Improved energy-norm a posteriori error estimates for singularly perturbed reaction-diffusion problems on anisotropic meshes
aa r X i v : . [ m a t h . NA ] A p r Improved energy-norm a posteriori errorestimates for singularly perturbedreaction-diffusion problemson anisotropic meshes
Natalia Kopteva
Abstract
In the recent article [7] the author obtained residual-type a posteriori errorestimates in the energy norm for singularly perturbed semilinear reaction-diffusionequations on unstructured anisotropic triangulations. The error constants in theseestimates are independent of the diameters and the aspect ratios of mesh elementsand of the small perturbation parameter. The purpose of this note is to improve theweights in the jump residual part of the estimator. This is attained by using a novelsharper version of the scaled trace theorem for anisotropic elements, in which thehat basis functions are involved as weights.
Consider finite element approximations to singularly perturbed semilinear reaction-diffusion equations of the form Lu : “ ´ ε △ u ` f p x , y ; u q “ p x , y q P Ω , u “ B Ω , (1.1)posed in a, possibly non-Lipschitz, polygonal domain Ω Ă R . Here 0 ă ε ď
1. Wealso assume that f is continuous on Ω ˆ R and satisfies f p¨ ; s q P L p Ω q for all s P R ,and the one-sided Lipschitz condition f p x , y ; v q ´ f p x , y ; w q ě C f r v ´ w s whenever v ě w , with some constant C f ě
0. Then there is a unique solution u P W ℓ p Ω q Ď W q Ă C p ¯ Ω q for some ℓ ą q ą C f ` ε ě C f ` ε ).For this problem, the recent articles [6, 7] gave residual-type a posteriori errorestimates on unstructured anisotropic meshes. In particular, in [7] the error wasestimated in the energy norm ~ ¨ ~ ε ; Ω , which is an appropriately scaled W p Ω q norm naturally associated with our problem, defined for any D Ď Ω by ~ v ~ ε ; D : “ Natalia KoptevaUniversity of Limerick, Limerick, Ireland; e-mail: [email protected] ! ε } ∇ v } D ` } v } D ) { . Linear finite elements were used to discretize (1.1) with apiecewise-linear finite element space S h Ă H p Ω q X C p ¯ Ω q relative to a triangulation T , and the the computed solution u h P S h satisfying ε x ∇ u h , ∇ v h y ` x f Ih , v h y “ @ v h P S h , f h p¨q : “ f p¨ ; u h q . (1.2)Here x¨ , ¨y denotes the L p Ω q inner product, and f Ih is the standard piecewise-linearLagrange interpolant of f h .To give a flavour of the results in [7], assuming that all mesh elements areanisotropic, one estimator reduces to ~ u h ´ u ~ ε ; Ω ď C ! ÿ z P N min t| ω z | , λ z u ›› ε J ∇ u h K ›› ; γ z ` ÿ z P N ›› min t , H z ε ´ u f Ih ›› ω z ` ›› f h ´ f Ih ›› Ω ) { , (1.3)where C is independent of the diameters and the aspect ratios of elements in T , andof ε . Here N is the set of nodes in T , J ∇ u h K is the standard jump in the normalderivative of u h across an element edge, ω z is the patch of elements surrounding any z P N , γ z is the set of edges in the interior of ω z , H z “ diam p ω z q , and h z » H ´ z | ω z | .A version of (1.3) obtained in [7] involves a somewhat surprising weight λ z “ ε H z h ´ z at the jump residual terms. The main purpose of this note is to improvethe jump residual part of the latter estimator and establish its sharper version with amore natural λ z “ ε H z . This will be attained by employing a novel sharper versionof the scaled trace theorem, in which the hat basis functions are involved as weights(see Remark 3.1). As the improvement that we present here applies to the jumpresidual terms only, we restrict our analysis to these terms.Note that the new shaper version of the jump residual part of the estimator worksnot only for (1.3) (see Theorem 4.3 below), but can be also combined with a shaperbound for the interior residual terms given by [7, Theorem 6.2]. The latter is more in-tricate and was obtained under some additional assumptions on the mesh, so we shallnot give it here. Comparing it to (1.3), roughly speaking, the weight min t , H z ε ´ u is replaced by a sharper min t , h z ε ´ u with a few additional terms included.Note also that a similar improved jump residual part of the estimator is also ob-tained in [8, (1.2)] using an entirely different (and more complicated in the context ofresidual-type estimation) approach for a version of (1.2) (with a special anisotropicquadrature used for the reaction term).Our interest in locally anisotropic meshes is due to that they offer an efficient wayof computing reliable numerical approximations of layer solutions. (In the contextof (1.1) with ε !
