A Study on Using Hierarchical Basis Error Estimates in Anisotropic Mesh Adaptation for the Finite Element Method

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1 A Study on Using Hierarcical Basis Error Estimates in Anisotropic Mes Adaptation for te Finite Element Metod Lennard Kamenski arxiv:06.603v3 [mat.na] 20 Jan 202 Abstract: A common approac for generating an anisotropic mes is te M-uniform mes approac were an adaptive mes is generated as a uniform one in te metric specified by a given tensor M. A key component is te determination of an appropriate metric wic is often based on some type of Hessian recovery. Recently, te use of a global ierarcical basis error estimator was proposed for te development of an anisotropic metric tensor for te adaptive finite element solution. Tis study discusses te use of tis metod for a selection of different applications. Numerical results sow tat te metod performs well and is comparable wit existing metric tensors based on Hessian recovery. Also, it can provide even better adaptation to te solution if applied to problems wit gradient jumps and steep boundary layers. For te Poisson problem in a domain wit a corner singularity, te new metod provides meses tat are fully comparable to te teoretically optimal meses. Keywords: mes adaptation, anisotropic mes, finite element, a posteriori estimate, ierarcical basis, variational problem, anisotropic diffusion. MSC 200: 65N30, 65N50. Introduction A common approac for generating an anisotropic mes is te M-uniform mes approac based on generation of a quasi-uniform mes in te metric space defined Department of Matematics, University of Kansas, lkamenski@mat.ku.edu. Tis researc was supported in part by te German Researc Foundation troug te grant KA 325/-. June 28, 20 by a symmetric and strictly positive definite metric tensor M. A scalar metric tensor will lead to an isotropic mes wile a full metric tensor will generally result in an anisotropic mes. In tis sense, te mes generation procedure is te same for bot isotropic and anisotropic mes generation. A key component of te approac is te determination of an appropriate metric often based on some type of error estimates. Typically, te appropriate metric tensor depends on te Hessian of te exact solution of te underlying problem, wic is often unavailable in practical computation, tus requiring te recovery of an approximate Hessian from te computed solution. A number of recovery tecniques are used for tis purpose, for example te gradient recovery tecnique by Zienkiewicz and Zu [35, 36], te tecnique based on te variational formulation by Dolejší [8], or te quadratic least squares fitting (QLS) proposed by Zang and Naga [34]. Generally speaking, Hessian recovery metods work well wen exact nodal function values are provided (e.g. interpolation problems), but unfortunately tey do not provide an accurate recovery wen applied to linear finite element approximations on non-uniform meses, as pointed out by te autor in [29]. Recently, conditions for asymptotically exact gradient and convergent Hessian recovery from a ierarcical basis error estimator ave been given by Ovall [3]. His result is based on superconvergence results by Bank and Xu [8, 9], wic require te mes to be uniform or almost uniform: assumptions wic are usually violated by adaptive meses. Hence, a convergence of adaptive algoritms based explicitly on te Hessian recovery cannot be proved in a

2 direct way, even if teir application is successful in practical computations [8, 26, 30, 33]. Tis explains te recent interest in anisotropic adaptation strategies based on some type of a posteriori error estimates. For example, Cao et al. [3] studied two a posteriori error estimation strategies for computing scalar monitor functions for use in adaptive mes movement; Apel et al. [6] investigated a number of a posteriori strategies for computing error gradients used for directional refinement; and Agouzal et al. [, 2, 3] and Agouzal and Vassilevski [4] proposed a new metod for computing metric tensors to minimize te interpolation error provided tat an edge-based error estimate is given. Recently, Huang et al. [24] presented