A Continuous Space-Time Finite Element Method for an Integro-Differential Equation Modeling Dynamic Fractional Order Viscoelasticity

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1 PREPRINT 29:17 A Coiuous Space-Time Fiie Eleme Mehod for a Iegro-Differeial Equaio Modelig Dyamic Fracioal Order Viscoelasiciy FARDIN SAEDPANAH Deparme of Mahemaical Scieces Divisio of Mahemaics CHALMERS UNIVERSITY OF TECHNOLOGY UNIVERSITY OF GOTHENBURG Göeborg Swede 29

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3 Prepri 29:17 A Coiuous Space-Time Fiie Eleme Mehod for a Iegro-Differeial Equaio Modelig Dyamic Fracioal Order Viscoelasiciy Fardi Saedpaah Deparme of Mahemaical Scieces Divisio of Mahemaics Chalmers Uiversiy of Techology ad Uiversiy of Goheburg SE Göeborg, Swede Göeborg, April 29

4 Prepri 29:17 ISSN Maemaiska veeskaper Göeborg 29

5 A CONTINUOUS SPACE-TIME FINITE ELEMENT METHOD FOR AN INTEGRO-DIFFERENTIAL EQUATION MODELING DYNAMIC FRACTIONAL ORDER VISCOELASTICITY FARDIN SAEDPANAH Absrac. A fracioal order iegro-differeial equaio wih a weakly sigular kerel is cosidered. A weak form is formulaed, ad sabiliy of he primal ad he dual problem is sudied. The coiuous Galerki mehod of degree oe is formulaed, ad opimal a priori error esimae is obaied by dualiy argume. A poseriori error represeaio based o space-ime cells is preseed such ha i ca be used for adapive sraegies based o dual weighed residual mehods. Some global a poseriori error esimaes are also proved. 1. Iroducio We sudy a iiial-boudary value problem, modelig dyamic fracioal order viscoelasiciy of he form we use o deoe he ime derivaive, 1.1 ρüx, σ u; x, + β s σ u; x, s ds = fx, i Ω, T, ux, = o Γ D, T, σu; x, = gx, o Γ N, T, ux, = u x, ux, = v x i Ω. Here u is he displaceme vecor, ρ is he cosa mass desiy, f ad g represe, respecively, he volume ad surface loads. The sress σ = σu; x, is deermied by wih σ = σ β sσ s ds, σ = 2µ ɛ + λ TrɛI, where I is he ideiy operaor, ɛ is he srai which is defied by ɛ = 1 2 u + u T, ad µ, λ > are elasic cosas of Lamé ype. The kerel is defied Dae: April 12, Mahemaics Subjec Classificaio. 65M6, 45K5. Key words ad phrases. fracioal order viscoelasiciy, Galerki approximaio, weakly sigular kerel, sabiliy, a priori esimae, a poseriori esimae. 1

6 2 F. SAEDPANAH by: 1.2 β = γ d d E α C 1+α,, α = γ α 1+αE α τ τ τ α τ which meas ha i is weakly sigular wih he properies 1.3 β, β L1R + = β d = γ E α E α = γ < 1, where γ, 1 is a cosa ad τ > is he relaxaio ime. Here E α is he Miag-Leffler fucio of order α, 1 ad defied by, E α z = = z Γ1 + α. There is a exesive lieraure o heoreical ad umerical aalysis of iegrodiffereial equaios modelig liear ad fracioal order viscoelasiciy, see, e.g., [1], [7], [8], [9], [1], [12], ad [13]. Exisece, uiqueess ad regulariy of soluio of a problem i he form of 1.1 has bee sudied i [7], [11]. A poseriori aalysis of emporal fiie eleme approximaio of a parabolic ype problem ad discoiuous Galerki fiie eleme approximaio of a quasi-saic ρü liear viscoelasiciy problem has bee sudied, respecively, i [1] ad [13]. The prese work exeds previous works, e.g., [1], [2], ad [13]. The oulie of his paper is as follows. I 2 we defie a weak form of 1.1 ad he correspodig dual adjoi problem, ad we sudy he sabiliy. I 3 we formulae a coiuous Galerki mehod of degree oe ad we obai sabiliy esimaes. The i 4 we obai a opimal a priori error esimae. We prese a a poseriori error represeaio based o he dual weighed residual mehod i 5, ad we prove some global a poseriori error esimaes. 2. Weak formulaio ad sabiliy We le Ω R d, d = 2, 3, be a bouded polygoal domai wih boudary Γ = Γ D Γ N where Γ D ad Γ N are disjoi ad measγ D. We iroduce he fucio spaces H = L 2 Ω d, H ΓN = L 2 Γ N d, ad V = v H 1 Ω d : v ΓD =. We deoe he orms i H ad H ΓN by ad ΓN, respecively. We also defie a biliear form wih he usual summaio coveio 2.1 av, w = 2µ ɛ ij vɛ ij w + λ ɛ ii vɛ jj w dx, v, w V, Ω which is coercive o V, ad we equip V wih he ier produc a, ad orm v 2 V = av, v. We defie Au = σ u, which is a selfadjoi, posiive defiie, ubouded liear operaor, wih DA = H 2 Ω d V, ad we use he orms v s = A s/2 v. We use a velociy-displaceme formulaio of 1.1 which is obaied by iroducig a ew velociy variable. Heceforh we use he ew variables u 1 = u, u 2 = u ad u = u 1, u 2 he pair of vecor valued fucios. Now we defie he biliear

7 CG METHOD FOR A FRACTIONAL ORDER VISCOELASTICITY MODEL 3 ad liear forms A : V W R, A τ : W V R, F : W R, J τ : W R, for τ R, by Au, w = T u 1, w 1 u 2, w 1 + ρ u 2, w 2 + au 1, w 2 β sa u 1 s, w 2 ds d + u 1, w 1 + ρ u 2, w 2, T T A τ w, z = w 1, ż 1 + aw 1, z 2 βs a w 1, z 2 s ds τ ρw 2, ż 2 w 2, z 1 d + w 1 T, z 1 T + ρ w 2 T, z 2 T, F w = J τ w = T T τ f, w 2 + g, w 2 ΓN d + u, w 1 + ρ v, w 2, w 1, j 1 + w 2, j 2 d + w 1 T, z1 T + ρw2 T, z2 T, where j 1, j 2 ad z1 T, z2 T represe, respecively, he load erms ad he iiial daa of he dual adjoi problem. I case of τ =, we use he oaio A, J for shor. Here 2.2 V = H 1, T ; V H 1, T ; H, V = H 1, T ; H H 1, T ; V, W = w = w 1, w 2 : w L 2, T ; H L2, T ; V, w i are righ coiuous i ime, W = w = w 1, w 2 : w L 2, T ; V L2, T ; H, w i are lef coiuous i ime, ad we oe ha V W, V W. The variaioal formulaio aalogue o 1.1 is he o fid u V such ha, 2.3 Au, w = F w, w W. Here he defiiio of he velociy u 2 = u 1 is eforced i he L 2 sese, ad he iiial daa are placed i he biliear form i a weak sese. A varia is used i [7] where he velociy has bee eforced i he H 1 sese, wihou placig he iiial daa i he biliear form. We also oe ha he iiial daa are reaied by he choice of he fucio space W, ha cosiss of righ coiuous fucios wih respec o ime. To obai he dual adjoi problem we oe ha A is he adjoi form of A. Ideed, iegraig by pars wih respec o ime i A, he chagig he order of iegrals i he covoluio erm as well as chagig he role of he variables s,, we have, 2.4 Av, w = A v, w, v V, w V. The he variaioal formulaio of he dual problem is o fid z V such ha, 2.5 A w, z = Jw, w W,

