Local Mesh Refinement for the Discretisation of Neumann Boundary Control Problems on Polyhedra

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1 Local Mesh Refnement for the Dscretsaton of Neumann Boundary Control Problems on Polyhedra homas Apel Johannes Pfefferer Max Wnkler October 20, 2014 Abstract hs paper deals wth optmal control problems constraned by lnear ellptc partal dfferental equatons. he case where the rght-hand sde of the Neumann boundary s controlled, s studed. he varatonal dscretsaton concept for these problems s appled and dscretsaton error estmates are derved. On polyhedral domans one has to deal wth edge and corner sngulartes whch reduce the convergence rate of the dscrete solutons,.e., one can not expect convergence order two for lnear fnte elements on quas-unform meshes n general. As a remedy a local mesh refnement strategy s presented and a pror bounds for the refnement parameters are derved such that convergence wth optmal rate s guaranteed. As a by-product fnte element error estmates n the H 1 Ω-, L 2 Ω- and L 2 Γ-norm for the boundary value problem are obtaned, where the latter one turned out to be the man challenge. AMS Subject Classfcaton: 49J20, 65N15, 65N30, 65N50 Keywords: optmal control, sngulartes, fnte element method, local mesh refnement, varatonal dscretsaton, error estmates nternatonal Research ranng Group GDK 1754 Optmzaton and Numercal Analyss for Partal Dfferental Equatons wth Nonsmooth Structures nsttut für Mathematk und Baunformatk, Unverstät der Bundeswehr München, Neubberg, Germany 1

2 1 ntroducton Local mesh refnement for the numercal soluton of boundary value problems s a well known technque to compensate lower convergence rates due to sngulartes contaned n the soluton of those problems, and has ntensvely been studed n the lterature, see e. g. [9, 33]. On polyhedral domans several papers explot a pror knowledge about sngulartes occurrng at edges and corners [3, 7, 12, 25]. he error estmates proven for graded meshes can also be used to derve error bounds for certan dscretsaton concepts appled to optmal control problems. t s the purpose of ths paper to derve such an a pror error estmate especally for boundary control problems governed by pure Neumann condtons. n contrast to dstrbuted control problems, fnte element error estmates on the boundary for the Neumann boundary value problem have to be proven. Usng standard technques, e. g. a trace theorem or the Aubn-Ntsche method, when dervng such error estmates wll yeld suboptmal convergence rates only. hus, we extend the technques from [6] to the three-dmensonal case. here are several publcatons whch deal wth error estmates for Neumann boundary control problems n two-dmensonal polygonal domans. n most cases one s nterested n the convergence rate of the dscrete control varable n the L 2 Γ-norm. For convex domans the results of Hnze/Matthes [21] mply the convergence rate max{mn{2, π/ω} ε, 3/2} on quas-unform meshes usng the varatonal dscretsaton approach, where ε > 0 s an arbtrary real number and ω the largest nteror angle at the corner ponts of the polygon. However, the proven convergence rate s suboptmal. hs result was mproved by Mateos/Rösch [26] who were able to show the convergence rate mn{2, 1 + π/2ω, 1/2 + π/ω} ε for convex and even non-convex domans. hs estmate s sharp for domans wth ω < 90 or ω > 180. Apel/Pfefferer/Rösch [5] used a mesh gradng technque whch ensures convergence order 3/2 for non-convex domans. Later