Publised in Journal for Computational and Applied Matematics, pp. 192-208, 2015 Variational Localizations of te Dual Weigted Residual Estimator Tomas Ricter Tomas Wick Te dual weigted residual metod (DWR) and its localization for mes adaptivity applied to elliptic partial differential equations is investigated. Te contribution of tis paper is twofold: first, we introduce a novel localization tecnique based on te introduction of a partition of unity. Tis new tecnique is very easy to apply, as neiter strong residuals nor jumps over element edges are required. Second, we compare and analyze (teoretically and numerically) different localization tecniques used for mes adaptivity wit respect to teir effectivity. Here, we focus on localizations in variational formulations tat do not require te evaluation of te corresponding differential operator in te classical strong formulation. In our matematical analysis, we sow for different localization tecniques (establised metods and our new approac), tat te local error indicators used for mes adaptivity converge wit proper order in te error functional. Several numerical tests substantiate our teoretical investigations. 1 Introduction In tis work, we investigate te dual weigted residual metod (DWR) and its localization for mes adaptivity applied to elliptic partial differential equations. Our goal is twofold: First, and most important, we introduce a new localization tecnique, given in weak form tat avoids bot te evaluation of strong residuals and jump terms over element edges. Tis metod is easy to implement and terefore suitable for coupled multipysics systems wit many different equations. Te second aim is ten to analyze different localization tecniques wit respect to teir effectivity. For some establised localization tecniques, tis as not yet been accomplised. Te DWR metod allows for estimating te error u u between te exact solution u V (for a function space V ) of te PDE and its Galerkin solution u V V in general (error) functionals J : V R. Tese functionals can be norms but also more general expressions, like point-values, (local) averages or tecnical expressions like (in te case of fluid dynamics) lift- or drag-coefficients. Error estimators based on te DWR metod always consist of residual evaluations, tat are weigted by adjoint sensitivity measures. Tese sensitivities are te solution to adjoint problems tat measure te influence of te error functional J. Institute for Applied Matematics, University of Heidelberg, tomas.ricter@iwr.uni-eidelberg.de Institute for Computational Engineering and Sciences, Te University of Texas at Austin, Austin, Texas 78712, United States twick@ices.utexas.edu 1
Te DWR metod goes back to Becker & Rannacer [8, 10] and is based on te pioneering work by Eriksson, Estep, Hansbo and Jonson [20]. It as been furter developed by various researcers [32, 1, 23] and as been applied to a vast number of application problems including fluid-dynamics [7], structural dynamics, and furter to complex multipysics problems like cemically reactive flows [13] or fluid-structure interactions [24, 39, 34, 38, 37]. A completely different field, were te use of strong residuals is to be avoided is kinetic teory, see [26] for an application of goal oriented error estimation to Boltzmann-type equations. In tis contribution to te DWR metod, we focus on more principle questions tat arise in its application. First, te adjoint weigts entering te error estimator usually must be approximated, as tey involve te unknown exact solution z V of an adjoint problem. Section 3 provides an overview of different approximation tecniques commonly used. Second, if used for adaptive mes refinement, te error estimator η J(u) J(u ) must be localized to positive error indicators η i η i, (1) wic describe te local error contribution η i of a mes element or a mes node, and tat can be used to establis adaptive mes refinement scemes. In te central Section 4, we describe different localization tecniques for te DWR estimator and discuss teir effectivity: a localization is called effective, if te sum of local indicators do not overestimate te error. Error indicators, tat igly overestimate te error will lead to adaptive meses, wic do not fit to te problem. Usually, for adaptive metods, one aims at sowing effectivity, suc tat te estimator bounds te error from below and above c 1 η i (u u ) c 2 η i. (2) i We cannot expect suc a sarp result, as we are not looking at norms only, but at errors in general functionals J( ). Te DWR estimator is an error approximation η J(u) J(u ), but usually not a rigorous estimate. Usually, it is straigtforward to bound te indicators by te estimator from below i η i η i. (3) Te main contribution of tis work is to provide insigt to te opposite direction. We are not able to bound te sum of indicators by te estimator η or even te functional error J(u) J(u ) itself, but we can sow, tat te error J(u) J(u ) and te indicators i η i satisfy a common upper bound. Tis as not been accomplised for some commonly used localization tecniques. Finally, we introduce a novel localization tecnique, tat is strikingly simple in its application and also permits a very simple proof to sow te effectivity of indicators (witin te limits just discussed). In Section 5, several numerical test cases are presented to discuss te performance of te different localization strategies. Finally, in Section 6, we conclude wit some furter remarks. Let us begin in te following second section by gatering te notation and sortly introducing te dual weigted residual metod for error estimation. 2 Te Dual Weigted Residual Metod for Error Estimation By Ω R d wit d = 2, 3 we denote a domain wit polygonal or polyedral domain. On Ω, we denote by (, ) te L 2 -inner product and by te corresponding L 2 -norm. By H r+1 (Ω) we denote te space of Lebesgue functions wit square integrable weak derivatives up to degree r + 1. In particular, by V := H 1 0 (Ω) we denote te space of H1 (Ω) functions wit trace zero on te boundary Ω. 2 Publised in Journal for Computational and Applied Matematics, pp. 192-208, 2015