8 4 F. SAEDPANAH ha is a weak formulaio of ρ z 2 + Az 2 T βs Az 2 s ds = j 1 j 2, wih iiial daa z1 T, z2 T, ad fucio j = j 1, j 2 ha is defied by Jw = T w, j d. I he aalysis below we use a posiive ype kerel ξ. Ideed, we recall 1.2, 1.3 ad we defie he fucio 2.6 ξ = γ ad i is easy o see ha 2.7 βs ds = βs ds = γe α, D ξ = β <, ξ = γ, lim ξ =, < ξ γ. Besides, ξ is a compleely moooe fucio, ha is, 1 j D j ξ,,, j N, sice he Miag-Leffler fucio E α, α [, 1] is compleely moooe, see, e.g., [5]. Cosequely, a impora propery of ξ is ha, i is a posiive ype kerel, ha is, i is coiuous ad, for ay T, saisfies 2.8 T ξ sφφs ds d, φ C[, T ]. Theorem 1. Le u be he soluio of 2.3 wih sufficiely smooh daa u, v, f, g. The for l R, T >, we have he ideiy, 2.9 ρ u 2 T 2 l + γ u 1 T 2 l T = ρ v 2 l γ u 2 l T T g, Al u 1 Γ N d ξ sa us, A l u ds d T f, Al u 1 d βa u, A l u 1 d 2ξT a u, A l u 1 T, where γ = 1 γ. Moreover, wih Γ N = or Γ N, g =, we have, 2.1 u 2 T l + u 1 T l+1 C v l + u l+1 + T f l d, for some C = Cρ, γ, T. Ad wih Γ N, g, l =, we have he esimae, 2.11 u 2 T + u 1 T 1 C v + u 1 + f L1,;H + g W 1 1,;H Γ N, for some C = CΩ, ρ, γ, T.

9 CG METHOD FOR A FRACTIONAL ORDER VISCOELASTICITY MODEL 5 Proof. Sice u is a soluio of 2.3, we obviously have u 2 = u 1. We recall γ = 1 γ, ad from 2.7 we have ha β s = D s ξ s, ξ = γ. These ad parial iegraio i he covoluio erm i A, yield 2.12 Au, w = T ρ u 2, w 2 + γau 1, w 2 + ξ sa u 1 s, w 2 ds + ξau, w 2 d + u 1, w 1 + ρ u 2, w 2. Puig his wih w = w 1, w 2 = A l+1 u 1, A l u 2 = A l+1 u 1, A l u 1 i 2.3 we obai, ρ u 2 T 2 l + γ u 1 T 2 l T = ρ v 2 l + γ u 2 l T T ξa u, A l u 1 d. = ρ v 2 l + γ u 2 l T T ξ sa u 1 s, A l u 1 ds d f, A l u 1 d + 2 f, A l u 1 d + 2 T T g, A l u 1 ΓN d g, A l u 1 ΓN d βa u, A l u 1 d 2ξT a u, A l u 1 T + 2γ u 2 l+1, where for he las equaliy we used parial iegraio ad D ξ = β. This, havig γ = 1 γ, implies he ideiy 2.9. Now we prove he simaes 2.1 ad Firs, i 2.9, we use 2.8, iegraio by pars i he erm wih he surface load g, ad he Cauchy-Schwarz iequaliy ad coclude, ρ u 2 T 2 l + γ u 1 T 2 l+1 ρ v 2 l γ u 2 l+1 T 2 + 2/C 1 max u 1 2 l + 2C 1 f l d T 2 T ġ, A l u 1 Γ N d + 2 gt, A l u 1 T Γ N 2 g, A l u Γ N + 2 u l+1 max T u 1 l+1 T β d + 2γ/C 2 u 2 l+1 + 2γC 2 u 1 T 2 l+1. This wih Γ N = or Γ N, g =, cosiderig β L1R + = γ, implies 2.1. Bu for he case ha he surface load g Γ N, we eed o resric o l =, due o he race heorem. Tha is, i his case wih l = ad usig u 1 ΓN CΩ u 1 1 by he race heorem, i a sadard way, we have he esimae u 2 T + u 1 T 1 C v + u 1 + f L1,;H + ġ L1,T ;H Γ N + g L,T ;H Γ N.

10 6 F. SAEDPANAH This ad he fac ha g L,T ;H Γ N C g W 1 1,T ;H Γ N, by Sobolev iequaliy, imply Remark 1. A ideiy, slighly differe from 2.9 has bee preseed i [2], by usig a fucio w, s = u u s ha mus belog o a weighed L 2 - space, iroduced i [6], see also [7] where he absrac framework by [6] has bee, specifically, applied o his problem. The proof preseed here avoids usig fucio w ad seems o be simple ad sraighforward. Remark 2. A impora propery, used here is ha he kerel is of posiive ype. This meas ha he echique preseed here ca be applied o he problems wih posiive ype kerels. For example, wih g =, we could cosider problems wih posiive ype kerels ad of he form ρü + Au β sbus ds = f, where B is a selfadjoi, posiive defiie liear operaor such ha, for some suiable cosa C, Bv, w CAv, w, v, w DA. For example, wih kerel β defied i 1.2, we should have 1 γc >. The a similar argume ca be applied wih l =, ad also wih l provided B ad A be comuaive. Theorem 2. Le z be he soluio of he dual problem 2.5 wih sufficiely smooh daa z1 T, z2 T, j 1, j 2. The for l R, < < T, we have he ideiy, 2.13 z 1 2 l + ρ γ z 2 2 l+1 + 2ρ T T = z T 1 2 l + ρ1 + γ z T 2 2 l ρ T r ξs raa l ż 2 r, ż 2 s ds dr T βt raa l z 2 r, z T 2 dr 2ρξT aa l z 2, z T 2. A l z 1, j 1 + A l+1 z 2, j 2 dr where γ = 1 γ. Moreover, for some cosa C = Cρ, γ, T, we have sabiliy esimaes T 2.14 z 1 l + z 2 l+1 C z1 T l + z2 T l+1 + j 1 l + j 2 l+1 dr. Proof. Sice z is a soluio of 2.5, we obviously have z 1 = ρż 2 ad, for [, T, z saisfies 2.15 A w, z = J w, v W. We recall γ = 1 γ, ad from 2.7 we have βs = D s ξs, ξ = γ. The, by parial iegraio wih respec o ime i he covoluio erm i A, we obai T T A w, z = w 1, ż 1 + γaw 1, z 2 ξs ra w 1, ż 2 s ds ξt ra w 1, z 2 T dr + w 1 T, z 1 T + ρ w 2 T, z 2 T. r