a mes adaptation metod based on ierarcical basis error estimates (HBEE). Te new framework is developed for te linear finite element solution of a boundary value problem of a second-order elliptic partial differential equation (PDE), but it is quite general and can easily be adopted to oter problems. A key idea in te new approac is te use of te globally defined HBEE for te reliable directional information: globally defined error estimators ave te advantage tat tey contain more directional information of te solution; error estimation based on solving local error problems, despite its success in isotropic mes adaptation, do not contain enoug directional information, wic is global in nature; moreover, Dobrowolski et al. [7] ave pointed out tat local error estimates can be inaccurate on anisotropic meses. A brief description of te metod is provided in Sect. 2. Te objective of tis article is to study te application of tis new anisotropic adaptation approac to different problems. Te first example deals wit a boundary value problem of a second-order elliptic PDE. Numerical results in [24] ave already sown tat te new mes adaptation approac can be a successful alternative to Hessian recovery in mes adaptation for a boundary value problem of a second-order elliptic PDE. In tis paper, we would like to investigate anoter interesting question, namely weter te proposed algoritm is able to provide optimal mes adaptation for a problem were te optimal mes design is known teoretically. For tis purpose, a Poisson problem in a domain wit a corner singularity is studied, wic requires proper mes adaptation in order to obtain an accurate finite element solution. Teoretically optimal meses are known for tis problem and are based on some a priori information of te domain and te solution. Section 3 investigates if te proposed metod is able to generate adaptive meses tat are comparable to te optimal ones witout any a priori information on te exact solution. A brief overview over te considered matematical problem, te properties of te optimal meses and te numerical results for te new metod are provided. Section 4 presents an anisotropic metric tensor for general variational problems developed by Huang et al. [25] using te HBEE and te underlying variational formulation and gives a numerical example for a nonquadratic variational problem. Te metric tensor is completely a posteriori: it is based solely on te residual, edge jumps, and te a posteriori error estimate. Te tird example is an anisotropic diffusion problem. Te exact solution of tis problem satisfies te maximum principle and it is desirable for te numerical solution to fulfill its discrete counterpart: te discrete maximum principle (DMP). Recently, Li and Huang [30] developed an anisotropic metric tensor based on te anisotropic non-obtuse angle condition, wic provides bot mes adaptation and DMP satisfaction for te numerical solution: te mes alignment is determined by te main diffusion drag direction, i.e. by te underlying PDE, and te Hessian of te exact solution determines te optimal mes density. In Sect. 5, te Hessian of te exact solution is replaced wit te Hessian of te ierarcical error estimator to obtain a new, completely a posteriori, metric tensor accounting for bot DMP satisfaction and mes adaptation. Concluding remarks on te numerical examples and some key components of te ierarcical basis error estimator are given in Sect Anisotropic mes adaptation based on ierarcical basis error estimator Consider te solution of a variational problem: find u V suc tat a(u,v) = f (v) v V (P) were V is an appropriate Hilbert space of functions over a domain Ω R 2, a(, ) is a bilinear form defined on V V, and F( ) is a continuous linear functional on V. For a given simplicial mes T, te linear finite element approximation u of u is te solution of te corresponding variational problem in a finite dimensional 2