on, Apel/Pfefferer/Rösch [6] proved that one can expect a convergence rate 2 up to a logarthmc factor f ω < 120. For larger nteror angles they used mesh gradng to retan ths convergence rate. he aforementoned results are also establshed for the postprocessng approach n [5, 6, 26]. o our knowledge, there are no publcatons that deal wth error estmates on the boundary of polyhedra so far. Solely n [23] optmal error estmates on quas-unform meshes for the normal dervatve are proven. However, estmates n the doman have been studed for varous mesh refnement strateges. For nstance Apel/Sändg/Whteman [7] consdered an sotropc refnement strategy for polyhedra wth the restrcton that the same mesh refnement parameter s used for all sngular ponts. A more general strategy whch allows to use a dfferent strength of refnement for each edge and corner has been nvestgated by Lubuma/Ncase [25]. Advanced refnement strateges usng ansotropc mesh gradng at sngular edges have been ntensvely studed by Apel/Srch [8] for prsmatc domans, where only edge sngulartes occur, and on general polyhedra by Apel/Lombard/Wnkler [3] and Băcuţă/Nstor/Zkatanov [12]. For all these approaches the requred error estmates on the boundary stll have to be proven. he am of ths paper s to derve a bound for the parameters used n the sotropc refnement strategy from [7] n order to guarantee an optmal convergence rate n H 1 Ω-, L 2 Ω- and L 2 Γ-norm. Usng these estmates we are then able to prove an optmal convergence rate for the varatonal dscretzaton approach [20] appled to a Neumann control problem. Let Ω be a polyhedron wth boundary Γ. We consder the followng ellptc Neumann 2

3 boundary control problem: Jy, u := 1 2 y y d 2 L 2 Ω + α 2 u 2 L 2 Γ mn! 1.1 subject to y + y = f n Ω, n y = u on Γ, 1.2 u U ad := {u L 2 Γ: a u b a. e. on Γ}, 1.3 wth a gven functon f L 2 Ω, regularsaton parameter α > 0 and constant control bounds a, b R. t s known that the state equaton 1.2 possesses a unque soluton y H 3/2+εreg Ω wth some fxed ε reg 0, 1 2 ] dependng on the geometry of Ω [14, Corollary 23.5]. We assume that the desred state s Hölder-contnuous,.e. y d C 0,σ Ω, wth a Hölder exponent σ 0, 1. he analyss of the contnuous problem s well developed cf. [22, Secton 1.5],[36, Secton 2.5]. here exsts a unque soluton ū L 2 Γ, and a necessary and suffcent optmalty condton s gven by ȳ + ȳ = f n Ω p + p = ȳ y d n Ω, Γ n ȳ = ū on Γ n p = 0 on Γ, px + αūxux ūxds x 0 u U ad, 1.4 where ȳ and p denote the state and adjont state related to ū. Note that the boundary value problems n 1.4 have to be understood n the weak sense. he varatonal nequalty s equvalent to the projecton formula where p Γ denotes the trace of p on Γ and ū = Π ad α 1 p Γ, 1.5 [Π ad v]x := max{ax, mn{bx, vx}} a. e. on Γ denotes the L 2 Γ-projecton onto the set of admssble controls. he paper s structured as follows: n Secton 2 we consder the varatonal dscretsaton approach appled to the problem An analogue to Formula 1.5 wll gve a relaton between the dscrete adjont state and the dscrete control. We refer to the books of röltzsch [36] and Hnze et al. [22] for a detaled dscusson of optmalty condtons and dscretsaton approaches for optmal control problems. he regularty of the soluton of the state equaton 1.2 s dscussed n Secton 3. here, weghted Sobolev spaces are ntroduced that allow to acqure the sngular parts n the soluton more accurately. Fnally, a modfed shft-theorem n these spaces s presented. Secton 4 approaches the fnte element error estmates for the state equaton requred to show the estmates for the optmal control problem More precsely, we derve estmates n L 2 Ω-, H 1 Ω- and L 2 Γ-norm, where the latter one s the frst man result of ths paper. he second man result an optmal error estmate for the Neumann control problem s presented n Secton 5 whch s confrmed numercally n Secton 6. 3

