Multigoal-Oriented Error Estimates for Non-linear Problems

Transcription

1 Multigoal-Oriented Error Estimates for Non-linear Problems B. Endtmayer, U. Langer, T. Wick RICAM-Report

2 Multigoal-Oriented Error Estimates for Non-linear Problems B. Endtmayer 1,2, U. Langer 2, and T. Wick 3 1 Doctoral Program on Computational Matematics, Joannes Kepler University, Altenbergerstr. 69, A-4040 Linz, Austria 2 Joann Radon Institute for Computational and Applied Matematics, Austrian Academy of Sciences, Altenbergerstr. 69, A-4040 Linz, Austria 3 Institut für Angewandte Matematik, Leibniz Universität Hannover, Welfengarten 1, Hannover, Germany Abstract In tis work, we furter develop multigoal-oriented a posteriori error estimation wit two objectives in mind. First, we formulate goal-oriented mes adaptivity for multiple functionals of interest for nonlinear problems in wic bot te Partial Differential Equation (PDE) and te goal functionals may be nonlinear. Our metod is based on a posteriori error estimates in wic te adjoint problem is used and a partition-of-unity is employed for te error localization tat allows us to formulate te error estimator in te weak form. We provide a careful derivation of te primal and adjoint parts of te error estimator. Te second objective is concerned wit balancing te nonlinear iteration error wit te discretization error yielding adaptive stopping rules for Newton s metod. Our tecniques are substantiated wit several numerical examples including scalar PDEs and PDE systems, geometric singularities, and bot nonlinear PDEs and nonlinear goal functionals. In tese tests, up to six goal functionals are simultaneously controlled. 1 Introduction A posteriori error estimation and mes adaptivity are well-developed metodologies for finite element computations, see, e.g., te monograps [49, 2, 7, 38, 25, 42] and te references terein. Specifically, goal-oriented error estimation is a powerful metod wen te evaluation of certain functionals of interest (often tese are tecnical quantities) is te main aim rater tan te computation of global error norms. Here, te dual-weigted residual (DWR) metod is often applied [9, 10]. Tanks to increasing computational resources, multipysics applications suc as multipase flow, porous media applications, fluid-structure interaction and electromagnectics are currently one main focus in applied matematics and engineering. Here, mes adaptivity (ideally combined wit parallel computing) can greatly reduce te computational cost wile measuring functionals of interest wit sufficient accuracy. Since in multipysics, several pysical penomena interact, it migt be desirable tat more tan one goal functional sall be controlled. However, only a few studies ave appeared yet. 1

3 A first metodology was proposed in [27, 26]. Oter studies can be found in [29, 39] and more recently in [48, 20, 31]. Until a few years ago, one principle problem in using te DWR metod was te fact tat te error estimator was based on te strong form of te equations [10] or te weak form using special interpolation operators working on patced meses [12]. In [44], te previous localization tecniques were analyzed in more detail and additionally a novel localization strategy based on a partition-ofunity (PU) was proposed. Te PU localization specifically allows for a muc simpler application of te DWR metod to multipysics and nonlinear, vector-valued equations [44, 51]. In addition, te PU-DWR metod works well wit oter discretization tecniques suc as BEM-based FEM [50] or te finite cell metod [46]. On te oter and, te metodology of te PU-DWR metod wit multiple goal functionals as recently been worked out for linear, scalar-valued problems in [20]. Te first goal of tis paper is to extend tis work to nonlinear problems and PDE systems. Here, our focus is on a careful design of te error estimator tat includes bot te primal part and te adjoint part. Te latter one is often neglected in te literature because te evaluation requires additional computational cost and renders te metod even more expensive. It is clear and well-known (see e.g., [10]) tat, in te linear case, te primal and adjoint residuals yield te same error values, but possibly different locally refined meses; see e.g. [44]. In our current work, we will see tat te adjoint estimator part is crucial to obtain good effectivity indices. Terefore, tis term sould not be neglected. Te second objective of tis paper is concerned wit balancing te discretization and te nonlinear iteration error. In recent years, tere as been publised some work on balancing te iteration error (of te linear or nonlinear solver) wit te discretization error [21, 8, 37, 40, 41]. We base ourselves on [40], and we employ specifically te PU localization. Consequently, te DWR metod is used to design an adaptive stopping criterion for Newton s metod tat is in balance wit te estimated discretization error. Te main aspects comprise a careful coice of te weigting functions to design an appropriate joined goal functional. Moreover, we provide all details for te nonlinear solver, wic is a Newton-type metod wit backtracking line-searc. Since we know a solution on te previous mes, we use tis solution as initial guess for Newton s metod yielding a nested iteration. Specifically, nested solution metods or nonlinear nested iterations were developed, for instance, in [10, 24]. We refer to [24, 43] for te analysis of nested iteration metods. In summary, te goals of tis work are two-fold: Design of te PU-DWR metod for multigoal-oriented error estimation for nonlinear problems and PDE systems. Balancing iteration and discretization errors for nonlinear multigoal-oriented error estimation and mes adaptivity. Te nonlinearities may appear in te PDE itself as well as in te goal functionals. Te outline of tis is paper is as follows: In Section 2, our setting is described. Next, in Section 3, we describe te metodology for one goal functional. Tis is followed by a detailed derivation of a multigoal-oriented approac presented in Section 4. Te key algoritms are formulated in Section 5. In Section 6 several numerical tests substantiate our developments. We summarize our work in Section 7. 2