3 subspace V V of piecewise linear functions over T : find u V suc tat a(u,v ) = f (v ) v V. (P ) For te adaptive finite element solution, T (and tus V ) is generated according to a given quantity of interest, for example te error of te solution in a cosen norm. Tis study follows te M-uniform mes approac [23] wic generates an adaptive mes as a uniform mes in te metric specified by a symmetric and strictly positive definite tensor M = M(x). Suc a mes is called an M-uniform mes. Once a metric tensor M as been cosen, a sequence of mes and corresponding finite element approximation are generated in an iterative fasion. Let {T (i) } (i = 0,,...) be an affine family of simplicial meses on Ω wit te corresponding space V (i) of continuous, piecewise linear functions over T (i). An adaptive algoritm starts wit an initial mes T (0). On every mes T (i) te variational problem (P ) wit V (i) is solved and te obtained approximation u (i) is used to compute a new adaptive mes for te next iteration step. Te new mes T (i+) is generated as a M-uniform mes wit a metric tensor M (i) defined in terms of u (i). Tis yields te sequence (T (0),V (0) ) u (0) M (0) (T (),V () )... Te conformity of te mes to te input metric is caracterized by te mes quality measure Q mes introduced in [24]. Q mes for any given mes but Q mes = if and only if te underlying mes is M-uniform (see [24, Sect. 4.] for more details). In te presented numerical examples, te mes adaptation process is repeated until te mes is M-uniform witin a given tolerance: Q mes +ε wit a cosen tolerance ε (ε = 0. is used in te numerical experiments trougout te paper). Mes generation software BAMG is used to generate new adaptive meses (bidimensional anisotropic mes generator developed by F. Hect [22]). 2. Adaptation based on a posteriori error estimates Typically, te metric tensor M depends on te Hessian of te exact solution of te underlying problem [20, 23]. As mentioned in te introduction, it is not possible to obtain an accurate Hessian recovery from a linear finite element solution in general [29], so tere is no way to prove a convergence of an adaptive algoritm based on te Hessian recovery in a direct way, even if its application is successful in practical computations [8, 26, 30]. An alternative approac developed in [24] employs an a posteriori error estimator for defining and computing M. Te brief idea is as follows. Assume tat an error estimate z is reliable in te sense tat u u C z. () for a given norm and tat it as te property Π z 0 (2) for some interpolation operator Π. Ten te finite element approximation error is bounded by te (explicitly computable) interpolation error of te error estimate z, viz., u u C z = C z Π z. (3) Now, it is known from te interpolation teory [27] tat te interpolation error for a given function v can be bounded by a term depending on te triangulation T and derivatives of v, i.e., v Π v C E (T,v), were C is a constant independent of T and v. Terefore, we can rewrite (3) as u u C E (T,z ). In oter words, up to a constant, te solution error is bounded by te interpolation error of te error estimate. Tus, te metric tensor M can be constructed to minimize te interpolation error of te z and does not depend on te Hessian of te exact solution. 2.2 Hierarcical basis a posteriori error estimate One possibility to acieve te property (2) is to use te ierarcical basis error estimator. Te general framework can be found among oters in te work of Bank and Smit [7] or Deuflard et al. [6]. Te approac is briefly explained as follows. 3

4 Let e = u u be te error of te linear finite element solution u V. Ten for all v V we ave a(e,v) = f (v) a(u,v). (E) Let V = V W be a space of piecewise quadratic functions, were W is te linear span of te quadratic edge bubble functions (a quadratic edge bubble function is defined as a product of te two linear nodal basis functions corresponding to te edge endpoints). Te error estimate z is defined as te solution of te approximate error problem: find z W suc tat a(z,w ) = f (w ) a(u,w ) w W. (E ) Te estimate z can be viewed as a projection of te true error onto te subspace W and relies on two assumptions: te strengtened CBS-inequality a(v,w ) γ v w wit γ < for all v V and w W and te saturation assumption u u q β u u wit β <, i.e. te assumption tat te quadratic finite element approximation u q is more accurate tan te linear approximation u. Now, if Π is defined