4 2 Dscretsaton of the optmal control problem For the dscretsaton of our optmal control problem a couple of approaches exst. n ths paper we wll consder the varatonal dscretsaton approach [20] only. hs approach employs dscretzaton of the state and adjont state varable but not of the control. he latter one s dscretsed mplctly by means of a dscrete verson of the projecton formula 1.5. n the followng, the boundary value problems n 1.4 are consdered n the weak sense. Hence, we defne the blnear form av, w := [ vx wx + vxwx] dx Ω and the nner products, Γ n L 2 Γ and, Ω n L 2 Ω, respectvely. he weak form of the state equaton 1.2 s then gven by ay, v = f, v Ω + u, v Γ v V := H 1 Ω. 2.1 t s well-known that the Lax-Mlgram lemma states the exstence and unqueness of a soluton n H 1 Ω. We are gong to dscretse problem 2.1 wth contnuous and pecewse lnear fnte elements. o ths end, let { h } h<1 be a famly of conformng tetrahedral meshes of Ω whch are assumed to be shape regular. he mesh parameter h denotes the maxmal dameter of all elements n h,. e. h := max h dam. he ansatz space for state and adjont state s defned by V h := { v h CΩ: v h P 1 for all h }. 2.2 Dscretsng the optmalty system 1.4 yelds now Fnd ȳ h, ū h, p h V h U ad V h : aȳ h, v h = f, v h Ω + ū h, v h Γ v h V h, av h, p h = ȳ h y d, v h Ω v h V h, 2.3 ū h = Π ad α 1 p h Γ. hs s a fnte-dmensonal system whch possesses a unque soluton and can be solved e. g. wth the prmal-dual actve set strategy, the sem-smooth Newton method or the projected gradent algorthm [22, 36]. n the followng, S : L 2 Γ L 2 Ω stands for the control-to-state mappng whch maps a gven control u L 2 Γ to the soluton of the state equaton 1.2 and we wrte y = Su. he operator S h : L 2 Γ V h L 2 Ω stands for ts dscrete verson defned by S h u := y h. Moreover, let S and S h denote the adjont operators to S and S h, respectvely. hs allows us to wrte p Γ := S Sū y d and p h Γ := S h S hū h y d. 2.4 he followng convergence result for the varatonal dscretsaton concept s proven n [21]: 4

5 Lemma 1. Let ū be the soluton of the contnuous optmal control problem and ū h the soluton of the dscrete problem gven by J h u := JS h u, u mn! s. t. u U ad. 2.5 hen, 2.3 forms a necessary optmalty system for 2.5, and there exsts a constant c > 0 such that ū ū h L 2 Γ + ȳ ȳ h L 2 Ω c S S h ū L 2 Ω + S S h ȳ y d L 2 Γ. 2.6 t remans to derve error estmates for the two terms on the rght-hand sde of 2.6 whch wll be done n Secton 4. For the frst term t s well-known that the convergence rate 2 s acheved on convex polyhedra [19] and we wll see that ths rate can be acheved wth approprate mesh refnement also for non-convex domans. he second error term n 2.6 s the fnte element error for the adjont equaton evaluated on the boundary Γ only. Estmates of ths knd are rather less standard and requre very techncal proofs. hs wll be consdered n the second part of Secton 4. Remark 2. Another often-used dscretsaton approach s the post-processng concept [29, 26]. he dea s to approxmate state and adjont state as usual by lnear fnte elements and the control varable by pecewse constant functons. he obtaned