1, see, e.g., [4, 9, 14] and references therein.) But such anisotropicmeshes are frequently constructed a priori or by heuristic methods, while the ma-jority of available a posteriori error estimators assume shape regularity of the mesh[1]. In the case of shape-regular triangulations, residual-type a posteriori error es-timates for equations of type (1.1) were proved in [16] in the energy norm, and mproved energy-norm a posteriori error estimates on anisotropic meshes 3 more recently in [3] in the maximum norm. The case of anisotropic meshes havinga tensor-product structure was addressed in [15] for the Laplace equation and in[5, 2] for problems of type (1.1), with the error estimators given, respectively, in the H norm and the maximum norm. For unstructured anisotropic meshes, a posteriorierror estimates can be found in [10, 12] for the Laplace equation in the H norm,and in [11, 12] for a linear constant-coefficient version of (1.1) in the energy norm.Note that the error constants in the estimators of [10, 11, 12] involve the so-calledmatching functions; the latter depend on the unknown error and take moderate val-ues only when the grid is either isotropic, or, being anisotropic, is aligned correctlyto the solution, while, in general, they may be as large as mesh aspect ratios. Thepresence of such matching functions in the estimator is clearly undesirable. It isentirely avoided in the more recent papers [6, 7, 8], as well as here.The paper is organized as follows. In § §
3, we respectively describe our tri-angulation assumptions and give a novel shaper version of the scaled trace theoremfor anisotropic elements. In §
4, we dervie the main result of the paper, a new shaperjump residual part of the estimator. A simplified version of this analysis is givenin § § Notation.
We write a » b when a À b and a Á b , and a À b when a ď Cb witha generic constant C depending on Ω and f , but not on either ε or the diametersand the aspect ratios of elements in T . Also, for D Ă ¯ Ω , 1 ď p ď 8 , and k ě
0, let } ¨ } p ; D “ } ¨ } L p p D q and | ¨ | k , p ; D “ | ¨ | W kp p D q , where | ¨ | W kp p D q is the standard Sobolevseminorm, and osc p v ; D q “ sup D v ´ inf D v for v P L p D q . We shall use z “ p x z , y z q , S and T to respectively denote particular mesh nodes, edgesand elements, while N , S and T will respectively denote their sets. For each T P T , let H T be the maximum edge length and h T : “ H ´ T | T | be the minimumheight in T . For each z P N , let ω z be the patch of elements surrounding any z P N , S z the set of edges originating at z , and H z : “ diam p ω z q , h z : “ H ´ z | ω z | , γ z : “ S z zB Ω , ˚ γ z : “ t S Ă γ z : | S | À h z u . (2.1)Throughout the paper we make the following triangulation assumptions. • Maximum Angle condition.
Let the maximum interior angle in any triangle T P T be uniformly bounded by some positive α ă π . • Local Element Orientation condition.
For any z P N , there is a rectangle R z Ą ω z such that | R z | » | ω z | . Furthermore, if z P N X B Ω is not a corner of Ω , then R z has a side parallel to the segment S z X B Ω . • Also, let the number of triangles containing any node be uniformly bounded.