11 CG METHOD FOR A FRACTIONAL ORDER VISCOELASTICITY MODEL 7 Puig his wih w = w 1, w 2 = A l z 1, A l+1 z 2 = ρa l ż 2, A l+1 z 2 i 2.15 we have T ha implies 1 2 D z 1 r 2 l 1 2 ρ γd z 2 r 2 l+1 + ρ T r ξs ra A l ż 2 r, ż 2 s ds ρξt ra A l ż 2 r, z2 T T dr = A l z 1, j 1 + A l+1 z 2, j 2 dr, z 1 2 l + ρ γ z 2 2 l+1 + 2ρ T T = z T 1 2 l + ρ γ z T 2 2 l ρ T r T ξt ra A l ż 2 r, z2 T dr. ξs ra A l ż 2 r, ż 2 s ds dr A l z 1, j 1 + A l+1 z 2, j 2 dr Now iegraio by pars i he las erm, havig D r ξt r = βt r by 2.7, gives he ideiy The, usig 2.8 i 2.13, similar o he proof of 2.1, i a sadard way, we coclude he iequaliy 2.14, ad his complees he proof. 3. The coiuous Galerki mehod Le = < 1 < < 1 < < < N = T be a pariio of he ime ierval [, T ]. To each discree ime level we associae a riagulaio Th of he polygoal domai Ω wih he mesh fucio, 3.1 h x = h K = diamk, x K, K T h, ad a fiie eleme space Vh cosisig of coiuous piecewise liear polyomials. For each ime subierval = 1, of legh k = 1, we defie iermediae riagulaio T h which is composed of muually fies meshes of he eighborig meshes Th, T 1 h defied a discree ime levels, 1, respecively. The mesh fucio h is he defied by 3.2 h x = h K = diamk, x K, K T h. Correspodigly, we defie he fiie eleme spaces V h cosisig of coiuous piecewise liear polyomials. This cosrucio is used i order o allow coiuiy i ime of he rial fucios whe he meshes chage wih ime. Hece we obai a decomposiio of each ime slab Ω = Ω io space-ime cells K = K, K T h prisms, for example, i case of Ω R2. We oe he differece bewee he mesh fucios h ad h, ad his is impora i our a poseriori error aalysis.

12 8 F. SAEDPANAH The rial ad es fucio spaces for he discree form are, respecively: V hk = U = U 1, U 2 : U coiuous i Ω [, T ], Ux, I liear i, 3.3 U, Vh 2, U, I V h 2, W hk = V = V 1, V 2 : V, coiuous i Ω, V, I Vh 2, V x, I piecewise cosa i. We oe ha global coiuiy of he rail fucios i V hk requires he use of hagig odes if he spaial mesh chages across a ime level. We allow oe hagig ode per edge or face. Remark 3. If we do o chage he spaial mesh or jus refie he spaial mesh from oe ime level o he ex oe, i.e., 3.4 V 1 h V h, = 1,..., N, he we have V h = V h. I he cosrucio of V hk we have associaed he riagulaio Th wih discree ime levels isead of he ime slabs Ω, ad i he ierior of ime slabs we le U be from he uio of he fiie eleme spaces defied o he riagulaios a he wo adjace ime levels. This cosrucio is ecessary o allow for rial fucios ha are coiuous also a he discree ime leveles eve if grids chage bewee ime seps. Associaig riagulaio wih ime slabs isead of ime levels would yield a varia scheme which icludes jump erms due o discoiuiy a discree ime leveles, whe coarseig happes. This meas ha here are exra degrees of freedom ha oe migh use suiable projecios for rasferig soluio a he ime levels, see [7]. The coiuous Galerki mehod, based o he variaioal fomulaio 2.3, is o fid U V hk such ha, 3.5 AU, W = F W, W W hk. The Galerki orhogoaliy, wih u = u 1, u 2 beig he exac soluio of 2.3, is he, 3.6 AU u, W =, W W hk. Similarly he coiuous Galerki mehod, based o he dual variaioal formulaio 2.5, is o fid Z V hk such ha, 3.7 A W, Z = JW, W W hk. The, Z also saisfies, for =, 1,..., N 1, 3.8 A W, Z = J W, W W hk. We oice ha, raher ha usig he dual formulaio of he discree problem 3.5, we formulaed he same fiie eleme mehod for he coiuous dual problem 2.5.

13 CG METHOD FOR A FRACTIONAL ORDER VISCOELASTICITY MODEL 9 From 3.5 we ca recover he ime seppig scheme, U 1, W 1 U 2, W 1 d =, 3.9 ρ U 2, W 2 + au 1, W 2 β sa U 1 s, W 2 ds d = f, W2 d + g, W 2 ΓN d, W1, W 2 W hk, U 1 = u h, U 2 = vh, for suiable choice of u h, v h V h as approximaios of he iiial daa u, v. Here, as a aural choice, we have 3.1 u h = P h u, v h = P h v. Typical fucios U = U 1, U 2 V hk, W = W 1, W 2 W hk are as follows: 3.11 m U i x, = Ui x = Ui,jϕ j x, j=1 U i x, I = ψ 1 U 1 x + ψ Ui x, m W i x, I = Wi,jϕ j x, j=1 where m is he umber of degrees of freedom i Th, ϕ j xm j=1 are he odal basis fucios for Vh defied o riagulaio Th, ad ψ is he odal basis fucio defied a ime level. Hece 3.9 yields U1 U1 1, W 1 k U U2 1, W 1 =, ρu2 U2 1, W 2 + k 2 U 1 + U1 1, W 2 au1 l 1, W 2 U 1 = u h, U 2 = v h. l=1 l=1 au1, l W 2 This implies he discree liear sysem, i l l 1 l l 1 β sψ l 1 s ds d β sψ l s ds d = f, W2 d + g, W 2 ΓN d, W1, W 2 Vh, M Ũ 1 k 2 M Ũ 2 = M 1, Ũ k 2 M 1, Ũ 1 2, ρm Ũ2 + k 2 ω,s Ũ1 = ρm 1, Ũ2 1 + k 2 + ω, 1S 1, Ũ1 1 Ũ 1 = u h, Ũ 2 = v h, 2 + S, Ũ1 ω, + + ω,l S l, Ũ1 l + B, l=1

14 1 F. SAEDPANAH where ω +, = ω,l = 1 l+1 l 1 β sψ l s ds d, ω, = β sψ l s ds d, B = Bj j = f, ϕj + g, ϕ j ΓN d, j M = M ij ij = ϕ i, ϕ j ij, S l, = S l, ij ij = aϕ l i, ϕ j ij, 1 β sψ l s ds d, M 1, = M 1, ij ij = ϕ 1 i, ϕ j ij, ad Ũ i = Ui,j m j=1 wih U i,j iroduced i We defie he orhogoal projecios R h, : V Vh, P h, : H Vh ad P k, : L 2 d P d, respecively, by 3.12 ar h, v v, χ =, v V, χ Vh, P h, v v, χ =, v H, χ Vh, P k, v v ψ d =, v L 2 d, ψ P d, wih P d deoig he se of all vecor-valued cosa polyomials. Correspodigly, we defie R h v, P h v ad P k v for = 1,, N, by R h v = R h, v, P h v = P h, v, ad P k v = P k, v I. Remark 4. I he case of assumpio 3.4, by Remark 3 ad he defiiio of he L 2 -projecio P k, we have V, P k V W hk, for ay V V hk. We iroduce he liear operaor A,r : V r h V h by av r, w = A,r v r, w, v r V r h, w V h. We se A = A,, wih discree orms v h,l = A l/2 v = v, A l v, v Vh ad l R, ad A h so ha A h v = A v for v Vh. We use Āh whe i acs o V h. For laer use i our error aalysis we oe ha P h A = A h R h.