as te vertex-based, piecewise linear Lagrange interpolation ten z satisfies te condition (2) since te edge bubble functions vanis at vertices. Ten, if assumption () olds, te finite element approximation error can be controlled by minimizing te interpolation error of z, i.e., te rigt-and side in (3). Note tat tis definition of te error estimate is global and te cost of its exact solution is comparable to te cost of te quadratic finite element approximation. To avoid te expensive exact solution in numerical computation, only a few sweeps of te symmetric Gauss-Seidel iteration are employed for te resulting linear system (until te relative difference of te old and te new approximations is under a given relative tolerance). In numerical experiments, tree to ten sweeps were enoug to acieve te relative tolerance of % and it proved to be fully sufficient for te purpose of mes adaptation. Te computational cost of suc approximation is comparable to te cost of te Hessian recovery: in te tests, te computation of HBEE was in average about two times slower tan te Hessian recovery metod. Since bot metods require only a fraction of te overall computational cost, using te fast HBEE approximation for mes adaptation instead of Hessian recovery will not significantly increase te overall computational time. 3 Poisson problem in a domain wit a corner singularity Consider te Diriclet problem for te Poisson equation { u = f in Ω, u = g on Ω, (4) wit Ω = {x R 2 : r <, 0 < θ < π/λ} for some 2 < λ <, f = 0 and g = rλ sin(λθ) in usual polar coordinates (r,θ). Te solution of (4) or, more precise, te solution of te corresponding variational problem (P)is given by u(r,θ) = r λ sin(λθ). Te gradient of te solution u as a singular beaviour near te corner (0,0) of te domain Ω and u H +λ ε (Ω) wit arbitrarily small ε [28]. 3. Optimal mes grading For a uniform or quasi-uniform mes, te poor regularity of u leads to te suboptimal rate of convergence of te linear finite element metod: u u H (Ω) Cλ ε, (5) u u L 2 (Ω) C2λ ε. (6) A number of tecniques were developed to regain te optimal convergence order by using specially adapted finite element spaces, see [5, Sect. 4.2] and te references terein for more details. In particular, it is possible to regain te optimal convergence rate by using te standard linear finite element space wit proper mes grading around te corner of te domain. Te basic idea is described among oters in [28] and is as follows. Te linear finite element error is bounded by te linear interpolation error u u 2 H (Ω) u Π u 2 H (Ω) C K T 2 K u 2 H 2 (K). Tus, in order to obtain te optimal convergence rate, te mes size k as to be balanced wit te size of u H 2 (K). Now, if mes elements in a neigbourood around te corner (0,0) ave te size K = Cr µ K, 4

5 were is te mes size far away from te corner and r K is te distance of an element K to te corner, ten te linear finite element error is u u H (Ω) C, u u L 2 (Ω) C2, provided tat µ < λ [5, 28]. Note, tat suc meses still ave N 2 number of elements, tus significantly increasing te accuracy of te approximation for a given number of mes elements. In tis example, we follow te approac described in te Sect. 2 and employ te metric tensor developed in [24] for a boundary value problem of a second-order elliptic PDE. In two dimensions and for te L 2 -norm of te error, te metric tensor M HB based on te HB error estimator is given element-wise as ( M HB,K = det I + ) 6 [ H K (z ) I + ] H K (z ), α α were H K (z ) is te Hessian of te quadratic ierarcical basis error estimate z on element K and α is a regularization parameter to ensure tat te metric tensor is strictly positive definite. Also, α can be used as a adaptation intensity control: if α, te mes becomes uniform; if α 0, te mes becomes more adaptive. Usually, α is cosen so tat about alf of te mes elements are concentrated in regions were det(m) is large (see [24] for more details on te coice of α ). Note, tat M HB is completely a posteriori and does not require any a priori information on te solution. 