Natalia Kopteva
Note that the above conditions are automatically satisfied by shape-regular triangu-lations.Additionally, we restrict our analysis to the following two node types definedusing a fixed small constant c (to distinguish between anisotropic and isotropicelements), with the notation a ! b for a ă c b .(1) Anisotropic Nodes , the set of which is denoted by N ani , are such that h z ! H z , h T » h z and H T » H z @ T Ă ω z . (2.2)Note that the above implies that S z contains at most two edges of length À h z (seealso Fig. 3, left).(2) Regular Nodes , the set of which is denoted by N reg , are those surrounded byshape-regular mesh elements.Note that most of our analysis applies to more general node types that wereconsidered in [6, 7]; see Remarks 3.2 and 4.3 for details. Our task is to get an improved bound for the jump residual terms (see I in (4.2)below). The key to this will be to employ the following sharper version of the scaledtrace theorem for anisotropic elements, which is the main result of this section. Lemma 3.1.
For any node z P N “ N ani Y N reg , any function v P W p ω z q , and anyedge S Ă γ z , one has } v φ z } S À } ∇ v } ω z ` } v } ω z " H ´ z if S Ă ˚ γ z , h ´ z if S Ă γ z z ˚ γ z , (3.1) | S | ´ } v φ z } S À } v } ω z } ∇ v } ω z ` } v } ω z " H ´ z if S Ă ˚ γ z , h ´ z if S Ă γ z z ˚ γ z , (3.2) where φ z is the hat basis function associated with z.Remark 3.1. Similar versions of the scaled trace theorem for anisotropic elementswere obtained in [6, Lemma 3.1] and [7, § S Ă γ z z ˚ γ z ), the weights at } ∇ v } p ; ω z aresharper. To be more precise, the version of (3.1) in [6, Lemma 3.1] has the weight H z { h z " } ∇ v } ω z , while the version of (3.2) given by [7, Corollory 3.2] alsoinvolves the weight H z { h z " } ∇ v } ω z . Importantly, for the shaper bounds ofLemma 3.1 to hold true, one needs to estimate } v φ z } p ; S rather than } v } p ; S boundedin [6, 7]. Note that this improvement is crucial for getting an improved weight in thejump residual part of our estimator. Remark 3.2.
An inspection of the proof of Lemma 3.1 shows that this lemma re-mains valid for the more general node types introduced in [7, § mproved energy-norm a posteriori error estimates on anisotropic meshes 5 To prove Lemma 3.1, we shall employ the following auxiliary result.
Lemma 3.2.
For any sufficiently smooth function v ě on a triangle T with verticesz, z and z and their respective opposite edges S, S and S , one has sin = p S , S q } v φ z } S À } ∇ v } T ` | S | ´ } v } T , (3.3a) | S | ´ } v φ z } S À | S | ´ } v φ z } S ` | S || T | ´ } ∇ v } T . (3.3b) Proof.
For (3.3a), let µ be the unit vector along S directed from z to z so that ∇φ z ¨ µ “ | S | ´ . Note that ∇ ¨ p v φ z µ q “ ∇ p v φ z q ¨ µ , so the divergence theoremyields ż B T p v φ z µ q ¨ ν “ ż T ∇ p v φ z q ¨ µ “ ż T ` φ z ∇ v ¨ µ ` | S | ´ v ˘ . Here, to evaluate the integral ş B T , note that µ ¨ ν “ S and φ z “ S , while µ ¨ ν “ sin = p S , S q on S , so ş B T p v φ z µ q ¨ ν “ sin = p S , S q ş S v φ z . The desiredbound (3.3a) follows.To get (3.3b), we modify the proof of [7, Lemma 7.1]. Set w “ v φ z and also A S w : “ | S | ´ ş S w for any edge S . Now, with the ζ -axis having the inward normaldirection to S , and ¯ h : “ | T || S | ´ , one gets A S w ´ A S w “ ¯ h ´ ş ¯ h ` w | S ´ w | S ˘ d ζ .This yields (3.3b) as φ z does not change in the direction normal to ζ . l Proof of Lemma 3.1.
