore, te assertion follows from setting ɛ = min{ ɛ, ɛ }. Te proof is complete.

31 6 CHAPTER 5. NUMERICAL EXPERIMENTS In tis section, we provide a two-dimensional numerical experiment to gauge te accuracy and reliability of te fully discrete IPDGFE metod developed in te previous sections. We use a square domain Ω = [, ] [, ] R, and u 0 x = tan d 0x ɛ, were d 0 x stands for te signed distance from x to te initial curve Γ 0. Te test uses te smoot initial curves Γ 0, ence te requirements for u 0 are satisfied. Consequently, te results establised in tis paper apply to te test example. In te test we first verify te spatial rate of convergence given in 3.8 and 3.0. We ten compute te evolution of te zero-level set of te solution of te Allen-Can problem wit ɛ = 0.05 and at various time instances. Test. Consider te Allen-Can problem wit te following initial condition: tan dx u 0 x = ɛ, if x + x 0.5, tan dx ɛ, if x + x < 0.5, ere dx stands for te distance function to te circle x + x = 0.5. Table 5.. Spatial errors and convergence rates L L error L L order L H error L H order / / / / Table 5. sows te spatial L and H -norm errors and convergence rates, wic are consistent wit wat are proved for te linear element in te convergence teorem.

32 7 Figure 5. Test : Snapsots of te zero-level set of u ɛ,, at time t = 0, 0, 3 0, 4 0 and ɛ = Figure 5. displays four snapsots at four fixed time points of te zero-level set of te numerical solution u ɛ,, wit four different ɛ. Once again, we observe tat as ɛ is small enoug te zero-level set converges to te mean curvature flow Γ t as time goes on.

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