15 CG METHOD FOR A FRACTIONAL ORDER VISCOELASTICITY MODEL 11 Theorem 3. Le Z be he soluio of 3.7 wih sufficiely smooh daa z T 1, z T 2, j 1, j 2. Furher, we assume 3.4. The for l R, we have he ideiy, Z 1 2 h,l + ρ γ Z 2 2 h,l+1 + 2ρ T T ξs a A l hż2, Ż2s ds d 3.13 = Z 1 T 2 h,l + ρ1 + γ Z 2 T 2 h,l ρ T A l hz 1, P k P h j 1 d + 2 T T T T A l+1 h βs a A l hp k P h j 2, Z 2 s ds d βt a A l hz 2, Z 2 T d 2ρξT a A l hz 2, Z 2 T. Z 2, P k P h j 2 d where γ = 1 γ. Moreover, for some cosa C = Cρ, γ, T, we have sabiliy esimae Z 1 h,l + Z 2 h,l+1 C P h z1 T h,l + P h z2 T h,l T + P h j 1 h,l + P h j 2 h,l+1 d. Proof. The soluio Z of 3.7 also saisfies 3.8, for = N 1,..., 1,. The recallig Remark 4 for he assumpio 3.4, we obviously have, 3.15 P k Z 1 = ρż2 P k P h j 2. Usig his i 3.8 ad recallig he iiial daa Z i T = P h zi T, i = 1, 2, we obai T W 1, Ż1 + aw 1, Z 2 = T βs a W 1, Z 2 s ds d + W 1 T, P h z1 T + ρ W2 T, P h z2 T T W 1, j 1 d + W 1 T, z T 1 + ρ W2 T, z T 2. The erms cocerig he iiial daa are caceled by he defiiio of he orhogoal projecio P h. Besides, for he covoluio erm we recall βs = D s ξs from 2.7 ad he parial iegraio yields, T T βs a W 1, Z 2 s ds d = + γ T T T ξs a W 1, Ż2s ds d ξt a W 1, Z 2 T d T a W 1, Z 2 d.

16 12 F. SAEDPANAH These ad γ = 1 γ imply ha he soluio Z saisfies, T W 1, Ż1 + γaw 1, Z 2 T + ξt aw 1, Z 2 T d = ξs a W 1, Ż2s ds T W 1, P h j 1 d. Now we se W 1 = A l h P kz 1, ad we have 3.16 T A l hp k Z 1, Ż1 + γaa l hp k Z 1, Z 2 T ξs a A l hp k Z 1, Ż2s ds + ξt a A l hp k Z 1, Z 2 T d = T A l hp k Z 1, P h j 1 d. We sudy he four erms a he lef side of he above equaio. For he firs erm we have 3.17 T T A l hp k Z 1, Ż1 d = 1 D Z 1 2 h,l d 2 = 1 2 Z 1T 2 h,l Z 1 2 h,l. Wih 3.15 we ca wrie he secod erm as 3.18 T T γ aa l hp k Z 1, Z 2 d = ρ γ aa l hż2, Z 2 d γ T aa l hp k P h j 2, Z 2 d = ρ γ 2 Z 2T 2 h,l+1 + ρ γ 2 Z 2 2 h,l+1 γ T aa l hp k P h j 2, Z 2 d.

17 CG METHOD FOR A FRACTIONAL ORDER VISCOELASTICITY MODEL 13 For he hird erm i 3.16, by virue of 3.15 ad iegraio by pars, we obai 3.19 T T ξs a A l hp k Z 1, Ż2s ds d = ρ + = ρ + + T T T T T T γ T T T ξs a A l hż2, Ż2s ds d ξs a A l hp k P h j 2, Ż2s ds d ξs a A l hż2, Ż2s ds d βs a A l hp k P h j 2, Z 2 s ds d ξt aa l hp k P h j 2, Z 2 T d T a A l hp k P h j 2, Z 2 d. Fially, for he las erm a he lef side of 3.16, we use 3.15 ad iegraio by pars o have 3.2 T ξt a A l hp k Z 1, Z 2 T d = ρ = ρ T T T ξt a A l hż2, Z 2 T d ξt a A l hp k P h j 2, Z 2 T d βt a A l hz 2, Z 2 T d ργ Z 2 T 2 h,l+1 + ρξt a A l hz 2, Z 2 T T ξt a A l hp k P h j 2, Z 2 T d. Puig i 3.16 we coclude he ideiy Now we prove he esimae We recall, from 2.8, ha ξ is a posiive ype kerel. The, usig he Cauchy-Schwarz iequaliy i 3.13, ad β L1R + =

18 14 F. SAEDPANAH γ, ξ γ, we ge Z 1 2 h,l + ρ γ Z 2 2 h,l+1 Z 1 T 2 h,l + ρ1 + γ Z 2 T 2 h,l T A l hz 1, P k P h j 1 d + 2 T T T T A l+1 h βs a A l hp k P h j 2, Z 2 s ds d 2ρ βt a A l hz 2, Z 2 T d 2ρξT a A l hz 2, Z 2 T Z 1 T 2 h,l + ρ1 + γ Z 2 T 2 h,l+1 Z 2, P k P h j 2 d + C 1 max T Z 1 2 h,l + 1/C 1 T P k P h j 1 h,l d T + C 2 max Z 2 2 h,l+1 + 1/C 2 P k P h j 2 h,l+1 d T + C 3 Z 2 T 2 h,l+1 + 1/C 3 max Z 2 2 h,l+1 T + C 4 Z 2 T 2 h,l+1 + 1/C 4 Z 2 2 h,l+1. Usig ha, for piecewise liear fucios, we have 3.21 max U i max U i, [,T ] N ad 3.22 T P k f d T f d, ad ha he above iequaliy holds for arbirary N, i a sadard way, we coclude he esimae iequaliy Now he proof is complee. 4. A priori error esimaes We defie he sadard ierpola I k wih I k v belog o he space of coiuous piecewise liear polyomials, ad 4.1 I k v = v, =, 1,, N. By sadard argumes i approximaio heory we see ha, for q =, 1, 4.2 T I k v v i d Ck q+1 T D q+1 2 v i d, for i =, 1, where k = max 1 N k. We assume he ellipic regulariy esimae v 2 C Av, v DA, so ha he followig error esimaes for he Riz projecio 3.12, hold rue 4.3 R h v v Ch s v s, v H s V, s = 1, 2. Hece we mus specialize o he pure Dirichle boudary codiio ad a covex polygoal domai. We oe ha he eergy orm V is equiale o 1 o V. 2

19 CG METHOD FOR A FRACTIONAL ORDER VISCOELASTICITY MODEL 15 Theorem 4. Assume ha Γ N =, Ω is a covex polygoal domai, ad 3.4. Le u ad U be he soluios of 2.3 ad 3.5. The, wih e = U u ad C = Cρ, γ, T, we have e 1 T Ch 2 u 2 + u 1 T 2 + T T + Ck 2 ü2 + ü 1 1 d. u 1 2 d Proof. We recall Remark 4 for he assumpio 3.4. We se e = U u = θ + η + ω wih θ 1 = U 1 I k R h u 1, η 1 = I k IR h u 1, ω 1 = R h Iu 1, θ 2 = U 2 I k P h u 2, η 2 = I k IP h u 2, ω 2 = P h Iu 2. Now, puig W = P k θ i 3.7 we have where by defiiio ad by parial iegraio T JP k θ = A P k θ, Z, JP k θ = Pk θ 1, j 1 + P k θ 2, j 2 d + P k θ 1 T, z1 T + ρ Pk θ 2 T, z2 T, A P k θ, Z = Aθ, P k Z + P k θ 1 T, Z 1 T θ 1 T, Z 1 T + ρ P k θ 2 T, Z 2 T ρ θ 2 T, Z 2 T. We se j 1 = j 2 = ad z T 2 =, P h z T 1 = θ 1 T, ad we recall ha Z i T = P h z T i, i = 1, 2. Hece usig he defiiio of he orhogoal projecio P h we have θ 1 T 2 = Aθ, P k Z, ha, usig θ = e η ω ad he Galerki orhogoaliy 3.6, implies, θ 1 T 2 = Aη, P k Z Aω, P k Z T = η 1, P k Z 1 + η 2, P k Z 1 + ρ η 2, P k Z 2 aη 1, P k Z 2 + β sa η 1 s, P k Z 2 ds d η 1, P k Z 1 ρ η 2, P k Z 2 T + ω 1, P k Z 1 + ω 2, P k Z 1 + ρ ω 2, P k Z 2 aω 1, P k Z 2 + β sa ω 1 s, P k Z 2 ds d ω 1, P k Z 1 ρ ω 2, P k Z 2. By he defiiio of η, ha idicaes he ierpolaio error, erms icludig η i, η i vaish. We also use he defiiio of ω, ha idicaes he projecio error,