First, note that (3.2) follows from (3.1) as | S | ´ } v φ z } S ď} v φ z } S ď } v φ z } S , while ∇ v “ v ∇ v . With regard to (3.1), it suffices to proveit for the case v ě
0, as if v changes sign on ω z , apply (3.1) with v replaced by v τ : “ ? v ` τ ě
0, where τ is a small positive constant (while | ∇ v τ | ď | ∇ v | ), andthen let τ Ñ ` so that v τ Ñ | v | .Thus it remains to show (3.1) for v ě
0. When S Ă ˚ γ z , this bound follows from asimilar bound on } v } S in [6, Lemma 3.1]. Now consider S Ă γ z z ˚ γ z . Then S is a longedge shared by two anisotropic triangles. Consider two cases; see Fig. 1. Case (i):If in at least one of these triangles, T , the angle at z is Á
1, then an application of(3.3a) yields } v φ z } S À } ∇ v } T ` h ´ z } v } T , and (3.1) follows. Case (ii): Other-wise, in any triangle T sharing the edge S , the other edge S originating at z is alsoof length » H z , while the edge opposite to z is of length » h z . Then an applica-tion of (3.3b) yields H ´ z } v φ z } S À H ´ z } v φ z } S ` H ´ z } ∇ v } T or, equivalently, } v φ z } S À } v φ z } S ` } ∇ v } ω z . Thus, a possibly repeated application of (3.3b) re-duces this case to case (i); see Fig. 1. l z S z S S z S S Fig. 1
Illustration to the proof of (3.1) in Lemma 3.1: case (i) (left); case (ii) with a single appli-cation of (3.3b) (centre); case (ii) with a double application of (3.3b) (right). Natalia Kopteva
Assuming ~ u h ´ u ~ ε ; Ω ą
0, let G : “ u h ´ u ~ u h ´ u ~ ε ; Ω ñ ~ G ~ ε ; Ω “ , g : “ G ´ G h , (4.1)where G h P S h is some interpolant of G . Now, a relatively standard calculation yieldsthe following error representation [7, § ~ u h ´ u ~ ε ; Ω À ÿ z P N ε ż γ z p g ´ ¯ g z q φ z J ∇ u h K ¨ ν ` ÿ z P N ż ω z f Ih p g ´ ¯ g z q φ z ` |x f h ´ f Ih , G y|“ : I ` II ` E quad , (4.2)which holds for any G h P S h and any set of real numbers t ¯ g z u z P N such that ¯ g z “ z P B Ω . (To be precise, ¯ g z will be specified later as a certain average of g “ G ´ G h near z .) Here φ z denotes the standard hat basis function correspondingto z P N .In the following proofs it will be convenient to use, with p “ ,
2, the scaled W p p D q norm defined by {{{ v {{{ p ; D : “ } ∇ v } p ; D ` p diam D q ´ } v } p ; D ñ {{{ v {{{ p ; ω z “ } ∇ v } p ; ω z ` H ´ z } v } p ; ω z . To illustrate our approach in a simpler setting, we first present a version of theanalysis for a simpler, partially structured, anisotropic mesh in a square domain Ω “ p , q . So, throughout this section, we make the following triangulation as-sumptions.A1. Let t x i u ni “ be an arbitrary mesh on the interval p , q in the x direction. Then, leteach T P T , for some i ,(i) have the shortest edge on the line x “ x i ;(ii) have a vertex on the line x “ x i ` or x “ x i ´ (see Fig. 2, left).A2. Let N “ N ani , i.e. each mesh node z satisfies (2.2).A3. Quasi-non-obtuse anisotropic elements.
Let the maximum angle in any trianglebe bounded by π ` α h T H T for some positive constant α .These conditions essentially imply that all mesh elements are anisotropic andaligned in the x -direction. They also imply that if x z “ x i , then ω z Ď ω ˚ z : “ p x i ´ , x i ` q ˆ p y ´ z , y ` z q , y ` z ´ y ´ z » h z , diam ω ˚ z » H z , (4.3)where p y ´ z , y ` z q is the range of y within ω z , while x ´ : “ x and x n ` : “ x n . mproved energy-norm a posteriori error estimates on anisotropic meshes 7 Remark 4.1.