3.2 Numerical example Consider te Diriclet boundary problem (4) wit λ = 4/7. In two dimensions, te number of elements N of a quasi-uniform mes is proportional to 2, so tere is a factor of /2 in te convergence rate if it is given in terms of te number of mes elements. Tus, according to (5) and (6), te expected orders of convergence in terms of te number of mes elements for quasi-uniform meses sould be at most 2/7 (i.e. λ/2) for te H and 4/7 (i.e. λ) for te L 2 norms of te error. For te optimally graded meses we sould expect orders 0.5 and, respectively. Examples of uniform and adaptive meses as well as close-up views near te corner are given in Figs. and 2. For te adaptive mes, te concentration of mes points (a) Quasi-uniform, 24 triangles. (b) M HB, 225 triangles. Figure : Corner singularity and optimal mes grading: mes examples. (a) Quasi-uniform: triangles. (b) M HB, triangles. Figure 2: Corner singularity and optimal mes grading: close-ups at (0,0). near te singularity can be clearly observed (Figs. b and 2b). We can also observe te gradual cange of mes elements size in te close-up of te adaptive mes (Fig. 2b); in particular, along te boundary edges from te domain corner to te outer boundary. Te mes grading is rater moderate, but it turns out to be enoug to acieve te optimal convergence order, as can be observed in te corresponding convergence plot (Fig. 3). Figure 3 sows te H and L 2 norms of te global linear finite element solution error on uniform and adaptive meses against te number of mes elements. As expected, te convergence order of te finite element solution on uniform meses is only 2/7 and 4/7 for te H and te L 2 norms of te error, respectively. Contrary, te convergence order of te error of te finite element solution obtained by means of te metric tensor M HB is /2 and. Tus, te considered metod is able to acieve te optimal mes grading in tis example relying on only on 5

6 were F(,, ) is a given smoot function, Ω R d (d =,2,3) is te pysical domain and V g is a properly selected set of functions satisfying te Diriclet boundary condition v(x) = g(x) x Ω for a given function g. Te corresponding variational problem is to find a minimizer u V g suc tat I[u] = min v V g I[v]. Figure 3: Corner singularity and optimal mes grading: global error for quasi-uniform and adaptive meses against te number of elements. te a posteriori information provided by te ierarcical basis error estimator. 4 Variational problems Wile it as attracted considerable attention from many researcers and been successfully applied to te numerical solution of PDEs, anisotropic mes adaptation as rarely been employed for variational problems, especially wen combined wit a posteriori error estimates. Recently, Huang and Li [26] developed a metric tensor for te adaptive finite element solution of variational problems. In te anisotropic case, it is semi-a posteriori: it involves residual and edge jumps, bot dependent on te computed solution, and te Hessian of te exact solution. In [25], tis result was improved to provide a metric tensor for variational problems based on te HBEE and te underlying variational formulation. Te new metric tensor is a posteriori in te sense tat it is based solely on residual, edge jumps, and HBEE. Tis is in contrast to most previous work were M depends on te Hessian of te exact solution and is semi-a posteriori or completely a priori; e.g., see [, 2, 4, 2, 23, 26]. 4. General variational problem and te anisotropic metric tensor Consider a general functional of te form I[v] = F(x,v, v)dx, Ω v V g A necessary condition for u to be a minimizer is tat te first variation of te functional vanises. Tis leads to te Galerkin formulation δi[u,v] (F u (x,u, u) v + F u (x,u, u) v)dx = 0 Ω (7) for all v V 0, were V 0 = V g wit g = 0 and F u and F u are te partial derivatives of F wit respect to u and u, respectively. Given a triangulation T for Ω and te associated linear finite element space V g, V g, te finite element solution u can be found by solving te corresponding Galerkin formulation: find u V g, suc tat (F u (x,u, u ) v + F u (x,u, u ) v )dx = 0 Ω for all v V 0,. Te a posteriori metric tensor M HB,K for general variational problems developed in [25] for te error measured in te H -semi-norm is given element-wise by ( M HB,K = + ( K /2 r α K L 2 (K) wit te residual )) 2 + γ /2 R L 2 (γ) γ K ( det I + ) 4 [ H K (z ) I + ] H K (z ) α α r (x) = F u (x) F u (x) x K K T and te edge jump R (x) = (F u (x) n γ ) K + (F u (x) n γ ) K 6