4 2 An abstract setting Let U and V be Banac spaces, and let A : U V be a (possibly) nonlinear operator, were V denotes te dual space of te Banac space V. We ave in mind nonlinear differential operators A acting between Sobolev spaces. We now consider te following weak formulation of te operator equation A(u) = 0 in V : Find u U suc tat A(u)(v) = 0 v V. (1) Te discretization of te nonlinear variational problem (1) can be performed by means of different metods. Our favored metod is te Finite Element Metod (FEM), see also Section 5.1. Te corresponding discrete problem reads as follows: Find u U suc tat A(u )(v ) = 0 v V, (2) were U and V are finite-dimensional subspaces of U and V, respectively. For te time being, let us assume tat bot problems are solvable. Later we will specify our assumptions imposed on A. We are primarily not interested in approximating a solution u of (1), but in te approximate computation of one or more possibly nonlinear functionals at a solution. An example for suc an operator A is given by te weak formulation of te regularized p-laplace equation (see also [19, 28, 47]) tat reads as follows: Find u U := W 1,p 0 (Ω) suc tat A(u)(v) := (ε 2 + u 2 ) p 2 2 u, v (L p (Ω)) L p (Ω) f, v (W 1,p 0 (Ω)) W 1,p 0 (Ω) = 0 (3) for all v V := W 1,p 0 (Ω), were ε denotes a fixed positive regularization parameter, f (W 1,p 0 (Ω)) = W 1,q (Ω) is some given source, wit p 1 + q 1 = 1 and fixed p > 1, and, denots te corresponding duality products. Here, Ω R d, d = 1, 2, 3, is a bounded Lipscitz domain, and W 1,p 0 (Ω) denotes te usual Sobolev space of all functions from te Lebesgue space L p (Ω) wit weak derivatives in L p (Ω) and trace zero on te boundary Ω, see, e.g., [1]. Te notation is used for te Euclidean norm of some vector. Te corresponding strong form is formally given by div((ε 2 + u 2 ) p 2 2 u) = f in Ω, u = 0 on Ω. In Subsection 6.2, te regularized p-laplace (3) serves as first example for our numerical experiments. Remark 2.1. We refer te reader to [23] for te investigation of te original p-laplace problem. 3 Te dual weigted residual metod for nonlinear problems in te case of a single-goal functional In tis section, we apply te DWR metod to nonlinear problems. Te general metod was developed in [10]. Te extension to balance discretization and iteration errors was undertaken in [37, 41, 40]. We base ourselves on te latter study [40], in wic algoritms for nonlinear problems ave been worked 3

5 out. Tis last paper, togeter wit [44, 20], form te basis of te current paper. We are interested in te goal functional evaluation J : U R wit u J(u), were u U is a solution of te primal problem (1). Examples for suc goal functionals are: point evaluation: J(u) := u(x 0 ), integral evaluation: nonlinear functional evaluation: J(u) := Ω J(u) := Ω u(x)ξ(x) dx, u(x)ξ(x)u(x 0 ) 2 dx u(y)φ(y) dy, Ω were ξ and φ are given functions from L 2 (Ω) and x 0 a given point in Ω. For te DWR approac we need to solve te adjoint problem: Find z V corresponding to u U suc tat A (u)(v, z) = J (u)(v) v U, (4) were u denotes a (primal) solution of te primal problem (1), and A (u) and J (u) denote te Frécetderivatives of te nonlinear operator or functional, respectively, evaluated at u. Later we also need te corresponding discrete solution of te adjoint problem. Tis reads as follows: Find z V corresponding to u U suc tat A (u )(v, z ) = J (u )(v ) v U, (5) wit u as a solution of (2). Similarly to te findings in [40, 10, 41] for te Galerkin case (U = V ), we derive an error representation in te following teorem: Teorem 3.1. Let us assume tat A C 3 (U, V ) and J C 3 (U, R). If u solves (1) and z solves (4) for u U, ten it olds for arbitrary fixed ũ U and z V : J(u) J(ũ) = 1 2 ρ(ũ)(z z) ρ (ũ, z)(u ũ) ρ(ũ)( z) + R (3), (6) were ρ(ũ)( ) := A(ũ)( ), (7) ρ (ũ, z)( ) := J (u) A (ũ)(, z), (8) and te remainder term R (3) := [J (ũ + se)(e, e, e) A (ũ + se)(e, e, e, z + se ) 3A (ũ + se)(e, e, e)]s(s 1) ds, (9) wit e = u ũ and e = z z. 4

6 Proof. For te completeness of te presentation we add te proof below, wic is very similar to [40]. First we define x := (u, z) X := U V and x := (ũ, ṽ) X. By assuming tat A C 3 (U, V ) and J C 3 (U, R) we know tat te Lagrange function L(ˆx) := J(û) A(û)(ẑ) (û, ẑ) =: ˆx X, is in C 3 (X, R). Assuming tis it olds L(x) L( x) = Using te trapezoidal rule [40], we obtain f(s) ds = 1 2 (f(0) + f(1)) for f(s) := L ( x + s(x x))(x x) we conclude 0 L ( x + s(x x))(x x) ds. 1 0 f (s)s(s 1) ds, From te definition of L we observe tat L(x) L( x) = 1 2 (L (x)(x x) + L ( x)(x x)) + R (3). J(u) J(ũ) = L(x) L( x) + A(u)(z) +A(ũ)( z) = 1 }{{} 2 (L (x)(x x) + L ( x)(x x)) + A(ũ)( z) + R (3). =0 It remains to sow tat 1 2 (L (x)(x x) + L ( x)(x x)) = 1 2 ρ(ũ)(z z) ρ (ũ, z)(u ũ). But tis is true since L (x)(x x) + L ( x)(x x) = J (u)(e) A (u)(e, z) A(u)(e ) + J (ũ)(e) A (ũ)(e, z) A(ũ)(e ). }{{}}{{}}{{}}{{} =0 =0 =ρ (ũ, z)(u ũ) = ρ(ũ)(z z) Remark 3.2. Instead of A C 3 (U, V ) and J C 3 (U, R) it is sufficient tat A C 2 (U, V ), J C 2 (U, R) and J, A exist and are bounded. Moreover one can furter relax tese assumptions. Indeed te boundedness of te derivatives is just needed in te set {w U w = (1 s)u + sũ} and just in direction u ũ. Remark 3.3. It migt appen tat A C 3 (U, V ) and J C 3 (U, R) do not old for te continuous spaces. Since te result olds for general Banac spaces U and V, it is sufficient to be sown for te discrete spaces U,u, V,z, were U,u := {w + cu w U, c R}, V,z := {v + cz v V, c R}. Remark 3.4. In accordance wit te literature, we denote te parts ρ(ũ)(z z) and ρ (ũ, z)(u ũ) by primal error estimator and adjoint error estimator, respectively. Te remainder term R (3), as in (9), is of te order O( e 2 U max( e U, e V )). Terefore, it can be neglected if {ũ, z} are close enoug to {u, z}. As in [40],we can identify η := 1 2 ρ(ũ)(z z) ρ (ũ, z)(u ũ), (10) 5