20 16 F. SAEDPANAH ad we coclude θ 1 T 2 = T η 2, P k Z 1 aη 1, P k Z 2 + T ω 1, P k Z 1 d ω 1, P k Z 1, β sa η 1 s, P k Z 2 ds d ha by he Cauchy-Schwarz iequaliy implies T 2 θ 1 T 2 C 1 max P kz /C 1 η 2 d T T 2 + C 2 max P kz /C 2 η 1 1 d T T + C 3 max P kz /C 3 β η1 1 d T T 2 + C 4 max P kz /C 4 ω 1 d T + C 5 P k Z /C 5 ω 1 2. Usig 3.21, β L1R + = γ, ad he sabiliy esimae 3.14 wih l =, i a sadard way, we have θ 1 T C ω 1 + T η 2 + η ω 1 d. Recallig θ 1 T = e 1 T ω 1 T, ad sabiliy of he projecios P h, R h wih respec o, 1, respecively, we have θ 1 T C R h Iu + R h Iu 1 T + T This complees he proof by 4.2, 4.3. I k Iu 2 + I k Iu R h I u 1 d. 5. A poseriori error esimaes Havig cerai regulariy o he daa, i.e., iiial daa u, v ad he force erms f, g, here are sill wo ypes of limiaio for higher global regulariy of a weak soluio of 1.1. Oe is due o he mixed Dirichle-Neuma boudary codiio. This ype of boudary codiio are aural i pracice, ad a pure Dirichle boudary codiio ca o be realisic i applicaios. Oher limiaio is he sigulariy of he covoluio kerel β. This meas ha eve wih he pure Dirichle boudary codiio, higher regulariy of a weak soluio is limied, see [7], [11], hough wih smooher kerels we ca ge higher regulariy. Besides, he sabiliy ad a priori error esimaes preseed i Theorem 3 ad Theorem 4 do o admi adapive meshes. These, ad oher geeral moivaios such as o pracical use of a priori error esimaes, call for adapive meshes based o a poseriori error aalysis. Here a space-ime cellwise error represeaio is give. The mai framework is adaped from [3], ad a geeral liear goal fucioal J is used. This error represeaio ca be used for goal-orieed adapive sraegies based o dual

21 CG METHOD FOR A FRACTIONAL ORDER VISCOELASTICITY MODEL 17 weighed residual mehod. For more deails o dual weighed residual mehod ad is pracical aspecs for differeial equaios, see [3] ad refereces herei. Theorem 5. Le u ad U be he soluios of 2.3 ad 3.5, ad J he liear fucioal defied i 2. The, wih e = U u, we have he error represeaio 5.1 Je = Θ,K + K Th =1 K T h 6 i=1 Θ i,k, where, wih z hk W hk beig a approximaio of he dual soluio z ad E hk z = z hk z beig he error operaor, 5.2 Θ,K = U 1 u, E hk z 1 K + ρ U 2 v, E hk z 2 K, Θ 1,K = U 1 U 2, E hk z 1 K, Θ 2,K = ρ U 2 f, E hk z 2 K, Θ 3,K = r h, E hk z 2 K, Θ 4,K = g h g, E hk z 2 K, T Θ 5,K = r h, β se hk z 2 d Θ 6,K = g h, s T s β se hk z 2 d K, K. Here K = K ad K = K are he space-ime cells, ad r h, g h are defied below, 5.6, 5.7, respecively. I should be oiced ha K is o he boudary of K. Proof. Usig he ideiy 2.4 ad he Galerki orhogoaliy 3.6 we have, 5.3 Je = A e, z = Ae, z = Ae, E hk z = AU, E hk z F E hk z = RU; E hk z, where RU; is he residual of he Galerki approximaio U as a fucioal o he soluio space V. The by he defiiio of A, F we have 5.4 Je = U 1 u, E hk z 1 + ρ U 2 v, E hk z 2 + T T U 1, E hk z 1 U 2, E hk z 1 + ρ U 2, E hk z 2 + au 1, E hk z 2 β sa U 1 s, E hk z 2 ds d f, E hk z 2 + g, E hk z 2 ΓN d.

22 18 F. SAEDPANAH Now, by parial iegraio wih respec o he space variable, we obai 5.5 T au 1, z 2 d = = = = = =1 =1 =1 K T h K T h + =1 E E Γ N E E I K T h =1 K T h au 1, z 2 K d σ U 1, z 2 K d [σ U 1 ], z 2 σ U 1, z 2 E d rh, z 2 K + g h, z 2 K d rh, z 2 K + g h, z 2 K where E, E Γ N are, respecively, he ses of he ierior edges ad he edges o he Neuma boudary, correspodig o he riagulaio T h. Here r h are he residuals represeig he jumps of he ormal derivaives σ U 1, ad deermied by, r h Γ = 2 [σ U 1 ] if Γ K \ Ω, if Γ Ω, ad g h is he coribuio from he Neuma boudary defied as σ U 1 if Γ K Γ N, 5.7 g h Γ = if Γ Ω. For he covoluio erm i 5.4 we firs chage he order of he ime iegrals, he similar o 5.5, we have, E 5.8 T β sa U 1 s, z 2 ds d = = = T T s =1 β sa U 1 s, z 2 d ds K T h =1 K T h T + g h, a U 1 s, r h, T s s T s β sz 2 d ds K β sz 2 d K β sz 2 d K.

23 CG METHOD FOR A FRACTIONAL ORDER VISCOELASTICITY MODEL 19 Now, usig 5.5, 5.8 ad space-ime cellwise represeaio of he oher erms i 5.4 we coclude he error represeaio 5.1. The error represeaio 5.1 leads us o he weighed a poseriori esimae, Je R,K ω,k + R i,kωi,k, K T h wih he residuals ad weighs defied as =1 K T h i=1 R,K = U 1 u 2 K + U 2 v 2 K 1/2, ω,k = E hk z 1 2 K + E hk z 2 2 K 1/2, R 1,K = U 1 U 2 K, ω1,k = E hk z 1 K, R 2,K = ρ U 2 f 2 1 1/2, K + 2 h K r h 2 K ω2,k = E hk z 2 2 K + h K E hk z 2 2 K + h T K β se hk z 2 d 2 1/2, K s R h 1 3,K = K g h g 2 1 1/2, K + h K g h 2 K ω3,k = hk E hk z 2 2 K + h T K β se hk z 2 d 2 1/2. K I order o evaluae he a poseriori error represeaio 5.1 or he a poseriori esimae 5.9, we eed iformaio abou he coiuous dual soluio z. Such iformaio has o be obaied eiher hrough a priori aalysis i form of bouds for z i cerai Sobolev orms or hrough compuaio by solvig he dual problem umerically. I his coex we provide iformaio hrough a priori aalysis ad we leave he ivesigaio o he secod case o a laer work. I he followig, he arge fucioal J will be he global L 2 -orm of he approximaio displaceme u 1 = u 1 x,. We firs prese a weighed global a poseriori error esimae, usig global L 2 -projecios P k, P h defied i 3.12, ad error esimaes of P h i a weighed L 2 -orm. We recall he weighed global error esimaes of he L 2 -projecio P h 3.12, see [4]. Firs we recall some oaio. Le T be a give riagulaio wih mesh fucio h, ad for ay simplex K T, ρ K deoe he radius of he larges ball coaied i he closure of K, ha is K. A family F of riagulaios T is called o-degeerae, if here exsis a cosa c such ha we have c = max max h K. T F K T ρ K Le S K = K T : K K ad δ T be a measure for he give riagulaio T defied by δ T = max max 1 h 2 K T K K S /h2 K K. We defie a measure, δ F, for a give family F, by 5.1 δ F = max T F δ T. s