2.1 DWR for te Poisson problem and linear goal functionals Next, by u V we denote te solution of te Poisson equation ( u, φ) = (f, φ) φ V, (4) for a given rigt and side function f H 1 (Ω). We consider te case of omogenous Diriclet boundary conditions on Ω only. Next, we denote by V := V (r) V a finite dimensional, piece-wise polynomial of degree r finite element subspace and by u V te finite element solution ( u, φ ) = (f, φ ) φ V. (5) Here we only consider finite element spaces of polynomial degree r 1 on sape-regular triangulations Ω, suc tat tere exists an interpolation operator i : V V on every element K Ω k (u i u) K c in r+1 k K r+1 u K u H r+1 (K), k = 0, 1, 2, (6) wit K := diam(k) and were K is te L 2 -norm on K. Te interpolation constant c in depends on te polynomial degree r and te triangulation Ω. Furter, on element boundaries K, we use te estimate u i u K c in r+ 1 2 r+1 u K, (7) wit = max K K. Adaptive meses are realized wit anging nodes, see [25] for details on te construction. We assume, tat te problem data, e.g. rigt and side f and domain Ω are suc, tat te following two a priori error estimates old for te finite element solution (u u ) c r r+1 u, u u c r+1 r+1 u. (8) For linear elements wit r = 1, tis is given for f L 2 (Ω) on polygonal convex domains or if te boundary is smoot (piece-wise C 2 ) wit only convex corners [21]. By J : V R we denote a linear continuous functional and by z V we denote te adjoint solution to (4) ( φ, z) = J(φ) φ V. (9) Existence and uniqueness of tis adjoint solution follows by standard arguments. Te regularity of z V depends on te regularity of te functional J. For J H 1 (Ω) it olds z H 1 (Ω). Given a more regular functional like te L 2 -error J(φ) = e 1 (e, φ) wit J L 2 (Ω), it olds z H 2 (Ω) on suitable domains (convex polygonal or smoot boundary wit C 2 -parametrization). By z V V we denote te corresponding adjoint finite element solution ( φ, z ) = J(φ ) φ V. (10) Key relation in te context of te dual weigted residual metod, see Becker & Rannacer [8, 10], is te following error identity wic is only based on Galerkin ortogonality by plugging-in i z: J(u) J(u ) = (f, z i z) ( u, (z i z)). (11) Te error in te functional J(u) J(u ) can be expressed in terms of a residual, tat is weigted by (te local) adjoint sensitivity information z i z. Furter, a second adjoint error identity is directly given as by introducing z ; and Galerkin ortogonality of te adjoint equation can be used suc tat J(u) J(u ) = J(u i u) ( (u i u), z ). (12) Publised in Journal for Computational and Applied Matematics, pp. 192-208, 2015 3
Here, te residuals of te adjoint equation are weigted wit te primal interpolation error. Te two error representations involve as unknown parts in te weigts te primal solution u V or adjoint z V solution. Section 3 deals wit approximation tecniques for te evaluation of tese interpolation weigts. Tis approximation is required to obtain an usable error estimator. Ten, in Section 4 we come to te localization of te error representations and te definition of different local error indicators. For te following we collect some useful inequalities. All tese inequalities are given for twodimensional domains. First, on a mes element K Ω, we frequently use te trace-inequality u K c tr 1 2 K and in addition, we recall te inverse estimate ( u K + K u K ), (13) 0 s k : k u K c s k K s u K v V, (14) wic - on sape regular meses - easily follows by equivalence of norms in discrete spaces, see [15]. 2.2 DWR for nonlinear problems and nonlinear functionals Te teory presented above is limited to linear partial differential equations and linear functionals. Here, we sortly recapitulate te full nonlinear DWR teory as presented by Becker & Rannacer [10]. Let J : V R be a differentiable error functional and let a( )( ) be a differentiable semilinear form, wic is linear in te second argument. Let u V be te solution to te nonlinear problem and let z V be te solution to te linearized adjoint problem a(u)(φ) = (f, φ) φ V, (15) a (u)(φ, z) = J (u)(φ) φ V, (16) were by a (u)(, ) we denote te Gâteaux derivative of a( )( ) in u V, and by J (u)( ) te Gâteaux derivative of J( ) in u V. Ten, it olds te following mixed error representation J(u) J(u ) = 1 2 { (f, z i z) a(u )(z i z) } + 1 2 { J (u )(u i u) a (u )(u i u, z ) } + R (3) (u u, z z ), (17) were bot primal and adjoint residual appear, eac tested wit interpolation weigts coming from te oter problem. Te error identity (17) includes a remainder term R (3), tat is of tird order in te errors u u and z z and stems from te application of te trapezoidal quadrature rule, see Becker & Rannacer [10]. We notice tat te primal estimator (11) is still valid for nonlinear problems, it is owever disturbed by a second order error term R (2) (u u, z z ). If a( )( ) describes a bilinear form, e.g. if we consider linear equations and if te goal functional J( ) is linear, te mixed error identity (17) is exact wit R (3) = 0 and te adjoint problem is defined by (9). As primal (11) and dual (12) error identities are exact in te linear case, te mixed error formula (17) is an equivalent formulation. Wile all tese error representations are exact for measuring te error, tey lead to different error indicators wic may produce adaptive meses of different quality. See Sections 4 and 5 for discussions on tis point. 4 Publised in Journal for Computational and Applied Matematics, pp. 192-208, 2015
In contrast to oter classical a posteriori error estimators for te energy norm, te DWR metod is not an estimator in te strict case. Considering linear problems, te DWR metod is an error identity. However, as discussed in te following section, it is not possible to evaluate te error relation (even in te linear case), as te weigts depend on te exact values of primal and dual solution. Hence, computational approaces deliver an approximation of te error identity η : η (u, z ) J(u) J(u ). Te quality of tis approximation procedure can be measured by te effectivity index eff, defined as eff := η(u, z ) J(u) J(u ). (18) For eff 1, te estimate is asymptotically exact. As functional errors carry a sign, no absolute values may be used in defining te effectivity index. Te following discussion sows, tat cancellation effects by different signs are te major cause for difficulties connected to te localization of te DWR metod. Local indices to be used for adaptivity will ave to be positive measures of te error contribution. 3 Approximation of te Weigts For evaluation of te error identities (11), (12) and (17), we need approximations of te interpolation errors z i z and/or u i u. Obtaining suc an approximation is te critical part in te DWR framework tat stands in te way of strict reliability. Examples ave been constructed [31], were te DWR estimator underestimated te error due to coarse approximation of te weigts z i z and u i u. A remedy is only given by spending sufficient effort on te estimation of tese weigts on fine meses [10, 17] or an additional control of te approximation error in z i z and u i u [31]. For simplicity of presentation, we consider te case of linear goal functionals trougout tis section. 