7 as te discretization error and η m := ρ(ũ)( z), (11) as te linearization error if we neglect te remainder term R (3). Since Teorem 3.1 is valid for arbitrary z and ũ it also olds for approximations u and z, even if tey are not computed exactly. Of course, formula (10) still contains an exact solution u. Since u is not known, we eiter use an approximation in an enriced discrete space (for example, in a finite element space, wit iger polynomial degree), or we use an interpolant I 2, suc as in [10], to obtain a more accurate solution u(2). If not mentioned oterwise, we use te approximation in te enriced (finite element) space. An enriced discrete space is also used to compute an approximation z (2) of z. If one would use te same finite-dimensional space as for te test space used in te discrete primal problem (2), ten A(u )(z ) = 0 for our approximate solution u of (2) (if te nonlinear problem is solved exactly). Terefore, te discrete adjoint problem reads as follows: Find z (2) V (2) suc tat A (u (2) )(v(2), z(2) ) = J (u (2) )(v(2) ) v(2) U (2), (12) were U (2) and V (2) denote te enriced finite dimensional spaces, and u (2) denotes te more accurate solution, obtained by solving (2) wit U = U (2) and V = V (2) or by interpolation u (2) = I 2 u. Wit tese approximations, te practical error estimator reads: η := 1 2 ρ(u )(z (2) z ) ρ (u, z )(u (2) u ). (13) For localization of te error estimator, we use te partition of unity (PU) tecnique wic is presented in [44]. Tis means tat we coose a set of functions {ψ 1, ψ 2,, ψ N } suc tat N i=1 ψ i 1. Inserting tis into (13) leads to wit η := N η i, (14) i=1 η i := 1 2 ρ(ũ)((z(2) z)ψ i) ρ (ũ, z)((u (2) ũ)ψ i). (15) We notice tat in te primal part of te error indicator z is replaced by i z (2) as in [10]. For instance, a typical partition of unity is given by te finite element basis. In tis case, we distribute η i to te corresponding elements wit a certain weigt as for example illustrated in Figure 1. 6

8 1 4 ηi 1 η i 4 ηi 1 4 ηi 1 4 ηi Figure 1: Equal distribution of te local error estimator using te Q 1 c basis function at te central vertex to te corresponding elements as in [44], see also Section Multiple-goal functionals Now let us assume tat we are interested in te evaluation of N functionals, wic we denote by J 1, J 2,..., J N. From Section 3, we know ow to compute a local error estimator for one functional. We could compute te local error estimators separately. However, we would ave to solve te adjoint problem (4) N times [27, 26]. Let us now assume tat a solution u of problem (1) and te cosen ũ U belong to N i=1 D(J i), were D(J i ) describes te domain of J i. Definition 4.1 (error-weigting function). Let M R N. We say tat E : (R + 0 )N M R + 0 is an error-weigting function if E(, m) C 1 ((R + 0 )N, R + 0 ) is strictly monotonically increasing in eac component and E(0, m) = 0 for all m M. Let J : N i=1 D(J i) U R N be defined as J(v) := (J 1 (v), J 2 (v),, J N (v)) for all v N i=1 D(J i). Furtermore, we define te operation N : R N (R + 0 )N as x N := ( x 1, x 2,, x N ) for x R N. Tis allows us to define te error functional as follows J E (v) := E( J(u) J(v) N, J(ũ)) It is trivial to see from te definition of E tat J E (v) R + 0 v N D(J i ). (16) i=1 for all v N i=1 D(J i). Remark 4.2. Te idea of te construction of JE (v) is tat E( J(u) J(v) N, J(ũ)) is a semi-metric (as in [45, 32]) on te set of equivalence classes ( J) 1 (R( J)) := {( J) 1 (x) : x R( J)}, were ( J) 1 (x) := {v N i=1 D(J i) : J(v) = x}, wit R( J) denotes te range of J, measuring te distance between te equivalence classes containing u and v. Hence, JE (v) represents a semi-metric distance wic ensures tat J E is monotonically increasing if J i (u) J i (ũ) is monotonically increasing. Remark 4.3. If we drop te monotonicity condition in te definition of E, ten, for example, E( J(u) J(v) N, J(ũ)) := N J i (u) J i (v), i=0 7

9 would be an error-weigting function, resulting in J E (ũ) = 0 iff J i (u) = J i (ũ) at least for one i {1, 2,, N}. Remark 4.4. Te derivation given in tis section olds for a general ũ suc tat J(ũ) M. particular, we are interested in ũ to be an approximation to u solving (2). In Te weak derivative of (16) in U at ũ is given by J E (ũ)(v) := N i=1 sign(j i (u) J i (ũ)) E ( J(u) x J(ũ) N, J(ũ))J i(ũ)(v) v D( J E (ũ)), (17) i wit x x, for x 0, sign(x) := 0 else (18) In [27, 26, 20], te functionals were combined as follows J c (v) := N ω i sign(j i (u) J i (ũ)) J i (v) J i (ũ) }{{} =:w i i=1 N v D(J i ). (19) i=0 Carefully inspecting [26], we see tat te following result can be establised: Proposition 4.1. If J c is defined as in (19) and J E as in (16), ten we ave wit E(x, J(ũ)) := N i=1 ω ix i J i (ũ). Proof. First we conclude tat J c (u) J c (ũ) = J E (ũ), (20) J c(ũ)(v) = J E (ũ)(v), v D( J c(ũ)) D( J E (ũ)), (21) D( J c(ũ)) = D( J E (ũ)) (22) J c (u) J c (ũ) = = N i=1 N i=1 ω i sign(j i (u) J i (ũ)) (J i (u) J i (ũ)) J i (ũ) ω i J i (u) J i (ũ) J i (ũ) = E( J(u) J(ũ) N, J(ũ)) = J E (ũ), wic already sows (20). Te weak derivative of J c is given by J c(ũ)(v) = N i=1 ω i sign(j i (u) J i (ũ)) J J i (ũ) i(ũ)(v). (23) From E x i ( J(u) J(ũ) N, J(ũ)) = ω i J i (ũ) for all i {1, 2,, N}, and because (23) and (17) coincide up to te sign, it olds tat (21) and (22) are valid. 8