7 (a) Metric based on residual, edge jumps, and Hessian recovery: 60 triangles, maximum aspect ratio 5. Figure 4: Variational problem: numerical solution. on γ T \ Ω (R = 0 if γ Ω). As in te previous section, α is a regularization parameter to ensure tat te metric tensor is strictly positive definite (see [24] for more details on te coice of α ). Note, tat δi[u,v] in (7) is linear in v but is nonlinear in u in general. Tus, a modification of te error problem (E ) for z in Sect. 2.2 is required. For tis purpose, denote by a (u ;, ) a bilinear form resulting from a linearization of δi[, ] about u wit respect to te first argument. Te error estimate z is ten defined as te solution of te approximate linear error problem: find z W suc tat for all w W [7]. 4.2 Numerical example a (u ;z,w ) = δi[u,w ] Consider an anisotropic variational problem defined by te non-quadratic functional [ ( I[u] = + u 2) ] 3/ u 2 y dx Ω wit Ω = (0, ) (0, ) and te boundary condition { u = on x = 0 or x =, u = 2 on y = 0 or y =. Tis example is discussed in [25, 26] and is originally taken from [0]; te analytical solution is not available, but a computed solution in Fig. 4 sows tat te mes adaptation callenge for tis example is te resolution of te sarp boundary layers near x = 0 and x =. (b) Metric based on residual, edge jumps, and HBEE: 43 triangles, maximum aspect ratio 5. Figure 5: Variational problem: adaptive meses and close-up views at (0,0). Adaptive meses obtained by means of Hessian recovery and HBEE are given in Fig. 5. Te bot metods ave correct mes concentration and provide good alignment wit te boundary layers. Anisotropic meses are comparable, altoug mes elements near te boundary layer in te HBEE-based adaptive mes ave a larger aspect ratio 2 tan elements of te mes obtained by means of te Hessian recovery. Tis could be due to te smooting nature of te Hessian recovery: usually, it operates on a larger patc, tus introducing an additional smooting effect, wic affects te grading of te elements size and orientation. Te global ierarcical basis error estimator does not ave tis andicap and, in tis example, te mes obtained by means of HBEE as a better refinement in te ortogonal direction along te steep boundary layers. Quadratic least squares fitting Hessian recovery [34], wic seems to be te most robust and reliable Hessian recovery metod [29, 32]. 2 In tis paper, aspect ratio is defined as te longest edge divided by te sortest altitude. An equilateral triangle as an aspect ratio of 2/

8 5 Anisotropic diffusion and te DMP Anisotropic diffusion problems arise in various areas of science and engineering, for example image processing, plasma pysics, or petroleum engineering. Standard numerical metods can produce spurious oscillations wen tey are used to solve tese problems. A common approac to avoid tis difficulty is to design a proper numerical sceme or a mes so tat te numerical solution satisfies te discrete counterpart (DMP) of te maximum principle satisfied by te continuous solution. A well known condition for te DMP satisfaction by te linear finite element solution of isotropic diffusion problems is te non-obtuse angle condition tat requires te diedral angles of mes elements to be non-obtuse [5]. In [30], a generalization of te condition, te so-called anisotropic non-obtuse angle condition, was introduced for te finite element solution of eterogeneous anisotropic diffusion problems. Te new condition is essentially te same as te existing one except tat te diedral angles are measured in a metric depending on te diffusion matrix of te underlying problem, i.e. te mes is aligned wit te major diffusion directions. Based on te new condition, a metric tensor for anisotropic mes adaptation was developed, wic combines te satisfaction of te DMP wit mes adaptivity: in two dimensions and for te error measured in te H -semi-norm, it is given element-wise by were ( M DMP,K = + ) 2 B H,K det(dk ) 2 D K α, (8) B H,K = det(d