The above conditions (in particular A3) imply that there is J À ω ˚ z Ă ω p J q z for all z P N , where ω p q z : “ ω z , and ω p j ` q z denotes the patch ofelements in/touching ω p j q z . This conclusion is illustrated on Fig. 2 (right). (Note that J “ α “ g z in (4.2) is related to the orientation of anisotropicelements, and is crucial in our analysis. Let ¯ g z “ z P B Ω , and, otherwise, for x z “ x i with some 1 ď i ď n ´
1, let ż x i ` x i ´ p g p x , y z q ´ ¯ g z q ϕ i p x q dx “ . (4.4)Here we use the standard one-dimensional hat function ϕ i p x q associated with themesh t x i u (i.e. it has support on p x i ´ , x i ` q , equals 1 at x “ x i , and is linear on p x i ´ , x i q and p x i , x i ` q ). Theorem 4.1
Let g “ G ´ G h with G from (4.1) and any G h P S h , while Θ : “ ε } ∇ g } Ω ` ÿ z P N ` ` ε H ´ z ˘ } g } ω z . (4.5) Then ~ u h ´ u ~ ε ; Ω À I ` II ` E quad , where the right-hand side terms are specified in (4.2) , and, under conditions A1–A3, | I | À ! Θ ÿ z P N “ min t| ω z | , ε h z u ` ε ˚ J z ˘ ` min t| ω z | , ε H z u ` ε J z ˘ ‰) { , (4.6) where ˚ J z : “ } J ∇ u h K } ; ˚ γ z and J z : “ } J ∇ u h K } ; γ z z ˚ γ z . Corollary 4.2
Under conditions A1–A3, one has (1.3) with λ z “ ε H z .Proof. To get the desired result, combine (4.6) with the bound [7, (5.8)] on II , thestraightforward bound | E quad | ď } f h ´ f Ih } Ω , and Θ À ~ G ~ ε ; Ω “ l x i ´ x i x i ` zz ˆ z ˆ z Á h ˆ z » h z À h z . Fig. 2
Partially structured anisotropic mesh (left); illustration for Remark 4.1 (right): for any fixededge z ˆ z and any edge z ˆ z intercepting the dashed horizontal line via ˆ z , the figure shows that h z À h z ,so there is a uniformly bounded number of edges of type z ˆ z , so ω ˚ z Ă ω p J q z with J À
1. Natalia Kopteva
Proof of Theorem 4.1.
Split I of (4.2) as I “ ř z P N p ˚ I z ` I z q , where˚ I z : “ ε ż ˚ γ z p g ´ ¯ g z q φ z J ∇ u h K ¨ ν , I z : “ ε ż γ z z ˚ γ z p g ´ ¯ g z q φ z J ∇ u h K ¨ ν . (4.7)First, consider ¯ g z , the definition of which (4.4) implies that H z | ¯ g z | À } g ϕ i }
1; ¯ S z ,where ¯ S z is the segment joining the points p x i ´ , y z q and p x i ` , y z q , so | ¯ S z | » H z .Versions of (3.1) and (3.2) then respectively yield H z | ¯ g z | À } ∇ g } ω ˚ z ` h ´ z } g } ω ˚ z , H z | ¯ g z | À } g } ω ˚ z p} ∇ g } ω ˚ z ` h ´ z } g } ω ˚ z q . (4.8)These two bounds will be used when estimating both ˚ I z and I z .We now proceed to estimating ˚ I z . Note that (3.1) implies that }p g ´ ¯ g z q φ z }
1; ˚ γ z À{{{ g {{{ ω ˚ z À | ω z | { {{{ g {{{ ω ˚ z , where we also used } ¯ g z φ z }
1; ˚ γ z » h z | ¯ g z | combined withthe first bound from (4.8). Similarly, }p g ´ ¯ g z q φ z } γ z À h z } g } ω ˚ z {{{ g {{{ ω ˚ z , wherewe employed (3.2) and the second bound from (4.8). Now, from the definition of ˚ I z in (4.7) combined with the two bounds on }p g ´ ¯ g z q φ z }
