Goal. Adaptive Finite Element Methods for Non-Stationary Convection-Diffusion Problems. Outline. Differential Equation

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1 Goal Adaptive Fiite Elemet Methods for No-Statioary Covectio-Diffusio Problems R. Verfürth Ruhr-Uiversität Bochum Tübige / July 0th, 017 Preset space-time adaptive fiite elemet methods for o-statioary covectio-diffusio equatios based o stable discretizatios ad a posteriori error estimates. A posteriori error estimates should yield upper ad lower bouds for the eergy orm of the error that are uiform with respect to all possible relative sizes of covectio to diffusio. Use a commo framework for various stabilizatio methods. 1/ 3 / 3 Outlie Discretizatio Cocludig Remarks Differetial Equatio t u ε u + a u + βu = f i Ω 0, T ] u = 0 o Γ 0, T ] u = u 0 i Ω 0 < ε 1, β 0, a R d, a = 1, f piecewise polyomial. Results hold for geeral f, variable coefficiets, ad mixed boudary coditios. The ε is a lower boud for the smallest eigevalue of the diffusio ad β is a lower boud for b 1 div a with b deotig the reactio. Geeral right-had sides f ad variable coefficiets give rise to additioal oscillatio terms. All estimates should be uiform w.r.t. ε. 3/ 3 4/ 3

2 Norms Fid u L 0, T ; H0 1Ω with tu L 0, T ; H 1 Ω such that u = u 0 i L ad for all t 0, T ad all v H0 1Ω t u, v + {ε u v + a uv + βuv} = fv Ω } {{ } Ω }{{} =Bu,v = l, v Eergy orm v = { ε v + β v } 1 ϕ, v Dual orm ϕ = sup v H0 1Ω\{0} v Error orm { u Xa,b = ess. sup u, t + t a,b + b a b a u, t dt t u + a u, t dt } 1 5/ 3 6/ 3 Discretizatio Discretizatio Meshes ad Spaces I = {t 1, t ] : 1 N I } partitio of [0, T ]. τ = t t 1. T, 0 N I, affie equivalet, admissible, shape regular partitios of Ω. Trasitio coditio: There is a commo refiemet T of T ad such that h K ch K for all K T ad all K T with K K. X H 1 0 Ω fiite elemet space correspodig to T. Discrete Problem Fid u T X, 0 N I, such that u 0 T 0 = π 0 u 0 ad, for = 1,..., N I ad all v X with U θ = θ u T + 1 θ u 1 u, v + BU θ, v + S U θ, v = l, v τ The stabilizatio term S is supposed to be liear i its secod argumet ad affie i its first argumet, it may deped o T ad o f. Solutio u I is cotiuous piece-wise affie ad equals u T at t. 7/ 3 8/ 3

3 Discretizatio Stabilizatios Streamlie diffusio method S u, v = K ϑ K K { tu ε u + a u + βu f}a v with ϑ K a ch K Cotiuous iterior pealty method S u, v = E ϑ E E J Ea uj E a v with ϑ E ch E Local projectio scheme S u, v = M ϑ M M κ M a u κ M a v with ϑ M a ch M ad I κ M projectio oto S l, 1 M Subgrid scale approach S u, v = K ϑ K K a Π u a Π v with ϑ K a ch K ad Π projectio oto Y X Basic Steps Error ad residual are equivalet. The residual splits ito a spatial ad a temporal residual. The orm of the sum of these is equivalet to the sum of their orms. Derive a reliable, efficiet ad robust error idicator for the temporal residual. Derive a reliable, efficiet ad robust error idicator for the spatial residual. All stabilizatios yield the same spatial error idicator. 9/ 3 10/ 3 Equivalece of Error ad Residual Proof of the Equivalece u I is cotiuous piece-wise affie ad equals u T at t. Residual: Ru I, v = l, v t u I, v Bu I, v Lower error-boud: Ru I L t 1,t ;H 1 u u I Xt 1,t Upper error-boud: u u I X0,t {4 u 0 π 0 u 0 +6 Ru I L 0,t ;H 1 } 1 Relatio of residual ad error: Ru I, v = t e, v + Be, v Lower error-boud: Defiitio of primal ad dual orm plus Cauchy-Schwarz iequality. Upper error-boud: Parabolic eergy estimate with v = e as test-fuctio. 11/ 3 1/ 3