soluton ū h of the fnte-dmensonal problem can be mproved by an applcaton of the projecton formula ũ h := Π ad α 1 p h. Usng the new estmates of the present paper t should be possble to mprove the convergence order for ths concept ether. Snce several other terms have to be estmated, a complete proof would exceed the scope of ths paper and s subject of a forthcomng one. n two space-dmensons optmal error estmates can be found n [6] whch rely on the optmal fnte element error estmates n L 2 Γ as well. 3 Regularty of weak solutons Let us frst specfy some notaton. hroughout the paper, Ω s an open bounded polyhedron wth corners x j, j C := {1,..., d }, edges M k, k E := {1,..., d}, and plane faces F l, l F := {1,..., d }. Furthermore, let X j := {k : x j M k } be the ndex set of all edges M k havng an endpont n x j. he boundary of Ω s denoted by Γ. he am of ths secton s to collect some regularty results for the soluton of the boundary value problem y + y = f n Ω, n y = g on Γ. 3.1 t s well known that f the nput data satsfy f L 2 Ω and g H 1/2 Γ, the soluton possesses the regularty y H 2 Ω provded that the boundary of Ω s smooth or convex and polyhedral, but ths s not the case for domans wth reentrant corners and edges. As a remedy one can use weghted Sobolev spaces, because these spaces contan weghts that compensate the sngulartes that occur at corners and edges. Later, our assumptons upon the fnte element mesh and the weghts wll depend on the sngular exponents of the occurrng sngulartes. herefore, we assocate to each edge M k such an exponent λ e k and to each corner x j an exponent λ c j, and summarse them to vectors λ e R d and λ c R d, respectvely. n case of the dfferental operator + and pure Neumann boundary condtons the edge sngular exponents λ e k are explctly gven by λe k := π/ω k cf. [18, Secton 2.5], where ω k 5

6 denotes the nteror angle between the two faces ntersectng at M k. t s known that the occurrng sngulartes can be descrbed n cylnder coordnates r k, ϕ k, z k where the z k -axs concdes wth M k and ϕ k = 0 on one of the faces: S e k r k, ϕ k, z k := r λe k k cosλe k ϕ k. Obvously, the sngular functons are ndependent of the z k -drecton. However, the corner sngulartes are descrbed n sphercal coordnates ρ j, ϕ j, θ j around x j and have the structure [18, heorem 2.6.3] S c j ρ j, ϕ j, θ j := ρ λc j j F jϕ j, θ j, where λ c j := 1/2 + λ c,2 j + 1/4. Here, λ c,2 j s the second-smallest egenvalue the frst one s always zero for the Neumann problem and F j the correspondng egenfuncton of the Laplace-Beltram operator Gj, where G j denotes the ntersecton of the polyhedral cone correspondng to the corner x j and the unt sphere centred at x j. We thus have to assume that the corners have at least dstance two from each other whch can be acheved wth approprate scalng. More precsely, λ c,2 j, F j solves the egenvalue problem Gj F = λf n G j, n F = 0 on G j. 3.2 For further detals we refer to [18, Secton 2.6]. Remark 3. he egenvalue problem 3.2 can be computed exactly for specal cases only. We want to menton Stephan and Whteman [35] who derved λ c,2 j = 40/9 λ c j = 5/3 for the three-dmensonal L-shape doman whch s consdered n the numercal experments n Secton 6. n general, the egenvalue λ c,2 j has to be computed approxmately. Walden and Kellogg [38] and Beagles and Whteman [10] present a way to solve 3.2 numercally usng a fnte dfference method combned wth Raylegh quotent mnmsaton for the dscrete egenvalue problem. n the latter