1; ˚ γ z , one concludes that | ˚ I z | À ˚ θ { z ˚ λ { z p ε ˚ J z q , ˚ θ z : “ ε min | ω z |{{{ g {{{ ω ˚ z , h z } g } ω ˚ z {{{ g {{{ ω ˚ z ( ˚ λ z . Set ˚ λ z : “ min t| ω z | , ε h z u . Then, to get the bound of type (4.6) for ř z P N ˚ I z , it remainsto show that ř z P N ˚ θ z À Θ . For the latter, in view ofmin t aa , bb u{ min t a , b u ď a ` b @ a , a , b , b ą , (4.9)one gets ˚ θ z À ε {{{ g {{{ ω ˚ z ` ε } g } ω ˚ z {{{ g {{{ ω ˚ z , which leads to ř z P N ˚ θ z À Θ , alsousing Remark 4.1.For I z , first, recall the bound | I z | À ε {{{ g {{{ ω ˚ z p ε J z q from [7, (5.12)], whichimplies | I z | À ε | ω z | { {{{ g {{{ ω ˚ z p ε J z q . An alternative bound on I z follows from }p g ´ ¯ g z q φ z } γ z z ˚ γ z À H z } g } ω ˚ z p} ∇ g } ω ˚ z ` h ´ z } g } ω ˚ z q , where the latter is ob-tained by an application of (3.2) for g , while the second bound from (4.8) is em-ployed for ¯ g z . Combining the two bounds on I z , we arrive at | I z | À θ { z λ { z p ε J z q , θ z : “ ε min | ω z |{{{ g {{{ ω ˚ z , H z } g } ω ˚ z p} ∇ g } ω ˚ z ` h ´ z } g } ω ˚ z q ( λ z . (4.10)Here set λ z : “ min t| ω z | , ε H z p ` ε h ´ z qu . Now, again using (4.9), one gets θ z À ε {{{ g {{{ ω ˚ z ` ε } g } ω ˚ z p} ∇ g } ω ˚ z ` h ´ z } g } ω ˚ z q{p ` ε h ´ z q , (4.11)and hence ř z P N θ z À Θ . Finally, to get the bound of type (4.6) for ř z P N I z , itremains to note that λ z “ min t| ω z | , ε H z r ` ε h ´ z su » min t| ω z | , ε H z u . l mproved energy-norm a posteriori error estimates on anisotropic meshes 9 Remark 4.2.
While the definition (4.4) for ¯ g z is quite different from a standardchoice (see, e.g., [13, Lecture 5]), its role may not be immediately obvious inthe proof of Theorem 4.1. To clarify this, note that it is crucial for the bound | I z | À ε {{{ g {{{ ω ˚ z p ε J z q quoted from [7, (5.12)]. To be more precise, the latter boundis obtained in [7] using the representation I z “ I z ` I z ` I z : “ ε ż γ z z ˚ γ z p g ´ ¯ g z q φ z J B x u h K ν x ` ε ż γ z z ˚ γ z r g ´ g p x , y z qs φ z J B y u h K ν y ` ε ż γ z z ˚ γ z r g p x , y z q ´ ¯ g z s φ z J B y u h K ν y , where J w K , for any w , is understood as the jump in w across any edge in γ z evaluatedin the anticlockwise direction about z . Importantly, here I z “ g z (as well as due to the partial structure of our mesh; in a more general case, theestimation of I z is more intricate). Theorem 4.3
Suppose that N “ N ani Y N reg and all corners of Ω are in N reg . Letg “ G ´ G h with G from (4.1) and any G h P S h , while Θ is defined by (4.5) . Then ~ u h ´ u ~ ε ; Ω À I ` II ` E quad , where the right-hand side terms are specified in (4.2) ,and | I | À ! Θ ÿ z P N min t| ω z | , ε H z u ›› ε J ∇ u h K ›› ; γ z ) { . (4.12) Corollary 4.4
Under the conditions of Theorem 4.3, one has (1.3) with λ z “ ε H z .Proof. To get the desired result, combine (4.12) with the bound [7, (6.2)] on II ,the straightforward bound | E quad | ď } f h ´ f Ih } Ω , and Θ À ~ G ~ ε ; Ω “ N “ N ani Y N reg ). l Remark 4.3.
In view of Remark 3.2, an inspection of the proof of Theorem 4.3shows that this theorem remains valid for the more general node types introducedin [7, § Proof of Theorem 4.3.