10 Remark 4.5. E(x, J(ũ)) := N i=1 ω ix i J i (ũ) is an error-weigting function wit M := {x R N : min( x ) > 0} provided tat ω i > 0 for all i = 1, 2,..., N. Remark 4.6. Proposition 4.1 does not use te property tat u solves (1). N i=0 D(J i). However, te goal is to measure te error to an exact solution. We just need tat u Since an exact solution u is not known, neiter J c nor J E can be constructed. As in Section 3, we use te approximation u (2) instead of an exact solution u to approximate J c or J E by J c and J E, respectively. Tis approximation reads as follows wit te derivative J E (ũ)(v) := N i=1 J E (v) := E( J(u (2) ) J(v) N, J(ũ)) v N D(J i ), (24) sign(j i (u (2) ) J i(ũ)) E ( J(u x (2) ) J(ũ) N, J(ũ))J i(ũ)(v) v D( J E (ũ)). i i=1 (25) Using tis approximation of te error functional, we can apply te metods for te single-functional case in Section 3 wit J = J E. Remark 4.7. We notice tat Teorem 3.1 formally does not old for J E since te sign-function enters. However, if E(, m) C 3 ((R + 0 )N, R + 0 ) and te functionals are sufficiently smoot, ten te singularities (due to te signum function) in iger derivatives of J E just appear if J i (u) = J i (u ), or more precisely J i (u (2) ) = J i(u ), since we use te better approximation u (2) instead of u. Alternatively, we can replace te signum function wit a sufficiently smoot approximation. 5 Algoritms We now describe te algoritmic realization of te previous metods wen we use te FEM as spatial discretization. To tis end, we first introduce te finite element (FE) discretizations tat we are going to use in our numerical experiments presented in Section 6. Ten we recapitulate te basic structure of Newton s metod including a line searc procedure. Afterwards, we state te adaptive Newton algoritm for multiple-goal functionals followed by te structure of te final algoritm. 5.1 Spatial discretization For simplicity, we assume tat Ω R d is a polyedral domain. Let T be a subdivision (trianglation) of Ω into quadrilateral elements suc tat K T K = Ω and K K = for all K, K T wit K K. Furtermore, let ψ K be a multilinear mapping from te reference domain ˆK = (0, 1) d to te element K T. We now define te space Q r c as wit Q r (K) := {v ˆK ψ 1 K Q r c := {v C(Ω) : v K Q r (K), K T }, (26) : v(ˆx) = d i=1 ( r β=0 c β,iˆx β i ), c β,i R}. Specifically, we use continuous tensor-product finite elements as described in [17] and [13]. We also refer te reader to [4] for te specific 9

11 approximation properties of tese finite element spaces. Let T l be te triangulation of refinement level l. Ten our finite element spaces are given by U l := U Qr c and V l := V Qr c, wereas te enriced finite element spaces are defined by U l,(2) := U Q r c and V l,(2) := V Q r c, were Q r c and Q r c are defined as in (26) wit T = T l and r > r. If U and V are spaces of vector-valued functions, ten intersection as to be understood component-wise wit possibly different r in eac component. Remark 5.1. Te algoritms presented in tis section are formulated for FEM [15, 13, 4, 17]. However, we are not restricted to a particular discretization tecnique, but we must be able to realize te adaptivity in an appropriate way. For instance, in isogeometric analysis (IGA) tat was originally introduced in [30] on tensor-product meses, local mes refinement is more callenging tan in te FEM. Truncated ierarcical B-splines (THB-splines) are one possible coice to create localized basises wic form a PU, see [22]. Higer-order B-splines of igest smootness even on coaerser meses can be used to construct enriced spaces U l,(2) and V l,(2) tat lead to ceap problems on te enriced spaces, see [33, 34, 35] for te successful use of tis tecnique in functional-type a posteriori error estimates. 5.2 Newton s algoritm Newton s algoritm for solving te nonlinear variational problem (2) belonging to refinement level l is given by Algoritm 1. Below we identify A(u l,k ) wit te corresponding vector wit respect to te cosen basis wen we compute A(u l,k ) l. Algoritm 1 Newton s algoritm on level l 1: Start wit some initial guess u l,0 U l, set k = 0, and set T OLl Newton > 0. 2: wile A(u l,k ) l > T OL l Newton do 3: Solve for δu l,k, 4: Update : u l,k+1 5: k = k + 1. = u l,k A (u l,k )(δul,k, v ) = A(u l,k )(v ) v V l. + αδul,k for some good coice α (0, 1]. Remark 5.2. In order to save computational cost we do not rebuild te matrices in every step. We rebuild te matrices if A(u l,k ) l / A(u l,k 1 ) l > 0.85 in Algoritm 1. Remark 5.3. Motivated by nested iterations, see, e.g., Section 6 in [10], and te analysis for nonlinear nested iterations as given in Section 9.5 from [24], we use T OL 1 Newton = 10 8 A(u 1,0 ) l and T OL l Newton = 10 2 A(u l,0 ) l for l > 1 as stopping criteria. Remark 5.4. Te parameter α can be obtained by means of a line searc procedure. To obtain a good convergence, we used α = γ L wit 0 < γ < 1, were te smallest L tat fulfills A(u l,k + αδul,k ) l < c(l, L max ) A(u l,k ) l, 10

12 wit 0.8 L = 0 c(l, L max ) := L = 1, L+1 ( L max ) L > 1 L = {0, 1, 2, L max 1} and L max = 200, is accepted. Tis coice of α was taken euristically to obtain a better convergence of te Newton metod in te numerical Example In Algoritm 1, we coose γ = 0.9, and in Algoritm 2, γ = We remark tat a standard backtracking line searc metod also works, see, e.g., [47], but te previous exotic coice yields better iteration numbers. 5.3 Adaptive Newton algoritms for multiple-goal functionals In tis section, we describe te key algoritm. Te basic structure of te algoritm is similar to tat presented in [40] and [21]. Our contribution is te extension to multiple-goal functionals. Algoritm 2 Adaptive Newton algoritm for multiple-goal functionals on level l 1: Start wit some initial guess u l,0 U l and k = 0. 2: For z l,0, solve A (u l,0 )(v, z l,0 (0) ) = (J E ) (u l,0 )(v ) v V l, wit (J (0) E ) constructed wit u l,(2) and u l,0 as defined in (25). 3: wile A(u l,k )(zl,k ) > 10 2 η l 1 do 4: For δu l,k, solve 5: Update : u l,k+1 6: k = k : For z l,k, solve = u l,k A (u l,k )(δul,k, v ) = A(u l,k )(v ) v V l. + αδul,k for some good coice α (0, 1]. A (u l,k )(v, z l,k (k) ) = (J E ) (u l,k )(v ) v U l, wit (J (k) E ) constructed wit u l,(2) and u l,k as in (25). 5.4 Te final algoritm Now let us compose te final adaptive algoritm tat starts from an initial mes T 1 and te corresponding finite element spaces V 1, U 1, U 1,(2) and V 1,(2), were U 1,(2) and V 1,(2) are te enriced finite element spaces as described in Section 5.1. Te refinement procedure produces a sequence of finer and finer meses T l wit te correponding FE spaces V l, U l, U l,(2) and V l,(2) for l = 2, 3,