3.1 Classical approximation of te DWR estimator First, we consider a classical approac, tat is based on applying Caucy-Scwarz inequality locally on every element for getting strict upper bounds, see e.g. [8, 9, 6]. Wit integration by parts on every mes element K Ω it olds: J(u) J(u ) = ( f + u, z i z ) K + n u (z i z) ds. (19) K Ω Following te usual procedure for residual based error estimators [36], we combine eac two boundary integrals over element edges to a normal jump and proceed wit Caucy Scwarz to get J(u) J(u ) K Ω K ( f + u K + 1 1 2 2 K [ ) ( nu ] K z i z K + 1 ) 2 k z i z K, (20) }{{}}{{} =:ρ K (u ) =:ω K (z) were by [ n u ] we denote te jump of te u derivative in normal direction. On te outer boundary Ω, we set [ n u ] = 0. Te residual part ρ K only contains te discrete solution u and te problem data. A similar estimator can be derived based on te adjoint form (12), were te residuals of te adjoint equation ρ K (u, z ) are weigted wit primal interpolation errors ω K (u). Publised in Journal for Computational and Applied Matematics, pp. 192-208, 2015 5
Remark 1 (Evaluation of te strong residuals). Te evaluation of te classical error localization can bring along enormous computational costs, as for iger order finite elements te assembly of second order differential operators on general mes elements is computationally expensive. Considering nonlinear problems, we furter ave to evaluate te adjoint residuals ρ K (u, z ). For some complex problems, e.g. like fluid-structure interactions [24, 39, 34], te strong adjoint residual formulation as not even been derived yet. Suc complex nonlinear problems are our key motivation in deriving localization tecniques tat do not require te evaluation of classical residuals. Remark 2 (Strong residuals for C 1 -continous approximations). Edge terms in strong residuals only appear for H 1 -conforming finite elements. By using C 1 -continuous approximations, it olds [ n u ] = 0 on te element edges in (19), suc tat no edge terms appear in (20). Traditionally, C 1 -continuous elements found little usage due to teir ig computational effort. Tis owever canges in te context of isogeometric analysis, see [19] or [27] wit application to goal oriented error estimation. Te error weigts ω K (z) involve te unknown adjoint solution z V, wic as to be approximated. Here, two possible approaces exist: one can directly approximate te interpolation error z i z using available discrete quantities only. Tis approac is described in te following section. As an alternative, one could first apply an interpolation error estimate ω K (z) c in r+1 r+1 z P (K), (21) were P (K) is te patc of all tose elements K Ω tat sare a common boundary wit K Ω, followed by an approximation of te (r + 1)-t derivative r+1 r+1 z P (K) r+1 K r+1 z L (P (K)), (22) wic is based on discrete recovery concepts. Tis approac is typical for te gradient recovery error estimator and it is directly applicable to te DWR metod. We refer to te literature [40, 41] Te drawback of te classical DWR-approac is due to te application of te Caucy-Scwarz inequality tat rules out possible cancellation effects by local ortogonality and interpolation error estimates, wic bot bring along unknown constants. Usually, error estimators based on tis approac result in an over-estimation of te true error. We provide examples in Section 5. 3.2 Variational approximation of te DWR estimator Te second possibility for te evaluation of te weigts is by a direct approximation of te interpolation error witout prior estimates. If an approximation for z i z is available, it can be bot used for te error estimate given in te classical form (20), but also directly for te error identities (11), (12) and (17), given in weak formulation. Here, every discrete approximation to ψ z i z must be finer tan te trial space V, as te residual is ortogonal on V. A first obvious possibility is to simply solve for an approximation z in a iger accurate space, e.g., z /2 V /2 : ( φ /2, z /2 ) = J(φ /2 ) φ /2 V /2, (23) were z i z z /2 i z /2. Alternatively, it would be possible to solve for z (2) V (2r) in a finite element space of iger polynomial degree. Bot approaces work very well in application and usually yield optimal error estimators wit te effectivity index eff (18) going to one: eff (u, z ) = (f, z(2) i z (2) ) ( u, (z (2) i z (2) )) 1 ( 0). (24) J(u) J(u ) 6 Publised in Journal for Computational and Applied Matematics, pp. 192-208, 2015
We refer te reader to Section 5 for an example. In te practical application, owever, tis approac is not feasible, since it means, tat only for getting a reliable and effective error estimator we ave to spend very ig numerical effort [10]. A modification of tis approac is to define local subproblems, tat live on a small subset of elements eac, and tat can be solved efficiently and in parallel using iger order finite elements. A tird possibility tat goes witout te solution of additional iger order problems uses a reconstruction of te already computed discrete approximations u V and z V. For tis reconstruction, te discrete solutions are simply reinterpreted using a iger order basis. First, we assume, tat te finite element space V is constructed in a patced manner, suc tat eac element is part of four elements arising from one common fater-element P Ω 2. Suc a patced finite element set allows for a reinterpretation of te finite element basis by combining four r-t order elements to one element of order 2r (in 2d). As te finite element spaces V (r) and V (2r) 2 ave te same number of unknowns in te same Lagrange points, a iger order reconstruction is directly given by an excange of te basis: z = i z i φ i i z i φ (2r),i 2 =: z (2), (25) were z R N stands for te coefficient vector and by φ (2r),i 2 we denote te basis functions of te finite element space of double degree on a mes wit double mes size, see Fig. 1. Details on te application of tis reinterpretation process on unstructured meses are given by Carpio et al. [16]. Tis reconstruction strategy is igly reliable and effective for a large class of problems, see [13, 33, 35]. Similar to gradient recovery error estimators, it is based on super-approximation results obtained by error expansion tecniques, see e.g. Blum and coworkers [12, 11]; owever it can not be rigorously sown on adaptive meses. Te success in numerical examples owever works in favor of tis costefficient approac. 