reference ths technque was appled to the Fchera doman whch denotes a doman around a corner at the ntersecton of three mutually orthogonal planes. n the numercal experments of ths reference the exponent λ c j , was computed approxmately for the Drchlet problem. However, the exponent λ c j depends on the type of boundary condton. For a pure Neumann boundary we have computed λ c j 0.84 wth the software package CoCoS [30, 31]. We ntroduce now weghted Sobolev spaces and use the defnton of Maz ya and Rossmann [28]. Let U j, j C, be a coverng of Ω wth d j=1 U j Ω and Ū j M k = f k / X j. By r k x we denote the dstance of x to the edge M k and by ρ j x the dstance to the corner x j. For some gven weghts β R d, R d and a non-negatve nteger l N 0 we defne the space W l,p β, Ω as the closure of C 0 Ω\{x1,..., x d } wth respect to the norm v W l,p := β, Ω d j=1 Ω U α l j ρ j x pβ j l+ α k X j rk ρ j x pk D α vx p dx 1/p, 3.3 6

7 f p [1,, and v W l, := max β, Ω j C α l ess sup ρ j x β j l+ α x Ω U j k X j f p =. Correspondng sem-norms of W l,p β, Ω are gven by v W l,p := β, Ω d j=1 Ω U α =l j v W l, := max β, Ω j C α =l ρ j x pβ j ess sup ρ j x β j x Ω U j k X j k X j rk ρ j x k rk x D α vx. 3.4 ρ j pk D α vx p dx k rk x D α vx. ρ j he related trace spaces W l 1/p,p β, Γ are nduced by the natural norm v l 1/p,p W β, Γ := nf{ u W l,p β, Ω : u Γ\{x 1,...,x d } = v}. 3.5 n ths paper we wll frequently use the weghted Sobolev spaces W l,p β, G on some subset G Ω. hese are defned analogously except that the weghts contaned n the norm defnton are stll related to the corners and edges of Ω. Note that the weghts related to corners and edges far away from G may be omtted. he regularty of the weak soluton of problem 3.1 n weghted Sobolev spaces s proven n [1, 13, 28]. heorem 4. Let be gven some functons f W 0,p 1 1/p,p β, Ω and g W β, Γ wth p 1,. Assume that the edge and corner weghts and β satsfy 2 2/p mn{2, λ e k } < k < 2 2/p for all k E, 2 3/p mn{1, λ c j} < β j < 3 3/p for all j C. hen, the weak soluton y H 1 Ω of the boundary value problem 3.1 satsfes D α y W 1,p β, Ω α = 1. 1/p, Moreover, f β, 0, the a pror estmate D α y W 1,p β, Ω + y L p Ω c f W 0,2 β, Ω + g W 1 1/p,p α =1 β, Γ 3.6 holds. Proof. he desred asserton s stated n heorem of [28] under the addtonal assumpton that λ = 1 and λ = 0 are the only egenvalues of the problem Gj v = λλ + 1v n G j, n v = 0 on G j, 7

8 that are contaned n the strp 1 Re λ 0. Note, that ths egenvalue problem s the same as 3.2 when nsertng the defnton of λ c j compare also [24, Equaton 2.3.3]. hat ths strp ndeed contans only the egenvalues 0 and 1 n our stuaton, and, that algebrac and geometrc multplcty are equal, has been dscussed n [24, Secton 2.3.1]. t remans to prove the a pror estmate 3.6 whch s not drectly stated n [28], but n the followng, we outlne how ths estmate can be concluded. o ths end, ntroduce the space { } H := v L p Ω: D α v W 1,p β, α = 1 wth the naturally nduced norm as stated on the left-hand sde of 3.6, and the operator + A := : H W 0,2 1 1/p,p β, Ω W β, Γ. n t s easy to observe that the operator A s lnear and bounded snce the estmates u W 0,p β, Ω c u W β, Ω, u W 0,p β, Ω c u L p Ω, n u 1 1/p,p W β, Ω c D α u W 1,p Ω, β, α =1 hold for arbtrary u H. More precsely, the frst estmate follows drectly from the norm defnton 3.3, the second one from a trval embeddng takng nto account that β, 0, and the thrd one from the defnton of the trace space 3.5. We also confrm that A s bjectve whch