K ) 2 D K D K H(u) 2 dx K K (9) and α is a regularization parameter to ensure tat te metric tensor is strictly positive definite (see [24] for more details on te coice of α ). Te diffusion matrix D in (8) provides te correct mes alignment and tus te satisfaction of te DMP wereas te Hessian of te exact solution in (9) provides adaptivity wit te solution profile. As noted in Sect. 2, if te finite element solution error can be bounded by te interpolation error of te ierarcical basis error estimator, te Hessian of te exact solution in (9) can be replaced by te Hessian of te HBEE z : u = 2 Γ in u = 0 Γ out (a) Boundary conditions. u y 0 0 x 0.5 (b) Numerical solution. Figure 6: Anisotropic diffusion example. B HB,K = det(d K ) 2 D K D K H(z ) 2 dx. K K (0) Wit tis coice, te mes will still satisfy te DMP because te mes is still aligned wit te major diffusion directions, but tis time te mes density is determined by te a posteriori error estimate z. 5. Numerical example Consider te BVP discussed in [30]: { (D u) = f in Ω, wit u = g f 0, Ω = [0,] 2 \ on Ω, [ 4 9, 5 ] 2, g = 0 on Γ out, g = 2 on Γ in, 9 were Γ out and Γ in are te outer and inner boundaries of Ω, respectively (Fig. 6a). Te diffusion matrix is given by [ ][ ][ ] cosθ sinθ cosθ sinθ D = sinθ cosθ 0 sinθ cosθ wit te angle of te primary diffusion direction θ = π sinxcosy (te primary diffusion direction is parallel to te first eigenvector of D). Tis example satisfies te maximum principle and te solution stays between 0 and 2 and as sarp jumps near 8

9 (a) Isotropic: 470 triangles, minu , maximum aspect ratio 2.7. (a) M DMP : 4253 triangles, no undersoots, maximum aspect ratio (b) M HB : 4353 triangles, minu , maximum aspect ratio Figure 7: Anisotropic diffusion: meses and contour plots of te numerical solution for te (a) isotropic and (b) HBEE-based (M HB ) metric tensors. te inner boundary. Te analytical solution is not available, but te numerical solution is provided in Fig. 6b. Te goal is to produce a numerical solution wic also satisfies DMP, i.e. stays between 0 and 2, and as a good adaptation. To empasize te compliance wit te DMP, te metric tensors M DMP+H and M DMP+HB based on (9) and (0), respectively, are also compared to a uniform mes and te anisotropic metric tensor M HB based solely on te HBEE (Sect. 3). Figures 7 and 8 sow meses and solution contours. No oversoots in te finite element solutions are observed for all cases, but Fig. 7 sows tat undersoots and unpysical minima occur in te solutions obtained wit te uniform mes (minu 0.059) and M HB (minu ). As expected, no undersoots can be observed for M DMP+H and M DMP+HB (Fig. 8). As for M DMP and M DMP+HB, te solution contours for bot metric tensors are almost undistinguisable and, altoug not as smoot, still quite comparable to te one (b) M DMP+HB : 438 triangles, no undersoots, maximum aspect ratio Figure 8: Anisotropic diffusion: meses and contour plots of te numerical solution for te DMP-compliant (a) Hessian recovery-based (M DMP+H ) and (b) HBEE-based (M DMP+HB ) metric tensors. obtained wit M HB (cf. Figs. 8 and 7b), tus providing a good adaptation to te sarp solution jump near te interior boundary (cf. te somewat smeared solution jump for te isotropic mes in Fig.7a, rigt). Te mes computed by means of HBEE is fully comparable wit te mes obtained using Hessian recovery. However, te maximum aspect ratio of te mes obtained by means of te HBEE is, again, sligtly larger. 6 Concluding remarks Numerical results sow tat a global HBEE can be a successful alternative to Hessian recovery in mes adaptation: te new metod performs successfully and is quite comparable wit te commonly used Hessian recovery metod. A fast approximate solution is as fast as Hessian recovery and proved to be sufficient to provide enoug directional information for te purpose of te mes adaptation. Moreover, as already observed in 9