4 Decompositio of the Residual Recall U θ = θ u T + 1 θ u 1 Temporal residual: R τ u I, v = BU θ u I, v Spatial residual: R h u I, v = l, v t u I, v BU θ, v Splittig: Ru I = R τ u I + R h u I Estimate for L t 1, t ; H 1 -orms: 1 { R τ u I + R h u I } 1 R τ u I + R h u I 13 R τ u I + R h u I Motivatio of the Lower Boud Stregtheed Cauchy-Schwarz iequality for v = c ad w = b t b a : b a Hece: vw = 1 cb a = 3 v a,b w a,b v + w a,b 1 3 { } v a,b + w a,b 13/ 3 14/ 3 Proof of the Lower Boud R h u I is piece-wise costat. R τ u I is piece-wise affie: R τ u I = ρ, v = Bu T, v. Choose v, w H0 1 Ω such that θ t t 1 τ ρ with v = R h u I, R h u I, v = R h u I, w = ρ, ρ, w = ρ. Isert 3 t t 1 τ v + t t τ w as test-fuctio i represetatio of Ru I. Estimatio of the Temporal Residual R τ u I = θ t t 1 τ ρ with ρ, v = Bu T, v. Upper boud: ρ u T a + u Follows from defiitio of ρ ad. Lower boud: 1 { u 3 a + u } ρ 15/ 3 16/ 3

5 Proof of the Lower Boud Set w = u T ad choose v H0 1 Ω with v = a w ad a w, v = a w Isert 1 w + 1 v i the defiitio of ρ : ρ, 1 w + 1 v = 1 ε w, w + 1 βw, w + 1 }{{} a w, w }{{} = 1 w =0 Estimatio of the Covective Derivative I Assume that a c c ε. Friedrichs iequality implies a u T, v a u T c Ω v. Hece a u T u c c c Ω T 1 ad a u T is equivalet to u T. + 1 ε w, v + 1 βw, v + 1 }{{} a w, v }{{} 1 w a w = 1 a w 17/ 3 18/ 3 Estimatio of the Covective Derivative II Assume that a ε. Auxiliary problem with aalytical ad discrete solutios Φ ad Φ : ε ϕ, ψ + β ϕ, ψ = a u T, ψ 1 3 { Φ T + Φ Φ } a u T Φ + Φ Φ Φ Φ is equivalet to robust residual error idicator η τ for. Hece a u T is equivalet to Φ + η τ. Estimatio of the Spatial Residual Spatial error idicator ηh : ηh = α K R K K + K { T } α S = mi ε 1 h S, β 1 ε 1 E E αe R E E R K ad R E are the usual elemet ad iterface residuals. Stadard argumets for statioary problems yield: R h u I c η h + I MR h u I, η h c R h u I. I M R hu I measures the cosistecy error of the stabilizatio. c, c oly deped o the polyomial degrees ad o the shape parameters of the partitios T. 1 19/ 3 0/ 3

6 Proof of the Upper Boud L -represetatio: R h u I, v = rv + Ω Σ jv Quasi-iterpolatio error estimate: v I M v K cα K v ωk Trace iequality: v E E K v K + h K E v K K v K Proof of the Lower Boud Isert ψ K R K i L -represetatio with stadard elemet cut-off fuctios ψ K. Isert ψ E,ϑ R E i L -represetatio with squeezed face cut-off fuctios ψ E,ϑ ad ϑ = ε 1 h 1 E α E = mi { 1, ε 1 h 1 E β 1 F 1 E,K F E,K }. Φ ϑ 1/ 3 / 3 Estimatio of the cosistecy error I M R hu I I MR h u I = R h u I, I M v sup v H0 1Ω v S u I, I M v = sup v H0 1Ω v Streamlie diffusio ad iterior pealty methods: I MR h u I cη h Local projectio scheme ad subgrid-scale approach: I M v ker κ M ad I M v ker Π hece I M R hu I = 0. A Posteriori Error Estimate Defie the space-time error estimator by: =η 1 τ {}}{ η = τ 1 η h }{{ + u T } + Φ + η τ. }{{} spatial temporal The } N e X0,T c { u 1 I 0 π 0 u 0 + η, η c e Xt 1,t. =1 c, c oly deped o the polyomial degrees ad the shape parameters of the partitios T. 3/ 3 4/ 3