4 Localizations of te error identity and effectivity of localizations In tis section, we discuss wit te localization of te error estimator. In most representations, te error estimator allows for a direct splitting into a sum J(u) J(u ) η = i η i. Te absolute values of te local quantities η i are te indicators used for refinement. Our discussion follows two goals: te localization procedure sould be simple in terms of implementation and numerical effort. Second, te localization sould be effective, suc tat te sum of local error indicators does not eavily overestimate te error. Similar to te effectivity index (18), we define te indicator index to measure te quality of te localization process: i ind := η i J(u) J(u ). (26) It is not possible to reac strict effectivity wit ind 1 in te context of goal-oriented errors. Te functional error J(u) J(u ) as a sign, and ence te error can vanis, altoug te solution sows a very large approximation error, for example by symmetry reasons. Te local estimator values η i migt ave canging sign, suc tat i η i may be a strong over-estimation. However, we aim at strategies, were ind is uniformely bounded in. By a priori estimates, a worst case bound for te functional error is given by te product of primal and dual energy errors J(u) J(u ) c (u u ) (z z ), (27) wit a constant c > 0 tat depends only on te continuity of te variational formulation, and wit c = 1 in te case of te Poisson equation. Tis bound is not sarp, as it neglects possible ortogonality Publised in Journal for Computational and Applied Matematics, pp. 192-208, 2015 7
of primal and dual errors. A functional error J(u) J(u ) can be zero, even if te energy errors are substantial. In te following, we call error indicators effective, if tis worst case estimate also olds for te indicators η i c (u u ) (z z ), (28) i wit a constant c > 0 tat may depend on te bilinear form of te equation, te finite element space and te sape regularity of te mes, but tat sould be robust wit respect to te mes size parameter. For practical approximations of te error identity, te weigts are given by interpolation errors. Hence, te concept of effectivity defined in (28) must be sligtly altered to also allow for bounds in te interpolation error (u i u) and (z i z). Finally, wenever formulations based on strong residuals are considered, a pure H 1 -estimate is not sufficient. Instead, we define a mes-dependent norm, tat is - assuming enoug regularity u H 2 (Ω) - equivalent in terms of convergence (u u ) u u in : φ := φ 2 Ω + K Ω { } 2 K 2 φ 2 K + 2 K φ 2 K 1 2. (29) Using tis norm, we can now define our concept of effective error indicators: η i c max{ u u, u i u } max{ z z, z i z }. (30) i Every localization can only be as accurate, as te approximation of te weigts permits as discussed in te previous section. For te following discussion, we assume tat te approximation of te weigts is sufficiently accurate. On coarse meses, tis is a simplification, as sown by te discussion of Nocetto et al. [31]. 4.1 Localization based on te classical (strong) formulation Te typical localization procedure [8, 10] for residual based error estimators is based on te classical formulation of te error estimator (20) by defining local element-wise indicator values η K := ρ K ω K. Reliability of tese indicators depends on te approximation properties of te interpolation weigts z i z. Te question of effectivity is a bigger concern, as te Caucy-Scwarz inequality as been used. We know, tat a functional error J(u) J(u ) can cange its sign and pass troug zero. Tis beavior cannot be represented by te classical localization. However, given te more subtle definition of effectivity (30), it olds: Lemma 1 (Effectivity of te localization based on te classical residual). Let u, z V H 2 (Ω) be te solution and adjoint solution, respectively. Furtermore, let u, z V = V (r) be te corresponding finite element solutions of degree r. Te classical error indicators η K = ρ K ω K given by (20) are effective, i.e., ρ K ω K c u u z i z, (31) K Ω K wit a constant c > 0. Proof. (i) We split te residual part ρ K = ρ i K + ρe K into inner part ρi K = f + u K and edge part ρ e K = 1 2 K [ nu ] K. First, it olds by standard a priori analysis using u = f: ρ i K = f + u K = (u u ) K 2 (u u ) K. (32) 8 Publised in Journal for Computational and Applied Matematics, pp. 192-208, 2015
On every edge, it olds for u V H 2 (Ω), tat [ n u] = 0 and ence wit (13) ρ e K = 1 2 K [ nu ] K = 1 2 K [ n(u u )] K ) c tr 1 K ( (u u ) K + K 2 (u u ) K. (33) Combining (32) and (33) gives te first part of te estimate. (ii) Next, for te weigts, we get by using (13) ω K = z i z K + 1 2 k z i z K c K ( (z i z) K + 1 K z i z K ). (34) (iii) Te result follows by combining te estimates for ρ K and ω K using Hölder s inequality. We can bound te sum of error indicators by te product of approximation error in te primal solution and interpolation error of te adjoint solution. Using te dual form of te error identity (12) we would get te opposite result. Starting wit te mixed identity (17) results in te sum of bot estimates, or - in terms of te concept of effectivity (30) - in te maximum value of approximation and interpolation error. For an evaluation of tis indicator (at a iger polynomial degree) one must assemble te strong residual of te equation, namely f Lu, were L is te second order differential operator in classical formulation. Tis evaluation can be very costly, wen parametric finite elements of iger order are used, see Remark 1. Finally, aving te full nonlinear case in mind, see Section 2.2, te error indicator consists of a primal and adjoint part, were ηk = ρ K ω K, wit ρ K = J (u ) L (u )z K, (35) were L (u ) is te linearized adjoint operator at u. For complex coupled problems it is sometimes not possible to assemble te adjoint operator in strong formulation, see e.g. [34]. 4.2 Localization based on filtering te variational formulation Braack and Ern [14] proposed a localization tecnique for te DWR estimator tat is fully based on te variational formulation and firmly linked to te approximation of te weigts using a iger order representation z V of te adjoint solution z V (r) (r), were V is te finite element space of degree r, see Section 3.2. Given a patced mes and finite element space setup, we can define te space V (r) 2 V (r) of double mes spacing and introducing i 2 (z i z) = 0. As te two interpolation operators commute i 2 i = i i 2 (considering a standard nodal interpolation), it olds z i z i 2 (z i z) = (id i 2 )(id i )z = (id i )(id i 2 )z = (id i )π 2 z, (36) were te patc-wise filtering operator is π 2 := id i 2 ; introduced by Braack and Ern. It remains to apply te approximation of te weigts as described in Section 3.2: Te operator i : V (r) V (2r) := V 2 is te patc-wise interpolation into te space of double polyno- on te patc mes, and for φ i V (r) it olds i i φ i = φi. Let P = {K 1,..., K p } := i φ i V (2r). Ten it olds, mial degree V (2r) 2 J(u) J(u ) ( f, (i id)π 2 z ) ( u, ( (i id)π 2 z ) ). (37) be a patc of elements K Ω and φ i, 2 (i id)φ i = φi, i φ i,, (38) Publised in Journal for Computational and Applied Matematics, pp. 192-208, 2015 9