13 Algoritm 3 Te final algoritm 1: Start wit some initial guess u 0,(2),u 0, set l = 1 and set T OL dis > 0. 2: Solve (2) for u l,(2) using Algoritm 1 wit te initial guess u l 1,(2) on te discrete space U l,(2). 3: Solve (2) and (5) using Algoritm 2 wit te initial guess u l 1 on te discrete spaces U l and V l. 4: Construct te combined functional J E as in (24). 5: Solve te adjoint problem (4) for J E on V l,(2). 6: Construct te error estimator η K by distributing η i defined in (15) to te elements. 7: Mark elements wit some refinement strategy. 8: Refine marked elements: T l l+1 T and l = l : If η < T OL dis stop, else go to 2. In step 3 of Algoritm 3, we replaced te estimated error η l by ηl 1 in Algoritm 2, because we want to avoid te solution of te adjoint problem on te space V l,(2). Since te error in te previous estimate migt be larger in general, we take 10 2 η l 1 instead of 10 1 η l, wic was suggested in [40]. Tus, η l 1 is not defined on te first level. Terefore, we set it to η 0 := Tis means tat we perform more iterations on te coarsest level. However, solving on tis level is very ceap. Remark 5.5. We notice tat step 2 in Algoritm 3 is costly, because we ave to solve a problem corresponding to an enriced finite element space. Remark 5.6. In step 7 of Algoritm 3, we mark all elements K were η K 1 T K T l l η K, were T l denotes te number of elements. Remark 5.7. Inspecting Algoritm 3, we need solve at eac refinement level four problems: two are solved in step 3, and one in step 2 and 5, respectively. On te one and, tis is costly in comparison to oter error estimators, e.g., residual-based, were only te primal problem needs to be solved. On te oter and, te adjoint solutions yield precise sensitivity measures for accurate measurements of te goal functionals. In addition, we control bot te discretization and nonlinear iteration error for multiple goal functionals. Finally, te proposed approac is noneteless muc ceaper for many goal functionals. A naive approac (for a discussion in te linear case of multiple goal functionals or for using te primal part of te error estimator only, we refer te reader again to [27, 26]) would mean to solve 2N + 2 problems (i.e., N + 1 for te primal part). 6 Numerical examples In tis section, we perform numerical tests for two nonlinear problems, were te first problem contains two model parameters. We consider different coices of tese parameters tat lead to different levels of difficulty wit respect to teir numerical treatment. Example 1 (p-laplacian): 12

14 a) Smoot solution wit omogeneous Diriclet boundary conditions and rigt and side on te unit square for p = 2 and p = 4 wit ε = 1 as regularization parameter, and an integral evaluation over te wole domain as functional of interest. b) Smoot solution wit inomogeneous Diriclet boundary conditions on te unit square wit a disturbed grid and p = 5 and p = 1.5 wit ε = 0.5 and a point evaluation as functional of interest. c) Solution wit corner singularities and omogeneous Diriclet boundary conditions on a ceese domain wit p = 4 and p = 1.33 wit a very small regularization parameter ε = 10 10, and two nonlinear and two linear functionals of interest. Example 2 (a quasilinear PDE system): Solution wit low regularity on a slit domain wit mixed boundary conditions, and one linear and five nonlinear functionals of interest. Te implementation is based on te finite element library deal.ii [5] and te extension of our previous work [20]. 6.1 Preliminaries Te following examples are discretized using globally continuous isoparametric quadrilateral elements as introduced in Section 5.1. If not mentioned oterwise, we use U (2) = Q r+1 c U and V (2) = Q r+1 c V for te enriced finite element spaces, if U = Q r c U and V = Q r c V is used for te original finite element spaces. In all numerical experiments we used r = 1 except in Section Case 1, were te used discretization is given explicitly. To solve te arising linear systems, we used te sparse direct solver UMFPACK [18]. Te error-weigting function E(x, J(u )) := N x i i=1 J i (u ) is used to construct J E as in (24). In our computations, we used te finite element function wic is 1 at te nodes wic do not belong to te Diriclet boundary and fulfills te boundary conditions at te nodes wic belongs to te Diriclet boundary as initial guess for u 0,(2) and u 0. To investigate ow well our error estimator performs in estimating te error, we introduce te effectivity indices for te functional J as follows: I eff := η J(u) J(u ), (27) I effp := ρ(ũ)(z(2) z ) J(u) J(u ), (28) I effa := ρ (ũ, z)(u (2) u ), (29) J(u) J(u ) were ρ is defined by (7), ρ as in (8), and η as in (13). We call (27) te effectivity index, (28) te primal effectivity index, and (29) te adjoint effectivity index. In te first part, we analyze te beavior of our algoritm for te regularized p-laplace equation (30). In Section 6.2.1, Case 1, we apply our algoritm to te linear problem given in [44], i.e., for p = 2. For Section 6.2.1, Case 2, we cose p = 4, ε = 1, and apply our algoritm to a nonlinear problem, and compare te refinement evolution for te different error estimators ρ(ũ)(z (2) z ), ρ (ũ, z)(u (2) u ) and η. 13

15 In Section 6.2.2, we solve te p-laplace equation for p = 5 and p = 1.5 on a disturbed grid, aiming for a point evaluation. We compare te results of our algoritm wit te results of global refinement and also to te different error estimators. Te examples in Section consider several reentrant corners, several nonlinear functionals, and a very small regularization parameter ε = In Section 6.3, we investigate te beavior of our algoritm for a quasilinear PDE system. 6.2 Example 1: p-laplace Let ε > 0 and p R wit p > 1, and let Ω be a bounded Lipscitz domain in R 2. We again consider te Diriclet problem for p-laplace equation, cf. Section 2, but now wit inomogeneous Diriclet boundary conditions: Find u suc tat: div((ε 2 + u 2 ) p 2 2 u) = f in Ω, u = g on Ω. (30) Te Frécet derivative A (u) at u of te nonlinear operator A corresponding to te p-laplace problem problem 30, cf. also Section 2, is given by te variational identity A (u)(q, v) = (ε 2 + u 2 l 2 ) p 2 2 q, v + (p 2)(ε 2 + u 2 l 2 ) p 4 2 ( u, q)l2 u, v q, v W 1,p 0 (Ω) Regular cases Here we consider a problem wit a smoot solution and a smoot adjoint solution. Case 1 (p = 2, i.e. Poisson problem): Tis is te same example as Example 1 in [44]. In tis example, te data are given by Ω = (0, 1) (0, 1), f = 1 and g = 0. We are interested in te following functional evaluation: J 1 (u) := Ω u(x) dx ± Tis reference value was taken from [44]. If we compare our results in Table 1 wit te results in [44], ten we observe tat tey are quite similar. Te estimated error η is almost te same, and te DOFs exactly coincide wit te DOFs in [44]. However, using just one polynomial degree iger for U (2), we obtain similar results wit less computational cost as is sown in Table 2. 14