24 2 F. SAEDPANAH We defie he error operaors E hk, E h, ad E k by 5.11 E hk v = P k P h Iv, E h v = P h Iv, E k v = P k Iv, ad we oe ha 5.12 E hk = E h + E k P h. Lemma 1. Assume ha he family F of raigulaios T be o-degeerae. The for sufficiely small δ F, here exiss a cosa C such ha for ay riagulaio T F we have, for all v H 2, h s E h v C s v, s = 1, 2, v H s, h 1 E h v C 2 v, v H 2, where deoes he usual gradie. For more deails o he pracical aspecs of δ F, see [4]. For he ex heorem we recall he mesh fucios h, h from 3.1, 3.2, ad we defie he oaios h mi, = mi h K, K T h h max, = max h K. K Th Theorem 6. Le u be he soluios of 2.3, ad U be he soluio of 3.5 wih a o-degeerae family F h of riagulaios Th, =, 1,..., N, wih sufficiely small δ Fh, such ha he weighed global error esimaes 5.13 ad 5.14 hold. The, wih e = U u, we have he weighed a poseriori error esimae e 1 T C h U1 u + h 2 U2 v + =1 h U 1 U 2 + h 2 ρ U 2 f + ζ + ζ,n h 3 K r h 2 K 1/ ζ K T h h 3 K g h g 2 K 1/2 + ζ,n h 3 K g h 2 K 1/2 where 5.16 ζ = K T h + k U 1 U 2 + k E k Ā h U 1 + k Ek β sāhu 1 s ds + k E k f + k h 2 mi, h2 max,, K T h ζ,n = K T h 1/2 h 1 K E kg 2 K d, 2 h mi, max j N h2 max,j. Proof. Le z V be he soluio of he dual problem 2.5. From he defiiio of he L 2 projecios P k, P h i 3.12 ad he es space W hk i 3.3 we have P k P h z W hk. Therefore, usig 5.3 ad he error operaors 5.11 we have, 5.17 Je = RU; E hk z = RU; E h z + RU; E k P h z,

25 CG METHOD FOR A FRACTIONAL ORDER VISCOELASTICITY MODEL 21 where we used We sudy he wo erms a he righ side of his equaio. For he firs erm we ca wrie, RU; E h z = U 1 u, E h z 1 + ρ U 2 v, E h z 2 + =1 U 1 U 2, E h z 1 + ρ U 2 f, E h z 2 + au 1, E h z 2 g, E h z 2 ΓN β sa U 1 s, E h z 2 ds d. Now by parial iegraio i space, similar o 5.5, 5.8, we have RU; E h z = U1 u, E h z 1 K + ρ U 2 v, E h z 2 K K T h + =1 K T h U 1 U 2, E h z 1 K + ρ U 2 f, E h z 2 K 5.18 = + r h, E h z 2 K + g h g, E h z 2 K T r h, βs E h z 2 s ds g h, 2 I i + i=1 T βs E h z 2 s ds 6 II i. i=1 K K d We he, for each erm, use he Cauchy-Schwarz iequaliy wice. Firs o he local elemes K, K, o obai local L 2 -orms, ad he o he sum over he elemes o obai global orms such ha he weighed global error esimaes 5.13, 5.14 ca be used. For I 1 we have 5.19 I 1 h U 1 u h 1 E hz 1 C h U 1 u z 1, ad i a similar way we have, 5.2 I 2 C h 2 U 2 v 2 z 2. For he ex erm, usig he error esimae 5.13, we have, 5.21 II 1 = =1 =1 K Th U 1 U 2, E h z 1 d K h U 1 U 2 h 1 C max [,T ] z 1 =1 E h z 1 d h U 1 U 2 d,

26 22 F. SAEDPANAH ad similarly we obai, 5.22 II 2 C max [,T ] 2 z 2 =1 h 2 ρ U 2 f d. For II 3, we firs have, II 3 =1 K T h h 3 K r h 2 K 1/2 K T h h 3 K E hz 2 2 K 1/2 d. The by a scaled race iequaliy ad he weighed global error esimaes 5.13, 5.14, we obai h 3 K E hz 2 2 K C h 4 K E hz K + h K E hz 2 2 K K T h K T h C h 4 mi, h4 max, h 2 E h z h mi, h2 max, h 1 E h z 2 2 C h 4 mi, h4 max, 2 z 2 2. These imply he esimae 5.23 II 3 C max [,T ] 2 z 2 =1 mi, h 2 h2 max, K T h h 3 K r h 2 K 1/2 d. I a similar way, we have 5.24 II 4 C max [,T ] 2 z 2 =1 mi, h 2 h2 max, K T h h 3 K g h g 2 K 1/2 d. Fially we sudy II 5 ad a similar resul will hold for II 6. To his ed, firs we oe ha, II 5 =1 K T h K T h h 3 K h 3 K r h 2 K T 1/2 βs E h z 2 s ds K 2 1/2 d.

27 CG METHOD FOR A FRACTIONAL ORDER VISCOELASTICITY MODEL 23 The, usig Mikowski s iequaliy, he Cuachy-Schwarz iequaliy, ad he fac ha β L1R + = γ, we have K T h h 3 K T βs E h z 2 s ds K T h γ K T h T h 3 K h 3 K 2 K T 2 βs E h z 2 s K ds T βs βs ds K T h T βs E h z 2 s 2 K ds h 3 K E hz 2 s 2 K ds, ha usig a scaled race iequaliy ad he error esimaes 5.13, 5.14, we have K T h h 3 K T C C C βs E h z 2 s ds T βs j K T h j= 2 + h mi, h2 max,j h 1 j j j= 2 K h 4 K E hz 2 s 2 2 K + h K E hz 2 s 2 K ds βs h 4 mi, h4 max,j h 2 j E h z 2 s 2 j 1 E h z 2 s 2 ds j 1 βs h 4 mi, h4 max,j 2 z 2 s 2 ds Cγ max [,T ] 2 z 2 s 2 h 4 max mi, j N h4 max,j. Hece we obai, 5.25 II 5 C max [,T ] 2 z 2 =1 mi, h 2 max j N h2 max,j K T h h 3 K r h 2 K 1/2 d. A similar esimae for II 6 holds, ha is, 5.26 II 6 C max [,T ] 2 z 2 =1 mi, h 2 max j N h2 max,j K T h h 3 K g h 2 K 1/2 d.

28 24 F. SAEDPANAH Puig i 5.18 we coclude, 5.27 RU; E h z C max max [,T ] 2 z 2, max z 1 [,T ] h U1 u + h 2 U2 v + =1 h U 1 U 2 + h 2 ρ U 2 f 2 + h mi, h2 max, K T h 2 + h mi, h2 max, h 3 K r h 2 K 1/2 h 3 K g h g 2 K 1/2 2 + h max K T h mi, j N h2 max,j 2 + h max mi, j N h2 max,j K T h K T h h 3 K r h 2 K h 3 K g h 2 K 1/2 1/2 d. Now we sudy he secod erm i 5.17, ha is, RU; E k P h z = RU; E k P h z ± T Pk f, E k P h z 2 + P k g, E k P h z 2 ΓN d = U 1 u, E k P h z 1 + ρ U 2 v, E k P h z 2 + U 1 U 2, E k P h z 1 =1 + au 1, E k P h z 2 β sa U 1 s, E k P h z 2 ds + ρ U 2 P k f, E k P h z 2 Pk g, E k P h z 2 ΓN + E k f, E k P h z 2 + E k g, E k P h z 2 ΓN d. Recallig he iiial codiio U i = P h u i, i = 1, 2, he firs wo erms o he righ side vaish. Besides, from he secod equaio of 3.9 we have, for W V h, ρ U 2, W P k f, W P k g, W d = ap k U 1, W a P k β su 1 s ds, W d.