suc tat for k = 0 and k = 1 te interpolation estimate gives k K k (i id)φ i P = k K k (φ i, i φ i, ) P c K φ i, P c K. (39) Applied to z, te filtering operator π 2 z is a strictly local algebraic process acting on te coefficient vector z R N : z i 2 z = z i (φ i i 2φ i ) =: (π 2 z) i φ i, (40) i i as te interpolations of te finite element bases functions i 2 φ i can be linearly combined by φj. Finally, Braack and Ern defined te local error indicators as J(u) J(u ) { ( ) ( ( ) ) } f, (i id)φ i u, (i id)φ i (π 2 z) i. (41) i }{{} =:ηi π We refer to Fig. 1 for a sketc of te different interpolation operators i 2, i and te filtering operator π 2 employed for tis approac. We notice tat tese local indicators ηi π are node-wise and not element-wise contributions. Te error indicators inerit te patc structure and it olds, tat η i = 0 for every second degree of freedom (in a tensor-product way), e.g. for all degrees of freedom tat belong to Lagrange points x i of te finite element space V as well as te finite element space V 2. If tese indicators are to be used for mes-refinement, a first step as to be a summation of all indicators associated to one patc. Ten, refinement is carried out on te patc-mes. Lemma 2 (Effectivity of te algebraic filter-approac). Let u V H 2 (Ω) and z V be solution to primal and adjoint problem, u V and z V be te corresponding finite element solutions of degree r. Te filtering indicator defined by (41) is effective N ηi π ( c u u (z z ) + (z i 2 z) ), (42) i=1 wit a constant c > 0 and were i 2 : H 2 (Ω) V (r) 2 is te interpolation operator into te finite element space of te same degree r on te patced mes wit mes-size 2. Proof. We consider te two dimensional case only. Te tree dimensional case follows by similar arguments. All indicators η i belonging to Lagrange points x i Ω 2 vanis. We must distinguis between indicators η e belonging to degrees of freedom on edges of a patc x e P and indicators η m belonging to inner points x m P, see Fig. 2. (In te tree dimensional case we would ave to add all indicators η f belonging to faces of elements.) Figure 1: Patc of four elements. Discrete solution u V (r), reconstruction i u V (2r) 2, coarse-mes interpolation i 2 u V (r) 2 and fluctuation operator π 2u (going from left to rigt). 10 Publised in Journal for Computational and Applied Matematics, pp. 192-208, 2015
(i) Inner points. Let x m P be te midpoint of a patc P. Ten, te basis functions φ i, i φ i and i 2 φ i all ave teir support in P. Let z = π 2z (x m ). { ( ) ( ( ) ) } ηi π = z f, (i id)φ i u, (i id)φ i P P ( ) = z f + u, (i id)φ i P K P n u, (i id)φ i were K P are all in te patc. On K P, te boundary integral vanises. By e i P we denote te interiour edges between te elements. Tey appear twice, suc tat using te normal jumps it olds ( ) ηi π = z f + u, (i id)φ i [ n u ] [ n u], (i id)φ i P e i. (44) e i P As in te proof to Lemma 1, we added [ n u] = 0 on all inner edges e i P : { ηi π z (u u ) P (i id)φ i P + c 1 K ( (u u ) P + K 2 (u u ) K ) K, (43) ( (i id)φ i K + K (i id)φ i K) }. (45) We proceed wit (39) to get } ηi π c z { K 2 (u u ) P + (u u ) P. (46) It remains to estimate te discrete fluctuation z. By inverse estimates, it olds z π 2 z L (P ) c 1 K π 2z P c π 2 z K. (47) Tis last inverse estimate works out, as for π 2 z = 0 it must follow tat π 2 z is constant on P and as z V 2 finally π 2 z = 0. By introducing ±z, and using te stability of te interpolation operator i 2, te estimator sum gets ηi π c (u u ) 2 + 2 K 2 (u u ) 2 K i K Ω 1 2 ( (z z ) + (z i 2 z) ). x m x e Figure 2: Te filter based error indicator on two patces P Ω. It olds η i = 0 for all outer points x i Ω 2. Splitting of te estimator into inner nodes x m and edge nodes x e. Publised in Journal for Computational and Applied Matematics, pp. 192-208, 2015 11
(ii) Boundary points. Let x e P be a node on te edge of two patces P 1 and P 2. Ten, it olds { ( ) ( ( ) ) } ηi π = z f, (i id)φ i u, (i id)φ i, (48) P 1 P 2 P 1 P 2 were again, z = π 2 z (x e ). Here, te test-function φ i and its interpolants ave teir support in te joint patc P 1 P 2 and all terms can be estimated as in step (i). Remark 3. Tis estimate sligtly differs from te effectivity concept (30). Te patc-structure enters by te interpolation (z i 2 z) onto te coarse mes. Tis term owever is of te same order in as z i z, just carrying a larger constant. Te benefit of tis localization strategy is te simplicity of implementation, if a patc structured mes is available. For evaluation of te estimator, two residuals must be calculated, r i := (f, φ i ) ( u, φ i ), r i := (f, φi, ) ( u, φ i, ), i = 1,..., N, (49) te first using te standard basis, te latter wit a iger order basis. Ten, given te filtered coefficient vector π 2 z, te estimator is given by te algebraic computation η π i = (r i r i )(π 2 z) i, i = 1,..., N. (50) All tese ingredients are usually available in standard finite element libraries. One drawback of tis localization is its interpretation, as te indicator values ηi π are neiter given in an element-wise way, nor strictly in a node-wise manner, as ηi π = 0 on all coarse Lagrange points. Te indicators must instead be regarded in a patc-wise sense wic comes at te cost of loosing granularity. Tis migt be an issue regarding 3D simulations as it will lead to meses, wic are up to a factor of 8 more complex as te optimal ones. 4.3 Localization using partition of unity (PU) Finally, we introduce a new localization approac based on te variational formulation tat combines te simplicity of te filter based approac (as it is given in terms of variational residuals) wit a very simple structure, wic does not require patced meses. Localization is simply based on introducing a partition of unity (PU) ψ i 1 into te error identity (11): J(u) J(u ) = N { } (f, (z i z)ψ i ) ( u, ((z i z)ψ i ). (51) } {{ } =:ηi PU i=1 Te resulting error indicators ηi PU are node-wise contributions of te error. Mes adaptivity can directly be carried out in a node-wise fasion: if a node is picked for refinement, all elements toucing tis node will be refined. Alternatively, one could also first assemble element wise indicators by summing up all indicators belonging to nodes of te element and ten carry out adaptivity in te usual element-wise way. Lemma 3 (Effectivity of te PU localization). Let u V be te solution to te Poisson equation, z V be te adjoint solution. u, z V = V (r) teir discrete counter-part. Furter, let ψ i 1 be a PU wit ψ i = O( 1 ). Te error indicators given by (51) are effective, i.e., 1 N ηi PU c (u u ) (z i z) 2 + 2 K z i z 2 K, (52) i=1 K Ω 2 12 Publised in Journal for Computational and Applied Matematics, pp. 192-208, 2015