10 [24], it could be of advantage for problems wit discontinuities, because Hessian recovery could result in unnecessarily ig mes density for suc problems. Te global HBEE seems to be less affected by tis issue and, depending on te underlying problem, can provide a more appropriate mes adaptation. Also, Hessian recovery can also cause a ligt mes smooting: meses obtained by means of Hessian recovery-based metod in Sect. 4 ave a smaller maximum aspect ratio tan meses obtained wit HBEE and terefore seem to be sligtly worse in terms of adaptation wit te steep boundary layers. For te corner singularity problem, te discussed mes adaptation metod proved to be able to capture te required information and to provide te optimal mes grading witout any a priori information on te solution. One of te key components of te metod is te reliability of te error estimator on anisotropic meses: error estimation wit ierarcical bases is usually based on te saturation assumption, wic basically states tat quadratic approximations provide finer information on te solution tan linear ones. Existing results on its validity require bounds on te elements aspect ratio [9]. It is still unclear if similar results can be acieved for general adaptive meses, but numerical results suggest tat aspect ratio bounds are not necessary if te mes is properly aligned. Moreover, it seems tat good mes adaptation does not require an accurate Hessian recovery or an accurate error estimator, but rater some additional information of global nature, altoug it is still unclear wic information exactly is necessary. Tis question is definitely of interest for future investigations. Acknowledgements. Te autor is very grateful to te anonymous referees for teir valuable comments and suggestions. References [] Agouzal, A., Lipnikov, K., Vassilevski, Y.: Generation of quasi-optimal meses based on a posteriori error estimates. In: Proceedings of te 6t International Mesing Roundtable, pp (2008) [2] Agouzal, A., Lipnikov, K., Vassilevski, Y.: Anisotropic mes adaptation for solution of finite element problems using ierarcical edge-based error estimates. In: Proceedings of te 8t International Mesing Roundtable, pp (2009) [3] Agouzal, A., Lipnikov, K., Vassilevski, Y.: Hessian-free metric-based mes adaptation via geometry of interpolation error. Comput. Mat. Mat. Pys. 50(), (200) [4] Agouzal, A., Vassilevski, Y.V.: Minimization of gradient errors of piecewise linear interpolation on simplicial meses. Comput. Metods Appl. Mec. Engrg. 99(33-36), (200) [5] Apel, T.: Anisotropic Finite Elements: Local Estimates and Applications. B. G. Teubner, Stuttgart (999) [6] Apel, T., Grosman, S., Jimack, P.K., Meyer, A.: A new metodology for anisotropic mes refinement based upon error gradients. Appl. Numer. Mat. 50(3-4), (2004) [7] Bank, R.E., Smit, R.K.: A posteriori error estimates based on ierarcical bases. SIAM J. Numer. Anal. 30(4), (993) [8] Bank, R.E., Xu, J.: Asymptotically exact a posteriori error estimators, Part I: Grids wit superconvergence. SIAM J. Numer. Anal. 4(6), (2003) [9] Bank, R.E., Xu, J.: Asymptotically exact a posteriori error estimators, Part II: General unstructured grids. SIAM J. Numer. Anal. 4(6), (2003) [0] Bildauer, M.: Convex variational problems. Linear, nearly linear and anisotropic growt conditions., Lecture Notes in Matematics, vol. 88. Springer Berlin / Heidelberg (2003) [] Boroucaki, H., George, P.L., Hect, F., Laug, P., Saltel, E.: Delaunay mes generation governed by metric specifications. Part I. Algoritms. Finite Elements in Analysis and Design 25(-2), 6 83 (997) [2] Boroucaki, H., George, P.L., Moammadi, B.: Delaunay mes generation governed by metric specifications. Part II. Applications. Finite Elem. Anal. Des. 25(-2), (997) [3] Cao, W., Huang, W., Russell, R.D.: Comparison of two-dimensional r-adaptive finite element metods using various error indicators. Mat. Comput. Simulation 56(2), (200) [4] Castro-Díaz, M.J., Hect, F., Moammadi, B., Pironneau, O.: Anisotropic unstructured mes adaption for flow simulations. Int. J. Numer. Met. Fluids 25(4), (997) [5] Ciarlet, P.G., Raviart, P.A.: Maximum principle and uniform convergence for te finite element metod. Comput. Metods Appl. Mec. Engrg. 2(), 7 3 (973) [6] Deuflard, P., Leinen, P., Yserentant, H.: Concepts of an adaptive ierarcical finite element code. Impact Comput. Sci. Engrg. (), 3 35 (989) 0

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