Split I of (4.2) as I “ ř z P N I z , where I z is defined as in (4.7),only with γ z z ˚ γ z replaced by γ z . It suffices to show that for some edge subset S ˚ Ă S with some quantities I S ; z associated with any S P S z X S ˚ (to be specified below),one has ÿ z P N ÿ S P S z X S ˚ I S ; z “ , (4.13a) | I z ` ÿ S P S z X S ˚ I S ; z | À ε {{{ g {{{ ω z ›› ε J ∇ u h K ›› ; γ z À ε | ω z | { {{{ g {{{ ω z ›› ε J ∇ u h K ›› ; γ z , (4.13b) | I z | ` ÿ S P S z X S ˚ | I S ; z | À ε ! H z } g } ω z p} ∇ g } ω z ` h ´ z } g } ω z q ) { ›› ε J ∇ u h K ›› ; γ z . (4.13c)Indeed, (4.13a) implies that I “ ř z P N I z “ ř z P N p I z ` ř S P S z X S ˚ I S ; z q , while(4.13b), (4.13c) yield | I z ` ÿ S P S z X S ˚ I S ; z | À θ { z λ { z ›› ε J ∇ u h K ›› ; γ z , θ z : “ ε min | ω z |{{{ g {{{ ω z , H z } g } ω z p} ∇ g } ω z ` h ´ z } g } ω z q ( λ z . (4.14)Here set λ z : “ min t| ω z | , ε H z p ` ε h ´ z qu . Then (4.14) becomes a version of (4.10)with ω ˚ z replaced by ω z , so proceeding as in the proof of Theorem 4.1 (i.e. againemploying (4.9)), one gets a version (4.11) with ω ˚ z replaced by ω z , which leads to ř z P N θ z À Θ . Now, to get the desired bound (4.12), it remains to note that λ z “ min t| ω z | , ε H z p ` ε h ´ z qu » min t| ω z | , ε H z u .So, to complete the proof, we need to establish (4.13). Relations (4.13a) and(4.13b) immediately follow from [8, (6.10), (6.11a), (6.11b)] for a certain choiceof t ¯ g z u z P N , the edge subset S ˚ Ă S and the quantities I S ; z associated with any S P S z X S ˚ . We need to recall their definitions to prove the remaining requiredbound (4.13c) (which is a sharper version of [8, ((6.11c)]).First, we recall the definition of t ¯ g z u z P N . In view of the Local Element Orienta-tion condition (see § z P N , introduce the following local notation.Let the local cartesian coordinates p ξ , η q be such that z “ p , q , and the unit vec-tor i ξ in the ξ direction lies along the longest edge ˆ S z P S z (see Fig. 3 (left)). For z P N ani X B Ω (hence z is not a corner of Ω ), let i ξ be either parallel or orthogonalto B Ω at z (depending on whether ω z is, roughly speaking, parallel or orthogonal to B Ω ).Next, split S z “ ˚ S z Y S ` z Y S ´ z , where ˚ S z “ t S Ă S z : | S | À h z u (so ˚ γ z “ ˚ S z zB Ω ). Here we also use S ˘ z : “ t S Ă S z z ˚ S z : S ξ Ă R ˘ u , where S ξ “ proj ξ p S q denotes the projection of S onto the ξ -axis. Now, let p ξ ´ z , ξ ` z q be the maximal inter-val such that p ξ ´ z , q Ă S ξ for all S P S ´ z and p , ξ ` z q Ă S ξ for all S P S ` z . Also, let ϕ z p ξ q be the standard piecewise-linear hat-function with support on p ξ ´ z , ξ ` z q andequal to 1 at ξ “
0. Note that if S ´ z “ H (and S ` z “ H ), then we set ξ ´ z “ ξ ` z “
0) and do not use ϕ z for ξ ă ξ ą mproved energy-norm a posteriori error estimates on anisotropic meshes 11 Next, for ξ P r ξ ´ z , ξ ` z s define a continuous function ¯ η z p ξ q as follows: (i) ¯ η z p ξ q islinear on r ξ ´ z , s and r , ξ ` z s ; (ii) ¯ η z p q “