7 Overview The adaptive Algorithm yields a solutio such that, for all times, its error is below a prescribed tolerace eps. The algorithm cosists of several modules that coarse ad refie spatial meshes. A a priori boud for the eergy of the discrete solutio guaratees that the fial time is actually reached withi a fiite umber of time-steps. The coarseig of spatial meshes leads to a icrease i the eergy of the discrete solutio which must effectively be cotrolled. Results hold for reactio-diffusio problems with domiat diffusio, ie. a = 0 ad β c r ε. Adaptive Space-Time Algorithm 1. Give τ 0 ad eps, choose ε 0, ε h,τ, ε such that ε 0 + T ε h,τ + ε = eps. Set = 0.. Determie T 0 ad π 0 u 0 such that u 0 π 0 u 0 ε Set τ = ε / f Ω 0,T + π 0u 0 miimal time-step. 4. Icremet by 1 ad set τ = mi{τ 1, T t 1 }. 5. Determie T = COARSENu 1,. 6. Determie u T, τ, T = ADAPTu 1, τ, T,, ε h,τ ad compute the eergy-icremet η, = π u 1 u 1 1 u τ π u If η, 0, go to 8. Otherwise determie T = REFINEη,, T, ad go to If t 1 + τ = T, stop. Otherwise go to 4. 5/ 3 6/ 3 Module COARSEN Module REFINE COARSENu 1, produces a partial coarseig of. This module may ot be based o a error idicator at all; its output may be idepedet of u 1 ad may equal. This module should remove as may degrees of freedom as possible while keepig the differece of u 1 to a suitable iterpolatio i the resultig fiite elemet space at a moderate size. This module is used i two differet ways depedig o its iput argumets. Give a partitio T ad a error idicator η = K η K 1, REFINEη, T produces a ew admissible partitio such that at least oe elemet i the subset argmax K T η K of T is refied. Give a partitio T, a associated error idicator η ad a secod partitio T, REFINEη, T, T has the same effect as REFINEη, T with the additioal coditio that at least oe elemet of T \ T which has previously bee coarseed is refied. 7/ 3 8/ 3

8 Module ADAPT 1. Solve the discrete problem for T ad τ ; compute η τ, η h.. If ητ + ηh ε h,τ, stop; otherwise go to If ηh > η τ, set T = REFINEηh, T ad go to 1; else set η 1 = u π u 1 ad η = 4. If τ > τ go to 5; else go to If η 1 > η, set τ = max{ 1 τ, τ }; else set T = REFINEη h, T, ad go to If ηh + η ε h,τ, stop; else go to 7. u 1 π u If η h > η, set T = REFINEη h, T ; else set T = REFINEη h, T,. Go to 1. Covergece Proof Difficulty: The goal η eps ivolves all times while, at ay itermediate time, oly iformatio up to that time is available. Basic steps: Prove a a priori boud for the discrete eergy depedig oly o the give data. Usig covergece results for statioary problems, prove the termiatio of the module REFINE. Prove the termiatio of the module ADAPT. 9/ 3 30/ 3 Cocludig Remarks Cocludig Remarks Modificatios ad Ope Problems Istead o parabolic eergy estimates, oe ca also base the a posteriori error aalysis o properties of the evolutio operator ad elliptic recostructio. This is a alterative way to decouple the temporal ad spatial error. For the spatial error oe may also use other a posteriori error idicators, e.g. auxiliary local discrete problems, Hdiv-liftigs,.... These, however, may ot be robust. Mildly oliear problems may be hadled similarly. The results, however, are less complete ad are based o the assumptio that the variatioal solutio is a regular oe i the sese of the implicit fuctio theorem. Cotrary to statioary problems, for time-depedet problems, the optimality of the adaptive algorithm is completely ope. Refereces Hadout of this talk C. Kreuzer, C. A. Möller, A. Schmidt, K.-G. Siebert Desig ad covergece aalysis for a adaptive discretizatio of the heat equatio IMA J. Numer. Aal. 3 01, o. 4, L. Tobiska, R. Verfürth Robust a posteriori error estimates for stabilized fiite elemet methods IMA J. Numer. Aal , o. 4, R. Verfürth A Posteriori Error Estimatio Techiques for Fiite Elemet Methods Oxford Uiversity Press / 3 3/ 3