29 CG METHOD FOR A FRACTIONAL ORDER VISCOELASTICITY MODEL 25 Hece, we coclude 5.28 RU; E k P h z = U 1 U 2, E k P h z 1 =1 ae k U 1, E k P h z 2 + a E k β su 1 s ds, E k P h z 2 + E k f, E k P h z 2 + E k g, E k P h z 2 ΓN d. For he las erm we have, =1 E k g, E k P h z 2 ΓN d =1 K T h 1/2 h 1 K E 1/2. kg 2 K h K E k P h z 2 K 2 K Th By a scaled race iequaliy ad local iverse iequaliy we have, K T h h K E k P h z 2 2 K C K T h C Ek P h z 2 2 K + h 2 K E k P h z 2 2 K Ek P h z 2 2 K + E k P h z 2 2 K K T h = C E k P h z 2 2. Hece, E k g, E k P h z 2 ΓN d C =1 K T h h 1 K E kg 2 K 1/2 Ek P h z 2. Cosiderig his i 5.28 ad usig he Cauchy-Schwarz iequaliy we have RU; E k P h z C =1 U 1 U 2 E k P h z 1 + E k Ā h U 1 E k P h z 2 + E k β sāhu 1 s ds E k P h z 2 + E k f E k P h z 2 + K T h 1/2 Ek h 1 K E kg K 2 P h z 2.

30 26 F. SAEDPANAH This, L 2 -sabiliy of he L 2 -projecio P h, ad a sadard error esimaio of he error operaor E k, coclude RU; E k P h z C max max ż 1, max ż 2 [,T ] [,T ] 5.29 =1 k U 1 U 2 + k E k Ā h U 1 + k Ek β sāhu 1 s ds + k E k f + k K T h 1/2 h 1 K E kg 2 K d. We ow se j 1 = j 2 = z2 T = ad z1 T = A 1/2 e 1 T. The, puig 5.27 ad 5.29 i 5.17, usig he sabiliy esimaes 2.14 ad a sadard argume, we coclude he a poseriori error esimae 5.15, ad his complees he proof. Remark 5. We oe ha for he error esimae 5.15 here are wo ypes of resricio o he riagulaios; Oe by ζ, ζ,n, ha measures he quasiuiformiy of he family of riagulaio, ad he oher by δ Fh, ha measure he regulariy of he family of riagulaios i a slighly differe sese. Alhough maybe o explicily, bu ζ, ζ,n ad δ Fh ca be relaed. I pracice we use fiiely may riagulaios, ha meas quasiuiformiy holds, hough possibly wih big ζ, ζ,n. This meas ha we sill ca use he a poseriori error esimae Bu whe δ Fh is o sufficiely small, he error esimae 5.15 does o hold. This calls for usig local ierpolas isead of global L 2 -projecio P h. I he ex heorem we prese a a poseriori error esimae usig ierpolaio, liear i space ad cosa i ime o each space-ime cell. We also oe ha a possible, bu o eccesserily opimal, way of igorig hese limiaios could be usig global error esimaes of P h wih global mesh size h max. Tha is a poseriori error esimae i he form, wih k = k for, e 1 T C h max, U 1 u + h 2 max, U 2 v T + Ch max U 1 U 2 d + Ch 2 max + T T ρ U 2 f + r h + g h g ΓN + g h ΓN d k U 1 U 2 + E k Ā h U 1 + E k β sāhu 1 s ds + E k f + E k g ΓN d, where r h K = h 1 K max K r h, g h K = h 1/2 K g h, ad g K = h 1/2 K g. We recall he decomposiio of he space-ime slab Ω = Ω io cells K = K, K T h. Le I hk be he sadard ierpola, such ha I hk v K be liear i space ad cosa i ime. We defie he error operaor E hk by E hk v = I hk Iv.

31 CG METHOD FOR A FRACTIONAL ORDER VISCOELASTICITY MODEL 27 A varia of he Bramble-Hilber lemma he implies he error esimaes, E hk v K C 5.3 h r K r v K + k v K, r = 1, 2, E hkv K C 5.31 h 2 K 2 v K + k v K, where K is a pach of space cells suiably chose aroud K. We recall he mesh fucio h from 3.2, ad we defie he oaio We will also use he fac ha h max, = max h K. K T h 5.32 v 2 Ω = v I 2 d k max v. Theorem 7. Le u ad U be he soluios of 2.3 ad 3.5. The, wih e = U u, we have he weighed a poseriori error esimae 5.33 e 1 T C h U1 u + h 2 U2 v + C h U 1 U 2 Ω + k U 1 U 2 Ω =1 k 1/2 + h 2 ρ U 2 f Ω + k ρ U 2 f Ω 1/2 1/2 + h 3 K r h 2 K + h 3 K g h g 2 K K T h + k K T h h 1 K r h 2 K 1/2 + k h 1 K g h g 2 K 1/2 K T h + k K T h h K r h 2 K 1/2 + k h K g h g 2 K 1/2 K T h + h2 max, + k + h2 max, + k + hmax, + k + hmax, + k j=1 K T j h j=1 K T j h h max,j j=1 K T j h h max,j j=1 K T j h K T h h 1 K β r h j 2 K 1/2 2 d h 1 K β g h j 2 K 1/2 2 d h 1 K β r h j 2 K 1/2 2 d h 1 K β g h j 2 K 1/2 2 d 1/2 1/2 1/2 1/2.

32 28 F. SAEDPANAH Proof. We wrie he error represeaio 5.1 as Je = Θ,K + Θ i,k = I + I i. K Th i=1 =1 K T h i=1 Firs we esimae I. To his ed, recallig Θ,K from 5.2, we use he Cauchy- Schwarz iequaliy ad he ierpolaio error esimae 5.3 o obai, U1 u, E hk z 1 K U 1 u K E hk z 1 K K T h K T h C U 1 u K h K z 1 K K T h C z 1 h 2 K U 1 u 2 K 1/2 K T h = C z 1 h U1 u. Similarly we have ρ U 2 v, E hk z 2 K C 2 z 2 h 2 U2 v. K T h From hese wo esimaes we coclude 5.35 I C max z 1, 2 z 2 h U1 u + h 2 U2 v. For he ex erm, usig he Cauchy-Schwarz iequaliy ad he error esimae 5.3, we have I 1 U 1 U 2 K E hk z 1 K =1 K T h C =1 K T h =1 K T h U 1 U 2 K hk z 1 K + k ż 1 K C h 2 K U 1/2 1 U 2 2 K z 1 2 K =1 K T h + C k U 1/2 1 U 2 2 K ż 1 2 K = C =1 ha usig 5.32 we have 5.36 K T h K T h 1/2 1/2 h U 1 U 2 Ω z 1 Ω + k U 1 U 2 Ω ż 1 Ω, I 1 C max =1 max z 1, max ż 1 [,T ] [,T ] k 1/2 h U 1 U 2 Ω + k U 1 U 2 Ω.