s equvalent to the exstence and unqueness of a soluton n H and follows from the frst part of ths theorem and the Lax-Mlgram Lemma. From the bounded nverse theorem [16, heorem 3.7] we conclude that the nverse mappng A 1 s also contnuous whch s equvalent to 3.6. he above theorem excludes p =, but ths case s needed snce we have to explot regularty n W 2, β, Ω n order to obtan optmal error estmates on the boundary. o overcome ths ssue we apply regularty results n weghted Hölder spaces. heorem 5. Let be gven a functon f C 0,σ Ω wth some σ 0, 1 and let g 0. Assume that the weghts β R d and R d satsfy k 0, 2 λ e k < k < 2 for all k E, β j 0, 2 λ c j < β j for all j C. hen, the weak soluton y H 1 Ω of 3.1 satsfes D α y W 1, β, Ω α = 1. Proof. Let us frst ntroduce the weghted Hölder spaces defned n [28, Secton 8.2]. We denote by U j,k := {x U j Ω: r k < 3ρ j x/2} for k X j a coverng of U j. Furthermore, a Hölder exponent σ 0, 1, a non-negatve nteger l N 0 and some weghts β R d, R d wth k 0 k E are gven. o each edge we assocate the nteger m k := [ k σ]

9 he weghted Hölder space C l,σ β, Ω denotes the space of l tmes contnuously dfferentable functons on Ω := Ω \ k E M k wth fnte norm d u C l,σ := β, Ω sup ρ j x β j l σ+ α x U j j=1 α l + d sup x,y U j=1 k X j α =l m j,k k x y <ρ j x/2 + d sup x,y U j=1 α =l j x y <ρ j x/2 ρ β j j x k X j k X j rk x max{0,k l σ+ α } D α ux ρ j x ρ j x β j k D α ux D α uy x y m k+σ k 3.7 rk x k D α ux D α uy ρ j x y x σ. We ntroduce weghts β := β + σ and := + σ and observe that the nequaltes f C 0,σ β, Ω c f C 0,σ σ, 0 Ω c f C 0,σ Ω hold, where the frst nequalty s a consequence of the embeddng theorem [28, Lemma 8.2.1] whch holds for arbtrary β j σ j C and k 0 k E, and the second one can be confrmed when nsertng β j = σ and k = 0 n the defnton of the norm 3.7. he assumptons upon β and mply that 2 λ e k < k σ < 2, k E, and 2 λc j < β j σ, j C, 3.8 and wth the regularty result from [27, heorem 5.1 and Remark 5.1] we obtan D α y C 1,σ β, Ω for all α = 1. t remans to show that D α u W 1, β, Ω c Dα u C 1,σ β, Ω α = 1. t suffces to bound the W 1, β, Ω-norm by the frst row n the norm defnton 3.7. Obvously, when nsertng β j = β j σ, the corner weghts concde. nsertng k = k σ 0 yelds for the edge weghts rk ρ j k = rk k σ ρ j c rk ρ j max{0, k σ 1+ α }, where we exploted k σ max{0, k σ 1 + α } for all α 1. Consequently, we have shown 3.8 and the asserton can be concluded. n order to derve fnte element error estmates we have to apply several embeddngs whch are summarzed n the followng lemma. A proof can e. g. be found n Lemma and Lemma of [28]. Lemma 6. Let be G Ω. he followng embeddngs hold: 1. Let 1 < q < p. Assume that the weghts satsfy β j + 3/p < β j + 3/q for j C, and 0 < k + 2/p < k + 2/q for k E. hen the contnuous embeddng holds. W l,p l,q β, G W β, G 9

10 2. Assume that p [1,, β j 1 + β j for j C, and k 1 + k, such as k, k > 2/p for k E. hen the contnuous embeddng W l+1,p β, G W l,p β, G holds. n case of β j < 1 + β j, j C, and k < 1 + k, k E, the embeddng s even compact. Remark 7. n [28] the case p = was excluded for the frst part of Lemma 6. However, by consderaton of the Hölder nequalty one can easly confrm the valdty of ths asserton for p =. 