wit a constant c > 0. Proof. Let supp(ψ i ) P i = K j for some elements K j Ω. It olds wit (f, φ) = ( u, φ) for all φ H 1 0 (Ω): η PU i = (f, (z i z)ψ i ) ( u, ((z i z)ψ i )) = ( (u u ), ((z i z)ψ i )) Pi (u u ) Pi ( (z i z) Pi ψ i L (P i ) + z i z Pi ψ i L (P i )). (53) Te result follows by using ψ L (P i ) = O( 1 ). In contrast to te classical localization and te filtering approac, te PU localization tecnique requires minimal regularity u, z H0 1 (Ω) only. A similar tecnique based on a PU as been used by Kuzmin & Korotov [28] to localize a strong residual formulation of te DWR estimator applied to 1D transport problems. Te introduction of a partition of unity into te strong formulation of te residual is also te fundamental basis for te family of flux-free error estimators [29, 18]. Here, te PU is used to define local sub-problems tat are used to construct robust error estimators. Te construction wit elp of a PU directly yields a localized form of te estimator. Tis tecnique is not only accurate for energy norm estimates but also robustly applied in te context of linear output functionals [29]. Furter, it is possible to design a convergent finite element metod based on flux-free error estimators [30]. In contrast to flux-free error estimators, we simply insert a partition of unity to localizing te standard DWR estimator. Realization As PU, we consider te space of piece-wise bilinear elements V (1) (witout restrictions on Diriclet boundaries) wit usual nodal basis {ψ i, i = 1,..., N (1) }. Te approximated local error indicator is ten given by η PU i := N (r) j=1 { (f, (φ (2),j 2 φ j )ψi ) Ω ( } u, ((φ (2),j 2 φ j )ψi )) z Ω j, (54) and it can be efficiently computed in an element-wise manner, as only few test-functions φ j, φ(2),j 2 and ψ i overlap on every element K Ω. On adaptive meses wit anging nodes, te evaluation of te PU indicator is straigtforward: First, te partition of unity is assembled on basis functions ψ i. In a second step, te contributions belonging to anging nodes are condensed in te usual way by distribution to te neigboring indicators, see [3] for details on andling anging nodes. Te benefit of tis localization tecnique is its simplicity and its accuracy according to Lemma 3 demonstrated in te numerical examples in Section 5. For te application, we only need evaluations of te rigt and side and te residual wit modified testfunctions. As PU we can simply use te standard nodal Lagrange basis of te continuous finite element space of lowest polynomial degree. Tis localization tecnique can be readily applied to general meses in two and tree dimensions. In contrast to te filtering approac, we do not require special mes structures, suc as patces. In particular for tree dimensional simulations, te use of patced meses can substantially increases te problem size. However, te problem of obtaining good approximations to te weigts z i z and u i u remains and ere, using reconstruction of patces still is one of te most efficient strategies. Te second advantage is te easy application of te localization to complex nonlinear systems, were te evaluation of strong residuals can be cumbersome. Once again we point out, tat te adjoint operator in strong formulation is not even known for some complex multipysics problems, see e.g. [22, 34]. Publised in Journal for Computational and Applied Matematics, pp. 192-208, 2015 13
Remark 4 (General elliptic problems). Te results stated in Lemmata 1, 2 and 3 can all be transferred to te case of general elliptic problems like transport reaction diffusion problems. We ave only used standard a priori results in te H 1 -seminorm and te L 2 -norm. Considering te general operator Lu := u + β u + αu, te adjoint operator reads L z := z β z + αz wit opposite transport direction. 5 Numerical Tests In tis final section, we substantiate our teoretical findings by tree numerical tests wit increasing complexity. In te first test, we use standard Poisson s problem wit a regular goal functional on a regular domain. Te second test case deals wit a low regularity problem on a L-saped domain wit singular rigt and side and functional. Finally, we consider a tree-dimensional nonlinear elasticity system to demonstrate te great flexibility and simple realization of te PU approac. In every testcase, we analyze different forms of te error estimator (primal, dual, mixed) and different localization tecniques and compare tem wit respect to estimator and indicator effectivities on bot uniform and adaptive meses. Te computations use quadrilateral and exadedra meses and are performed wit Gascoigne 3D [5] and deal.ii [4]. #el J(u u (1) ) η J(u u (2) ) η J(u u (3) ) η 16 8.11 10 4 8.08 10 4 2.23 10 6 2.23 10 6 5.16 10 8 5.14 10 8 64 2.04 10 4 2.04 10 4 1.77 10 7 1.77 10 7 3.20 10 9 3.19 10 9 256 5.11 10 5 5.11 10 5 1.34 10 8 1.34 10 8 0.20 10 9 0.20 10 9 1024 1.28 10 5 1.28 10 5 0.98 10 9 0.98 10 9 < T OL < T OL order 2.00 3.78 4.00 Table 1: Configuration 1: error and error estimator on uniform meses. From left to rigt: linear, quadratic and cubic finite elements. Te last line sows te estimated order of convergence. 5.1 Configuration 1: A regular Poisson example In tis first example, we consider Poisson s equation u = 1 on a unit square Ω := (0, 1) 2 wit a omogenous Diriclet conditions on Ω. As target functional we evaluate te average of te solution J(u) = u dx. (55) Tis functional corresponds to te adjoint problem z = 1, again wit z = 0 on Ω. Hence it olds u = z and te regularity of u, z H 3 ɛ (Ω) for ɛ > 0 is limited by te edges of te unit square [21]. For suc a regular problem we expect te a priori estimate Ω J(u) J(u ) c (u u ) (z z ) c min{2r,4 2ɛ}, (56) were r 1 is te polynomial degree of te finite element space V. By accurate computations on very fine meses using extrapolation we identify te reference value J = 0.03514425375 ± 10 10. We compute primal and adjoint solution z (r), u(r) V (r) by using finite elements of degree r = 1, r = 2 and r = 3. Te interpolation weigts are eiter approximated by using global finite element solutions u (2r) and z (2r) V (2r) of double polynomial degree, or obtained by local patc-wise reconstruction u (2r) 2, z(2r) 2 V (2r) 2, see Section 3. 14 Publised in Journal for Computational and Applied Matematics, pp. 192-208, 2015