0; (iii) p ξ , ¯ η z p ξ qq P ω z for all ξ P p ξ ´ z , ξ ` z q .(For example, one may choose ¯ η z p ξ q so that tp ξ , ¯ η z p ξ qq : ξ P p ξ ´ z , qu lies on anyedge in S ´ z , while tp ξ , ¯ η z p ξ qq : ξ P p , ξ ` z qu lies on any edge in S ` z ; see Fig. 3(left).)We are now prepared to specify ¯ g z . Let ¯ g z : “ z P B Ω or z P N reg (as for thelatter, ξ ´ z “ ξ ` z “ ż ξ ` z ξ ´ z “ g p ξ , ¯ η z p ξ qq ´ ¯ g z ‰ ϕ z p ξ q d ξ “ . (4.15)Also, let ¯ S ´ z : “ tp ξ , ¯ η z p ξ qq : ξ P p ξ ´ z , qu and ¯ S ` z : “ tp ξ , ¯ η z p ξ qq : ξ P p , ξ ` z qu , i.e.¯ S ˘ z is the segment joining p , q and p ξ ˘ z , ¯ η z p ξ ˘ z qq .We can now proceed to getting a bound of type (4.13c) for | I z | . First, con-sider ¯ g z , the definition of which (4.15) implies that H z | ¯ g z | À } g ϕ z }
1; ¯ S ´ z Y ¯ S ´ z , where | S ´ z Y ¯ S ´ z | » H z . Using (3.1) and (3.2) then yields a version of (4.8), only with ω ˚ z replaced by ω z (as now ¯ S ´ z Y ¯ S ` z Ă ω z ). Next, we get }p g ´ ¯ g z q φ z } γ z À H z } g } ω z p} ∇ g } ω z ` h ´ z } g } ω z q , which is obtained by an application of (3.2) for g , while the second bound from (4.8) is employed for ¯ g z . Combining this with thedefinition of I z immediately yields a bound of type (4.13c) for | I z | .To establish a bound of type (4.13c) for | I S ; z | , we now recall the definitions ofthe edge subset S ˚ Ă S and the quantities I S ; z associated with any S P S z X S ˚ from [7]. Let S ˚ : “ Y z P N ani zB Ω ˚ S z , and for any S P S ˚ with endpoints z and z ,define I S ; z : “ ε α S µ zz ¨ i ξ ˚ J S , α S : “ ż ξ ` S r g p ξ ˚ , ¯ η S p ξ ˚ qq ´ ¯ g S s ϕ S p ξ ˚ q d ξ ˚ . (4.16)Here J S is the standard signed version of | J ∇ u h K | on S , µ zz is the unit vec-tor directed from z to z , and i ξ ˚ is the unit vector along the ξ ˚ -axis. The lo-cal cartesian coordinates p ξ ˚ , η ˚ q are associated with S and coincide with thelocal coordinates p ξ , η q associated with either z P N ani zB Ω or z P N ani zB Ω (atleast one of them is always in N ani zB Ω ). The above α S is defined by a version ξη ¯ η z p ξ q ϕ z p ξ q ξ ` z ξ ´ z ξ ˚ η ˚ ϕ S p ξ ˚ q ξ ` S ξ ´ S zz ¯ η S p ξ ˚ q µ zz i ξ ˚ Fig. 3
Local notation associated with a node z P N ani (left), and an edge S P S ˚ with endpoints z and z (right).2 Natalia Kopteva ş ξ ` S ξ ´ S r g p ξ ˚ , ¯ η S p ξ ˚ qq ´ ¯ g S s ϕ S p ξ ˚ q d ξ ˚ “ ϕ S p ξ ˚ q is associated with the interval p ξ ´ S , ξ ` S q ; the latter is the projection of ω z X ω z (which includes at most two triangles) onto the ξ ˚ -axis. The piecewise-linear function ¯ η S p ξ ˚ q is defined similarly to ¯ η z p ξ q under the restriction that anypoint p ξ ˚ , ¯ η p ξ ˚ qq P ω z X ω z (see Fig. 3(right)).Under this definition, a bound of type (4.13c) for | I S ; z | is established similarly toa similar bound for | I z | . (Note also that µ zz ` µ z z “ I S ; z ` I S ; z “ l References
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