4 Fnte element error estmates for the boundary value problem n ths secton, we wll derve fnte element error estmates for the boundary value problem y + y = f n Ω, n y = g on Γ. 4.1 Let us recall, e. g. from [7], the famly of graded meshes we are gong to nvestgate. Denote the dstance of h to the edge M k by r k, := dst, M k and the mnmum dstance to all sngular ponts by r := mn k E r k,. For a gven mesh gradng parameter µ 0, 1] and refnement radus R > 0 the trangulaton has to satsfy the condton h 1/µ, f r = 0, h hr 1 µ, f 0 < r < R, 4.2 h, f R r, for all h. hs s a natural refnement condton whch ensures that adjacent elements have approxmately the same sze. n case of a quas-unform mesh we smply set µ = 1. he smaller ths parameter s, the stronger the mesh s refned locally. f a mesh s refned accordng to 4.2, the number of elements contaned n the trangulaton s of order Oh 3 unless µ < 1/3, compare also the dscussons of Apel et al. [7]. Remark 8. he reader wll already fnd L 2 Ω-error estmates for the refnement strategy 4.2 n [7], but ths reference deals only wth Drchlet and certan mxed problems. he technques used n ths reference do not apply n our case snce usually for the Drchlet problem dfferent weghted Sobolev spaces are employed to descrbe the regularty, see e.g. [28] for more sophstcated dscussons. he frst step s to derve some local nterpolaton error estmates that are used later to prove the convergence order of the fnte element method both n the doman and on the boundary. We wll use two dfferent nterpolaton operators. For approxmaton errors n L Ω, the usual Lagrange nterpolant h : C Ω V h s the preferred operator due to ts stablty n L Ω. However, the Lagrange nterpolant s not always suffcent for our purposes. hus, at some places we apply the quas-nterpolant Z h : W 1,p Ω V h, 10

11 p [1, ], as orgnally ntroduced by Scott and Zhang [34], whch s defned even for nonsmooth functons n Sobolev spaces W 1,p Ω by [Z h v]x := n a ϕ x. =1 Here, the functons ϕ denote the nodal bass functons. More precsely, when {x } n =1 are the nodes of the trangulaton h, we have ϕ x j =,j for all, j = 1,..., n. he dfference to the Lagrange nterpolant s the choce of the coeffcents a whch are defned by a := Π σ vx wth Π σ : L 2 σ P 1 σ, v Π σ v, w h L 2 σ = 0 w h P 1 σ. t remans to specfy the subsets σ. We wll follow the choce used already n [34]: f x s an nteror node, we choose σ = wth some h contanng x. f x s a boundary node, we choose σ = F Γ where F x s a face of some h. An essental advantage of Z h over the usual Lagrange nterpolant s the better stablty property whch s proved n [34]: Lemma 9. For some h let S denote the patch of elements around,.e. S := nt. h hen, for any nteger l 1, p, q [1, ], and non-negatve nteger m, the followng stablty estmate holds: Z h v W m,q c 1/q 1/p h m l h j v W j,p S v W l,p S. 4.3 j=0 Let ˆ be the standard reference tetrahedron havng vertces 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1 see Fgure 1a. We denote by F the affne lnear transformaton from ˆ to a world element h. he weghted Sobolev spaces on a reference settng are defned analogous to W l,p β, Ω wth slght modfcatons of the weght functons that we defne by ˆρ := ˆx, ˆrˆx := dstˆx, ˆM, where ˆM := {te 3 : t 0, 1} s the reference edge. For smplfcaton we assume throughout ths paper that each element touches at most one corner of Ω and has at most one edge whch s contaned n an edge of Ω. f an element volates ths condton t has to be bsected approprately. Analogous to 3.3 we defne a norm of W l,p β, ˆ wth β, R by û W l,p β, ˆ := α l ˆ p ˆρˆx ˆrˆρ pβ l+ α ˆx ˆD α ûˆx p dˆx 1/p. 11