16 l DOFs J(u) J(u ) η I eff I effp I effa E E E E E E E E E E E E Table 1: Section 6.2.1, Case 1. Display of exact error J(u) J(u ), estimated error η, and effectivity indices for U = Q 3 c and U (2) = Q 6 c. l DOFs J(u) J(u ) η I eff I effp I effa E E E E E E E E E E E E E E Table 2: Section 6.2.1, Case 1. Display of exact error J(u) J(u ), estimated error η, and effectivity indices for U = Q 3 c and U (2) = Q 4 c. Case 2 (p = 4, ε = 1): We use te same setting as above, but wit p = 4 and ε = 1. Te finite element spaces are given by U = Q 1 c and U (2) = Q 2 c. We are interested in te following functional evaluation J 1 (u) := u(x) dx ± Ω Tis reference value was computed on a fine grid wit DOFs (9 global refinement steps). In tis example, we compare te refinements for different error estimators. 15

17 l DOFs J(u) J(u ) I eff I effp I effa E E E E E E E E E Table 3: Section 6.2.1, Case 2. Refinement is only based on te primal part of te error estimator η. l DOFs J(u) J(u ) I eff I effp I effa E E E E E E E E E Table 4: Section 6.2.1, Case 2. Refinement is only based on te adjoint part of te error estimator η. l DOFs J(u) J(u ) I eff I effp I effa E E E E E E E E E Table 5: Section 6.2.1, Case 2. Refinement for te error estimator η. 16

18 In tis example, we obtain quite good effectivity indices for te refinements based on te te primal part of te error estimator, cf. Table 3, te adjoint part of te error estimator, cf. Table 4, and te full error estimator η, cf. Table 5. Furtermore, te convergence rates are also very similar. One migt conclude tat te adjoint error estimator is not required to obtain good effectivity indices. However, in te following examples, we observe tat tis is not te case for less regular solutions and adjoint solutions Semiregular cases As in te regular cases, we consider a smoot solution, but a low regular adjoint solution. Tis example is motivated by an example in [47]. We coose te rigt-and side and te boundary conditions suc tat exact solution is given by u(x, y) = sin(6x + 6y). Te computation was done on te unit square Ω = (0, 1) (0, 1) on a sligtly perturbed mes (generated wit te deal.ii [6, 5] command distort_random wit 0.2 on a 4 times globally refined grid unit square). Te resulting mes is sown in Figure 6. Te functional of interest is J(u) = u(0.6, 0.6). We consider te following two cases: Case 1 (p = 5, ε = 0.5), Case 2 (p = 1.5, ε = 0.5). In bot cases, te metod also worked for te perturbed meses. For te case p = 5 and ε = 0.5, we observe from Figure 3 tat te adjoint solution almost vanises in te set outside te domain wic is covered by te condition u = 0, and contains te point (0.6, 0.6). Tis was not observed in Case 2. However, te condition u = 0 seems to be important in bot cases. Te adaptively refined meses sown in Figure 2 and Figure 7 ave more refinement levels in tese regions. In Figure 4 and Figure 5, we observe tat we get te same convergence rate as in te case of uniform refinement. Since te solution is smoot, a global refinement already attains te optimal convergence rate. However, we get a reduction of te number of DOFs tat are needed to obtain te same error. Furtermore, we monitor tat te effectivity index is better on finer meses. Te reason migt be te neglected remainder term from Teorem 3.1. From Table 6 and Table 7, we conclude tat tis does not necessarily old for te primal and te adjoint error estimator separately. 17

19 Figure 2: Section 6.2.2, Case 1. Primal solu- Figure 3: Section 6.2.2, Case 1. Adjoint solu- tion and mes after six adaptive refinements. tion on te mes as given in Figure Error (adaptive) η Error (uniform) 1 O(DOFs ) 0.01 u(0.6, 0.6) u (0.6, 0.6) u(0.6, 0.6) u (0.6, 0.6) 0.1 Error (adaptive) η Error (uniform) 1 O(DOFs ) e-05 1e e-05 1e-06 1e-07 1e e DOFs e+06 DOFs Figure 4: Section 6.2.2, Case 1. Error vs DOFs Figure 5: Section 6.2.2, Case 2. Error vs DOFs for p = 5 and ε = 0.5. for p = 1.5 and ε = 0.5. l DOFs J(u) J(u ) Ief f Ief f p Ief f a l DOFs J(u) J(u ) Ief f Ief f p Ief f a E E E E E E E E E E E E E E E E E E E E E E E E E E Table 6: Section 6.2.2, Case 1. Effectivity in- Table 7: Section 6.2.2, Case 2. Effectivity in- dices for p = 5 and ε = 0.5. dices for p = 1.5 and ε =

20 Figure 6: Disturbed initial mes for Case 1 and Case 2 of Section Figure 7: Marked elements (in red) at refinement level l = 7 for Case 2 of Section Low regularity cases As in Section 6.2.1, we consider omogeneous Diriclet conditions and f = 1 as rigt-and side for te p-laplace equation (30). However, ere bot te solution and adjoint solution ave low regularity. Te initial mes is given as in Figure 12, wic was constructed using te deal.ii [6, 5] command ceese. Wit tis data, we ave singularities on eac of te reentrant corners. Furtermore, in tis example, we cose te regularization parameter ε to be 10 10, wic makes te problem very ill-conditioned (in fact it is practically te original p-laplace problem) were u = 0, but it is very close to te unregularized p-laplace problem as in [36] and[23]. We are interested in te following four goal functionals: J 1 (u) :=(1 + u(2.9, 2.1))(1 + u(2.1, 2.9)), ( 2 J 2 (u) := u(x, y) u(2.5, 2.5) d(x, y)), Ω J 3 (u) := u(x, y) d(x, y), (2,3) (2,3) J 4 (u) :=u(0.6, 0.6). Tese functionals will be combined to J E as formulated in (24). Case 1 (p = 4, ε = ): First we consider a case were p > 2. Te following values, wic were computed on a fine grid (8 global refinements, Q 2 c elements, DOFs) on te cluster RADON1 1, are used to compute te reference values: Ω (2,3) (2,3) u(x, y) d(x, y) ± , u(x, y) d(x, y) ± 10 5, u(2.9, 2.1) ± 10 5, u(2.9, 2.1) ± 10 5, u(0.6, 0.6) ± , u(2.5, 2.5) ± ttps:// 19