Figure 3: Configuration 1: Visualization of te local error indicators on similar meses: (left) classical formulation, (middle) filtering approac, (rigt) PU. All computations wit biquadratic finite elements. 0.0001 1e-05 1e-06 1e-07 1e-08 relative error 1e-09 unknowns 1e-10 100 1000 N 3 uniform Q2 uniform Q3 adaptive Q3 N 2 10000 100000 # dof s J(u) J(u (3) ) ηpu eff PU 169 8.51 10 07 8.47 10 07 0.99 317 1.12 10 07 1.37 10 07 1.23 937 5.56 10 09 7.54 10 09 1.36 1 813 1.14 10 09 1.41 10 09 1.24 3 877 5.28 10 11 8.05 10 11 1.52 7 057 1.61 10 11 2.07 10 11 1.29 Figure 4: Configuration 1. Left: error slopes on uniform (Q2 and Q3 elements) vs. adaptive meses (Q3 elements). Rigt: error, estimator and effectivity index for adaptive Q3-elements. In Table 1 we sow te functional error J(u) J(u (r) ) and te estimated error η on a sequence of uniform meses for different polynomial degrees. It can be seen, tat te error estimator sows perfect effectivity eff 1 even on very coarse meses. As te problem and te functional are linear, all tree versions of te error identity; namely, primal, dual and mixed result in te same findings. Tis is also found numerically, ence, only one value η is given in te table. We find no difference, weter te weigts are approximated using iger order simulations or by te reconstruction process. Hence, just one value is given. Localization and adaptivity Table 1 furter sows, tat going beyond second order finite elements does not result in an increased approximation order on uniform meses. Tis is due to te limited regularity u, z H 3 ɛ (Ω). Hence, we next consider localization of te error estimator and adaption of te meses. For mes adaption, we follow a simple equalization strategy tat aims at balancing te element wise error indicators, suc tat a mes element K Ω is being refined, if te error indicator η K is above average. Wile te classical localization tecnique described in Section 4.1 directly gives element-wise indicators η K, we agglomerate te adjacent node-wise values to element wise values in te case of te PU approac in Section 4.3 and to patc-wise values for te filtering approac of Section 4.2. In Fig. 3 we plot te error indicators η K, ηi π and ηpu i as function over te domain Ω. By construction, te classical indicator values η K are all positive, wile te two variational settings ηi π and ηi PU sow Publised in Journal for Computational and Applied Matematics, pp. 192-208, 2015 15
bot negative and positive values. Furtermore, te projection based indicators ηi π sow patc-wise fluctuations, wile te PU approac yields smoot node-wise contributions. Next, in Fig. 4, te error slopes obtained on uniform and adaptive meses using quadratic and cubic finite elements are displayed. For refinement we used te PU localization. Furter, we indicate error, estimator and effectivity index eff on a sequence of locally adapted meses. Here optimal order of convergence in terms of unknowns wit respect to te relation N 2 is recovered. In addition, we observe good effectivities. For tis simple and regular problems, all tree localization tecniques result in te same finite element meses. Hence, we always considered te PU metod only. Lastly, localizations based on te primal, dual or mixed formulation all result in te same adaptive meses. 5.2 Configuration 2: Poisson problem wit low regularity As second test-case, we consider Poisson s equation on an L-saped domain Ω L = ( 1, 1) 2 \ ( 1, 0) 2, were te rigt and side is given by a Dirac in x 0 = ( 0.5, 0.5) u = δ x0 in Ω L, u = 0 on Ω L. (57) As functional of interest, we consider te point evaluation in x 1 = (0.5, 0.5) suc tat J(φ) = φ(x 1 ). Te adjoint problem corresponds to solving Poisson s equation z = δ x1 wit a Dirac rigt and side in x 1. Bot te primal problem and te adjoint problem lack te required minimal regularity for te standard finite element teory, suc tat a regularization by averaging is required, e.g. by averaging over a small subdomain: J ɛ (φ) = 1 2πɛ 2 φ(x) dx, (58) x x 1 <ɛ were ɛ > 0 is a small parameter not depending on. As reference functional quantity we identify te value J = 2.134929 10 3 ± 10 7. (59) Due to limited regularity of primal and adjoint solution, we cannot expect ig order convergence. Adaptivity based on good localization is important for an accurate approximation. We start by comparing te different localization tecniques discussed in Section 4. Table 2 sows values obtained on a sequence of uniform meses using piece-wise bilinear finite elements. Here, we provide te number of mes elements, te error as well as effectivity index eff and indicator index ind, see (18) and (26), for te tree different localization tecniques based on te strong residual, te filtering approac and te PU metod. DoFs J(u) J(u Q1 ) effk ind K eff π ind π eff PU ind PU 48 1.45 10 4 1.80 1.80 1.06 2.92 1.06 1.74 192 4.66 10 5 2.04 2.04 1.24 3.27 1.24 2.19 768 1.46 10 5 1.98 1.98 1.08 2.80 1.08 2.03 3 072 5.31 10 6 1.83 1.83 1.02 2.28 1.02 1.77 12 288 2.02 10 6 1.66 1.66 1.02 1.85 1.02 1.50 Table 2: Configuration 2: calculations on uniform meses, comparing te effectivity index and te indicator index for te tree different localization tecniques using te classical formulation η K, te filtering approac ηπ and te PU tecnique ηpu. 16 Publised in Journal for Computational and Applied Matematics, pp. 192-208, 2015