33 CG METHOD FOR A FRACTIONAL ORDER VISCOELASTICITY MODEL 29 I he same way we obai 5.37 I 2 C max max [,T ] 2 z 2, max ż 2 [,T ] k 1/2 h2 ρ U 2 f Ω + k ρ U 2 f Ω. =1 Now for I 3, we use he Cauchy-Schwarz iequaliy, a race iequaliy, ad he error esimaes 5.3, 5.31 o obai, I 3 =1 K T h C C =1 K T h =1 K T h h 1/2 K r h K h1/2 K E hkz 2 K h 1/2 K r h K Ehk z 2 K + h K E hk z 1 K h 1/2 K r h K 2 h 2 K 2 z 2 K + k ż 2 K + h K k ż 2 K 1/2 C h 3 K r h 2 K 2 z 2 Ω =1 K T h + k K T h + k K T h h 1 K r h 2 K 1/2 ż2 Ω h K r h 2 K 1/2 ż2 Ω, ha usig 5.32 we have 5.38 I 3 C max max [,T ] 2 z 2, max =1 k 1/2 + k K T h [,T ] ż 2, max ż 2 [,T ] h 3 K r h 2 K 1/2 + k h K r h 2 K 1/2. K T h h 1 K r h 2 K 1/2 K T h

34 3 F. SAEDPANAH Ad similarly 5.39 I 4 C max max [,T ] 2 z 2, max =1 k 1/2 + k K T h [,T ] ż 2, max h 3 K g h g 2 K 1/2 h 1 K g h g 2 K 1/2 ż 2 [,T ] K T h + k h K g h g 2 K 1/2. K T h Fially we sudy I 5, I 6 which iclude he covoluio erms. We fid a esimae for I 5 ad a similar argume holds for I 6. Firs, recallig he defiiio of Θ 5,K from 5.2, we ca wrie I 5 as, I 5 = =1 K T h Θ 5,K = T β s =1 s K T h rh s, E hk z 2 d ds. K The we chage he order of he ime iegrals ad we obai, I 5 = = =1 j β s j=1 j 1 K T j h =1 j=1 K T j h rh s, E hk z 2 ds d K β rh j, E hk z 2 d, K where β v j = j j 1 β svs ds.

35 CG METHOD FOR A FRACTIONAL ORDER VISCOELASTICITY MODEL 31 Now by a race iequaliy ad he Cauchy-Schwarz iequaliy i he sum over he riagles, ad furher Cuachy-Schwarz iequaliy i he iegral over, we have, I 5 C =1 j=1 K T j h 1/2 h 1 K β r h j 2 K E hk z 2 + h max,j E hk z 2 d 1/2 = C E hk z 2 h 1 K β r h j K 2 d C + C + C =1 =1 =1 j=1 E hk z 2 K T j h h max,j j=1 1/2 K T j h E hk z 2 2 d 1/2 2 h 1 K β r h j K 2 d =1 j=1 K T j h E hk z 2 2 d h max,j j=1 K T j h 1/2 h 1 K β r h j 2 K 1/2 d 1/2 h 1 K β r h j 2 K 1/2 2 d 1/2. Sice by he error esimae 5.3 we have E hk z 2 2 d = E hk z 2 2 K d = E hk z 2 2 K K T h C K T h K T h h4 K 2 z 2 2 K + k 2 ż 2 2 K C h4 max, 2 z 2 2 Ω + k2 ż 2 2 Ω, ad similarly by 5.31 E hk z 2 2 d = E hk z 2 2 K d = E hk z 2 2 K K T h C K T h K T h h2 K 2 z 2 2 K + k 2 ż 2 2 K C h2 max, 2 z 2 2 Ω + k2 ż 2 2 Ω,

36 32 F. SAEDPANAH he, recallig 5.32, we coclude he esimae I 5 C max max [,T ] 2 z 2, max ż 2, max ż 2 [,T ] [,T ] N k 1/2 h2 max, + k 5.4 =1 + C =1 j=1 K T j h k 1/2 hmax, + k h max,j j=1 h 1 K β r h j 2 K 1/2 2 d K T j h 1/2 h 1 K β r h j 2 K 1/2 2 d 1/2 The same esimae holds for I 6 wih r h be replaced by g h. We ow se j 1 = j 2 = z2 T = ad z1 T = A 1/2 e 1 T. The, puig , ad he couerpar of 5.4 for I 6, i 5.1, usig he sabiliy esimaes 2.14 ad a sadard argume, we coclude he a poseriori error esimae 5.33, ad his complees he proof. Remark 6. We ca compue r h K = ψ 1 r h 1 + ψ r h 2 K d 1/2 k1/2 r h 1 2 K + r h 2 K k1/2 rh 1 K + r h K. Remark 7. We oe ha he las a poseriori error esimae preseed i 5.33, does o have he resricios ha were meioed i Remark 5. Refereces 1/2 1. K. Adolfsso, M. Eelud, ad S. Larsso, Space-ime discreizaio of a iegro-differeial equaio modelig quasi-saic fracioal-order viscoelasiciy, J. Vib. Corol 14 28, K. Adolfsso, M. Eelud, S. Larsso, ad M. Racheva, Discreizaio of iegro-differeial equaios modelig dyamic fracioal order viscoelasiciy, LNCS , W. Bagerh ad R. Raacher, Adapive Fiie Eleme Mehods for Differeial Equaios, Lecures i Mahemaics ETH Zürich, Birkhäuser Verlag, M. Boma, Esimaes for he L 2 -projecio oo coiuous fiie eleme spaces i a weighed L p-orm, BIT 46 26, A. Erdélyi, W. Magus, F. Oberheiger, ad F. G. Tricomi, Higher Trascedeal Fucios. Vol. III, Rober E. Krieger Publishig Co. Ic., 1981, Based o oes lef by Harry Baema, Repri of he 1955 origial. 6. R. H. Fabiao ad K. Io, Semigroup heory ad umerical approximaio for equaios i liear viscoelasiciy, SIAM J. Mah. Aal , S. Larsso ad F. Saedpaah, The coiuous Galerki mehod for a iegro-differeial equaio modelig dyamic fracioal order viscoelasiciy, IMA J. Numer. Aal., o appear..

37 CG METHOD FOR A FRACTIONAL ORDER VISCOELASTICITY MODEL W. McLea, V. Thomée, ad L. B. Wahlbi, Discreizaio wih variable ime seps of a evoluio equaio wih a posiive-ype memory erm, J. of Compu. Appl. Mah , A. K. Pai, V. Thomée, ad L. B. Wahlbi, Numerical mehods for hyperbolic ad parabolic iegro-differeial equaios, J. Iegral Equaios Appl , B. Riviére, S. Shaw, ad J. R. Whiema, Discoiuous Galerki fiie eleme mehods for dyamic liear solid viscoelasiciy problems, Numer. Mehods Parial Differeial Equaios 23 27, F. Saedpaah, Exisece ad uiqueess of he soluio of a iegro-differeial equaio wih weakly sigular kerel, Prepri 29:16, Dep. of Mahemaical Scieces, Chalmers Uiversiy of Techology ad Uiversiy of Goheburg. 12. S. Shaw ad J. R. Whiema, Applicaios ad umerical aalysis of parial differeial Volerra equaios: a brief survey, Compu. Mehods Appl. Mech. Egrg , , A poseriori error esimaes for space-ime fiie eleme approximaio of quasisaic herediary liear viscoelasiciy problems, Compu. Mehods Appl. Mech. Egrg , Deparme of Mahemaical Scieces, Chalmers Uiversiy of Techology ad Uiversiy of Goheburg, SE Göeborg, Swede address: fardi@chalmers.se