21 Considering te accuracy of te functional evaluations above, we observe tat te relative errors in te functionals J 1, J 2, J 3 and J 4 are less tan Our algoritm yields te results sown in Table 8. In Figure 8, we can see tat te absolute error in te error functional J E bounds te relative errors of te functionals J 1, J 2, J 3 and J 4. Furtermore, we observe tat J 2 is te dominating functional and J 1 is te one wit te smallest error on most refinement levels. Terefore, we compare te convergence of tis functionals in Figure 10. For uniform refinement, we obtain an error beavior of approximately O(DOFs 3 4 ) for J 2 and O(DOFs 3 5 ) for J 1, wereas we obtain excellent convergence rates of O(DOFs 1 ) for bot functionals using our refinement algoritm. We are not aware of a full convergence analysis on adaptive meses for pointwise estimates for te p-laplacian, but mention two related studies [16] for p > 2 sowing a posteriori estimates for te W 1,p norm and [14] wit pointwise a priori estimates for te p-laplacian. Te bad convergence of J 2 migt result from te fact tat te point (2.5, 2.5) is te intersection of two lines, were te problem is ill-conditioned, and also leads to a kink in te solution at tis point (see Figure 13). Tis kink is not visible in te case p = 1.33 (see Figure 14). Comparing te number of Newton steps in Table 9 and Table 10, we observe tat te number of Newton steps is less tan for Algoritm 2. However, te additional computational cost as to be considered, but we face a problem wit nonlinear functionals, several reentrant corners and a very small regularization parameter ε = Furtermore, tese tables also suggest tat we sould compute bot te primal and te adjoint error estimator to obtain a better approximation of te error. J l DOFs I 1 (u) J 1 (u ) eff J 1 (u) J 2 (u) J 2 (u ) J 2 (u) J 3 (u) J 3 (u ) J 3 (u) J 4 (u) J 4 (u ) J 4 (u) E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E-05 Table 8: Section 6.2.3, Case 1. Relative errors for te goal functionals on several refinement levels (l) and effectivity index I eff tat is computed for J E (24). 20

22 J 1 (adaptive) J 2 (adaptive) J 3 (adaptive) J 4 (adaptive) J E (adaptive) O(DOFs 3 5 ) O(DOFs 1 ) J 1 (adaptive) J 2 (adaptive) J 3 (adaptive) J 4 (adaptive) J E (adaptive) O(DOFs 3 5 ) O(DOFs 1 ) error error e e+06 DOFs Figure 8: Section 6.2.3, Case 1. Error vs DOFs for p = 4, ε = DOFs Figure 9: Section 6.2.3, Case 2. Error vs DOFs for p = 1.33 and ε = l DOFs Error in J E I eff I effp I effa Newton steps E E E E E E E E E E E E E E Table 9: Errors in J E, effectivity indices and number of Newton steps for p = 4 and Algoritm 2. l DOFs Error in J E I eff I effp I effa Newton steps E E E E E E E E E E E E E E Table 10: Errors in J E, effectivity indices and number of Newton steps for p = 4 and Algoritm 2 were A(u l,k )(zl,k ) > 10 2 η l 1 is replaced by A(u l,k ) l > Case 2 (p = 1.33, ε = ): We are interested in te same goal functional as in Case 1 but wit p = Te following values, wic are computed on a fine grid (8 global refinements, Q 2 c elements, DOFs) on te cluster RADON1, are used to compute te reference values: 21

23 1 J1 (adaptive) J1 (uniform) J3 (adaptive) J3 (uniform) 3 O(DOFs 4 ) O(DOFs 1 ) 0.1 relative error 0.1 relative error 1 J1 (adaptive) J1 (uniform) J2 (adaptive) J2 (uniform) 3 O(DOFs 5 ) O(DOFs 1 ) e e e+06 DOFs e+06 DOFs Figure 10: Section 6.2.3, Case 1. Error vs Figure 11: Section 6.2.3, Case 2. Error vs DOFs for p = 4 and ε = DOFs for p = 1.33 and ε = Z u(x, y) d(x, y) ± , Ω Z u(x, y) d(x, y) ± , u(2.9, 2.1) ± , u(0.6, 0.6) ± , u(2.5, 2.5) ± (2,3) (2,3) u(2.9, 2.1) ± , Considering again tat te accuracy of te functional evaluations is valid, we observe tat te relative error of J2 is less tan and te relative error of J1, J3, J4 is less tan As in Case 1, we compare te relative errors of te functionals in Figure 9. Here we see tat te error in JE bounds te relative errors. However, we loose control of te single functionals as long as tey do not dominate te error, as for J2 in Figure 9. In Case 2, J3 and J1, are tese functionals. In te error plot 3 given in Figure 11, we observe tat te error approximately beaves like O(DOFs 4 ) for a uniformly refined mes, and O(DOFs 1 ) for adaptive refinement, as for p = 4. It turns out tat te regions of refinement (except for corner singularities and te point evaluations) ave almost a complementary structure for p = 1.33 and p = 4 as we can conclude from Figure 16 and Figure 17. Figure 12: Initial mes. Figure 13: 6.2.3: Solution for Figure 14: Section 6.2.3: Solu- p = 4. tion for p =