N J J(u ) eff PU ind PU 12 7.98 10 5 0.87 0.95 48 1.25 10 5 0.81 0.82 84 4.98 10 6 0.86 0.87 240 1.85 10 6 0.86 0.86 276 6.67 10 7 0.96 0.97 N J J(u ) eff PU ind PU 12 4.15 10 5 0.87 0.96 48 6.39 10 6 0.83 0.83 84 2.43 10 6 0.85 0.87 120 8.91 10 7 0.92 0.97 324 2.80 10 7 1.17 1.16 Table 3: Configuration 2: effectivity of te estimator and te indicators (partition of unity) on adaptive meses wit N elements. Left: discretization wit piece-wise cubic, rigt wit piece-wise quartic finite elements. 0.1 uniform Q1 uniform Q2 uniform Q3 adaptive Q1 adaptive Q2 adaptive Q3 0.01 0.001 relative error 0.0001 unknowns N 3 N 2 N 1 100 1000 10000 100000 Figure 5: Configuration 2: relative error over number of unknowns for computations using linear, quadratic and cubic finite elements. Uniform vs. adaptive mes refinement. Te two approximation tecniques based on te variational formulation, η π and ηpu result in a better effectivity index. Here, te formulation based on te strong residual and application of Caucy- Scwarz sows a small overestimation of te error. For te local error indicators, we can only discover a small difference between te tree tecniques under investigation. Te PU metod yields a sligtly better constant, due to te prevention of local oscillation, tat is typical for te algebraic filtering approac. Table 3 sows te functional error, te estimator effectivity and te indicator effectivity on a sequence of locally refined meses using ig order finite elements. Localization and refinement is based on te PU metod. It sows, tat te estimator and te localization are igly accurate for finite elements of ig polynomial degree. In Fig. 5, we sow te convergence of te functional values on sequence of uniform and adaptive meses using linear, quadratic and cubic finite elements. First, we identify te necessity of local mes adaptation, as an increasing polynomial degree does not result in better approximation order on uniform meses. Using adaptivity and localization based on te PU metod, we recover te optimal order of convergence (based on te relation N = 2 ) for all polynomial degrees. We owever also see, tat using isotropic adaptive meses is not sufficiently able to resolve te singularities. Te optimal order is only recovered on very fine meses. Publised in Journal for Computational and Applied Matematics, pp. 192-208, 2015 17
Figure 6: Configuration 2: Comparisons of patced meses on refinement level 6 using primal (left), dual estimator (middle) and te mixed estimator (rigt). We compare from top to bottom te Q 1 and Q4 discretizations. In particular, te localization of te error estimator using iger order polynomials is observed. Ten, we use tis test-case to study te differences in te localization beavior of primal, dual, and mixed estimators. For linear problems, all tree versions of te error estimator (11), (12) and (17) are equivalent. Teir localizations owever will depend on a different weigting of approximation and interpolation errors as discussed in Section 4. By te pollution effect, wic is supposed to get stronger for iger order finite elements, see [2], we may experience different localizations. In Fig. 6 we sow locally refined meses obtained from localizations based on te tree different error representation formulas; primal, adjoint and mixed - all meses differ. Te refinement of primal and adjoint formulation is mirrored at te line x = y troug te midpoint. Te meses corresponding to te mixed formulation can be regarded as a union of te two one-sided error representations, leading to symmetric meses. Te effect of different meses gets stronger wit iger polynomial degree, as is expected by te analysis of te pollution effect. Even toug tis example is atypical, we stress te importance of a correct balancing of primal and adjoint residuals for adaptive mes refinement. 18 Publised in Journal for Computational and Applied Matematics, pp. 192-208, 2015
Figure 7: Configuration 2: Meses on refinement level 6 witout patc structure for Q 1 (left) and Q 4 (rigt) discretizations using te primal (non-symmetric) error estimator. 1 0.1 wit patc Q1 witout patc Q1 wit patc Q4 witout patc Q4 0.01 0.001 relative error N 1 0.0001 N 4 1e-05 10 100 unknowns 1000 10000 100000 Figure 8: Configuration 2: Relative error over number of unknowns for te primal error estimator. Te rate of convergence for te same order of finite element discretization is te same for patced and non-patced meses. However, te error constant is sligtly better using non-patced meses. Next, we demonstrate tat te PU error estimator does not rely on a patced mes structure. To see tis, we run and compare four different settings; namely Q 1 and Q 4 discretizations using patced and non-patced meses. For estimating te error, te primal formulation (11) is considered. Omiting te patc structure allows us to realize a sarper refinement towards singularities. On te oter and, witout te patc structure, we cannot use te simple reconstruction tecnique for te weigts tat Publised in Journal for Computational and Applied Matematics, pp. 192-208, 2015 19
DOF s J(u) error η eff ind 3 072 136 070 4 230 730 0.17 3.00 8 448 134 626 2 786 215 0.08 2.63 22 224 133 487 1 647 556 0.33 2.54 57 504 132 808 968 523 0.54 2.48 152 592 132 358 518 325 0.63 2.56 378 384 132 152 312 235 0.76 2.30 871 800 132 023 183 152 0.83 2.31 Table 4: Configuration 3: convergence istory on a sequence on locally refined meses wit number of unknowns, functional value J(u), error J(u) J(u ), effectivity eff = η / J(u) J(u ) and indicator index ind = η i / J(u) J(u ) for te PU localization. as been described in Section 3. In Fig. 7 we sow meses for Q1 and Q4 elements on refinement level 6. Tese meses must be compared to te left column in Figure 6, were te same computations ave been carried out - but ere using patced meses. In Fig. 8 we compare te error slopes for Q1 and Q4 elements on adaptive meses wit and witout patces. We first identify te same order of convergence for bot cases. Te computations witout patces give a sligtly better error constant, as te strong singularities in tis test-case can be better resolved. Comparing te meses in Fig 7 wit te left column in Fig. 6 sows, tat by omiting te mes structure, te regularity of te meses in lessened. In particular te Q1 computations introduce many islands of refined cells, wic lead to a large number of anging ndoes. However, by consulting te error plots in Figure 8, we cannot detect a negative influence on te resulting error wit respect to te number of unknowns (including all anging nodes). 5.3 Configuration 3: Application to a nonlinear system Te benefit of a variational localization of te error identity is in particular given for complex nonlinear systems of partial differential equations, were assembling te strong formulation of te system is too costly. To exploit te localization tecnique we consider a nonlinear elasticity problem. For te construction sown in Fig. 9, we compute te deformation u : Ω R 3 of an elastic beam Ω = (0, L) (0, D) (0, H) wit lengt L = 2, dept D = 1 and eigt H = 0.5 under a given volume force f = 100e 3. Te beam is attaced u = 0 on parts of te frontal boundary Γ D = (0 L/2) {0} (0, H). All oter boundary parts are free. As quantity of interest, we measure te D = 1 L = 2 H = 0.5 Γ D f = 100e 3 Figure 9: Configuration 3: Deformation of partially fixed elastic beam Ω b under gravity. Left: sketc of te configuration. Rigt: locally adapted mes. 20 Publised in Journal for Computational and Applied Matematics, pp. 192-208, 2015