24 Figure 15: Section 6.2.3: Local error estimator after 6 uniform refinements for p = 4. Figure 16: Section 6.2.3: Mes after 11 adaptive refinements for p = 4 ( DOFs). Figure 17: Section 6.2.3: Mes after 11 adaptive refinements for p = 1.33 ( DOFs). 6.3 Example 2: A quasilinear PDE system In tis second numerical test, we furter substantiate our approac for a nonlinear, coupled, PDE system. We consider te following nonlinear boundary value problem: Find u = (u 1, u 2, u 3 ) suc tat u 1 + u 2 + u 3 = 1, in Ω, u 2 + g 1 (1 u 2 ) g 1 (u 3 ) = 0, in Ω, div(g 2 (u 1 + u 2 ) u 3 ) + g 1 (u 3 ) g 1 (u 1 ) = 0, in Ω, is fulfilled in a weak sense, were x u 1 (x, y) = 1 u 2 (x, y) = u 3 (x, y) = sign(y) 2 + y 2 x on Γ D, u 1. n = u 2. n = g 2 (u 1 + u 2 ) u 3. n = 0 on Γ N. Here sign denotes te signum function as defined in (18). Te functions g 1 and g 2 are given by g 1 (t) := e t sin(t 1) and g 2 (t) := e t2 t, respectively. Obviously a solution is given by u 1 (x, y) = x 1 u 2 (x, y) = u 3 (x, y) = sign(y) 2 + y 2 x in Ω. Te computational domain is a slit domain as in [20, 51, 3] and visualised in Figure 18. Te boundary conditions above introduces a discontinuity on te slit-boundary ( 1, 0) {0} and consequently a discontinuity in te solution. Te construction of tis example was motivated by [3, 11]. Let J A, J B, J C, J D, J E, J F be defined as follows: J A (u) :=u 3 ( 0.5, 0.01), J D (u) := Φ D (x, y) u(x, y) d(x, y), Ω J B (u) := u 1 ( 0.01, 0.01), J E (u) := u 1 ( 0.9, 0.9), J C (u) := Φ C (x, y) u(x, y) d(x, y), Ω J F (u) :=u 2 ( 0.9, 0.1), were Φ C (x, y) := (0, 0, χ C (x, y)) and 2χ D (x, y) Φ D (x, y) := ( 4χ D (x, y), x, 4χ D (x, y)), 1 sign(y) 2 + y 2 x 23

25 wit y x x < y χ C (x, y) := 0 x y 1 x, y > 0 and χ D (x, y) :=. 0 else We are now interested in te six goal functionals J 1 (u) :=J B (u)j D (u), J 4 (u) :=J B (u)j E (u), J 2 (u) :=J A (u)j C (u), J 6 (u) :=J 3 B(u)J E (u), J 3 (u) :=J A (u)j C (u)j F (u), J 6 (u) =J C (u). For te functional J B, we can not expect optimal convergence rates for uniform refinement due to te singularity at te slit tip. Consequently, te same is true for te functionals J 1, J 4 and J 5 as monitored in Figures 21,20 and 22. For uniform refinement, we got a relative error in J 1 of about wit DOFs as visualized in Figure 21. To acieve a relative error less tan our adaptive algoritm just needs DOFs ( ). If we use a similar number of DOFs ( ), ten a relative error of is acieved. Figures 21, 20 and 22 migt also lead to te conclusion tat we obtain a convergence rate O(DOFs 1 ) for all given functionals, were te functionals for uniform refinement just converge wit approximately O(DOFs 1 2 ). Tis means, to obtain a relative error in J 1 of about for uniform refinement, we would need approximately DOFs. Tis would mean just storing te solution would require approximately 400 Terabyte. Terefore, obtaining tis accuracy by means of uniform refinement would even be a ard task on te supercomputer Sunway TaiuLigt 2, wic is number one te of TOP500 3 list from November We remark tat I eff, illustrated in Figure 23, as no importance on course meses since te approximations properties are bad anyway. On finer meses, we see excellent beavior. 2 ttp:// 3 ttps:// 24

26 1 0.5 Γ N Γ D Figure 18: Example 2: Te slit domain Ω wit Γ D (red) and Γ N (blue). Figure 19: Example 2: Adaptively refined mes for J E after 24 refinements ( DOFs) relative error e-05 1e-06 1e-07 J 2 (adaptive) J 2 (uniform) J 4 (adaptive) J 4 (uniform) O(DOFs 1 2 ) O(DOFs 1 ) e+06 1e+07 DOFs relative error e-05 1e-06 1e-07 J 1 (adaptive) J 1 (uniform) J 3 (adaptive) J 3 (uniform) O(DOFs 1 2 ) O(DOFs 1 ) e+06 1e+07 DOFs Figure 20: Example 2: Error vs DOFs. Figure 21: Example 2: Error vs DOFs relative error e-05 1e-06 1e-07 J 5 (adaptive) J 5 (uniform) J 6 (adaptive) J 6 (uniform) O(DOFs 1 2 ) O(DOFs 1 ) e+06 1e+07 DOFs l I eff for J E I effp for J E I effa for J E1 Figure 22: Example 2: Error vs DOFs. Figure 23: Example 2: Effectivity of te Error estimators. 25

27 7 Conclusions In tis work, we ave furter developed adaptive scemes for multigoal-oriented a posteriori error estimation and mes adaptivity. First, we extended te existing metods to nonlinear problems. Second, we combined te estimation of te discretization error wit an estimation of te nonlinear iteration error in order to obtain adaptive stopping rules for Newton s metod. In te key Sections 4 and 5, we formulated an abstract framework and its algoritmic realization. In Section 6, tese developments were substantiated wit several numerical tests. Here, we studied te regularized p- Laplace problem and a nonlinear, coupled PDE system. Our findings demonstrate te performance of te algoritms and specifically tat te adjoint part of te error estimator, wic is often neglected in te literature because of its iger computational cost, must be taken into account in order to acieve good effectivity indices. In view of te geometric singularities, nonlinearities in bot te PDE and te goal functionals, our results sow excellent performance of our algoritms. 8 Acknowledgments Tis work as been supported by te Austrian Science Fund (FWF) under te grant P Goal- Oriented Error Control for Pase-Field Fracture Coupled to Multipysics Problems. Te first autor tanks te Doctoral Program on Computational Matematics at JKU Linz te Upper Austrian Goverment for te support wen starting te preparation of tis work. Te tird autor was supported by te Doctoral Program on Computational Matematics during is visit at te Joannes Kepler University Linz in Marc References [1] R. Adams and J. Fournier. Sobolev Spaces. Pure and Applied Matematics. Elsevier Science, [2] M. Ainswort and J. T. Oden. A posteriori error estimation in finite element analysis. Pure and Applied Matematics. Jon Wiley & Sons, New York, [3] J. Andersson and H. Mikayelyan. Te asymptotics of te curvature of te free discontinuity set near te cracktip for te minimizers of te Mumford-Sa functional in te plain. a revision. arxiv: v2, [4] D. N. Arnold, D. Boffi, and R. S. Falk. Approximation by quadrilateral finite elements. Mat. Comp., 71(239): , [5] W. Bangert, D. Davydov, T. Heister, L. Heltai, G. Kanscat, M. Kronbicler, M. Maier, B. Turcksin, and D. Wells. Te deal.ii library, version 8.4. J. Numer. Mat., 24(3): , [6] W. Bangert, R. Hartmann, and G. Kanscat. deal.ii a general purpose object oriented finite element library. ACM Trans. Mat. Softw., 33(4):24/1 24/27,