12 ˆx 3 ˆM F 1 y ŷ ĉ ˆx ˆ ˆx 2 M y x rx M c y rx x ˆx 1 a Reference element c b Element touchng an edge and a corner c Element touchng an edge n a sngle pont Fgure 1: he reference element ˆ and the dfferent postons of the orgnal element. Let us brefly dscuss the relaton between the weghts n the reference settng and the orgnal weghts. n the followng ˆx ˆ s arbtrary and we set x = F ˆx. For the case llustrated n Fgure 1b that one edge of s contaned n an edge M of Ω, we defne the ponts y = arg mn z M x z, ŷ = arg mn ˆx ẑ. ẑ ˆM Note that y and F ŷ are n general dfferent ponts. For the transformaton from ˆ to we have to explot the property rx = x y h ˆx F 1 y h ˆx ŷ = h ˆrˆx, 4.4 where the second equvalence holds due to the assumed shape regularty of h. Moreover, f touches also the corner c := x j, we observe that ρx = x c h ˆx ĉ = h ˆρˆx, 4.5 snce c = F ĉ. he case llustrated n Fgure 1c where touches the edge M only n a sngle pont c s treated slghtly dfferent. Here, we get the property rx = x y x c h ˆx ĉ = h ˆρˆx, 4.6 and the edge weght becomes a corner weght n the reference settng. Note, that we dd not consder weghts related to corners and edges that are not touched by, snce these weghts are not needed for our analyss. n order to prove nterpolaton error estmates, the key step s the applcaton of the Bramble-Hlbert Lemma on a reference element and we wll need the followng verson n weghted Sobolev spaces. Lemma 10. Let be gven some functon v W l,q β, ˆ wth q 1,, a postve nteger l, and weghts β, R satsfyng β < 4 3/q, 2/p < < 3 2/q

13 hen, a polynomal p P l 1 ˆ of order l 1 and some c > 0 exst such that holds. v p W l,q β, ˆ c v W l,q β, ˆ Proof. t suffces to show that the norm equvalence u W l,q β, ˆ u W l,q β, ˆ + α l 1 ˆ D α uxdx. 4.8 holds and to nsert u = v p wth some p P l 1 ˆ such that the second part on the rghthand sde vanshes. n Lemma 2.2 of [8] the equvalence 4.8 has been shown for a slghtly dfferent space. However, the key steps of the proof theren are the embeddngs W l,q β, ˆ c W l 1,q β, ˆ and W 1,q β, ˆ W 1,1 1,1 ˆ L 1 ˆ. 4.9 he frst embeddng s stated n part two of Lemma 6 and holds under the assumpton 2/q <. he second embeddng n 4.9 s also a consequence of Lemma 6 and holds under the assumpton 4.7. he local nterpolaton error estmates depend on the poston of the element h wherefore we ntroduce the quanttes ρ j, := dstx j, j C, r k, := dstm k, k E, r := mn k E r k,. Lemma 11. Let h, U j for some j C, and u W 2,p β, S wth some p 6/5, ] be gven. Assume that the weght exponents satsfy 0 β j < 5/2 3/p, and 0 k < 5/3 2/p for all k X j. hen, for l = 0, 1, the followng nterpolaton error estmates hold: u Z h u H l ch 2 l 1/2 1/p u W 2,p β, S ρ β j j, k X j h k h β j rk, ρ j, k, f ρj,s > 0, r k,s > 0 k X j, ρ k β j j,, f ρ j,s > 0, r k,s = 0,, f ρ j,s = Furthermore, let κ := max{max k E k, max j C β j } denote the largest weght. hen, the estmate above smplfes to { u Z h u H l ch 2 l 1/2 1/p u W 2,p β, S r κ, f r S > 0, h κ, f r 4.11 S = 0. Proof. he estmate 4.10 n case of r S > 0 follows from a standard nterpolaton error estmate snce the regularty u W 2,p S can be exploted. We may ntroduce the weghts afterwards and obtan u Z h u H l ch 2 l 1/2 1/p ρ β k j rk, j, u W 2,p β, S k X j ρ j,

14 Let us consder the case that S touches a sngular pont. For arbtrary frst-order polynomals w P 1 S we have [Z h w] = w and wth the trangle nequalty we get the decomposton u Z h u H l u w H l + Z h u w H l We frst consder the second part of Usng an nverse nequalty, the transformaton to the reference element and the stablty of Z h n H 1 S ˆ compare Lemma 9 we get Z h u w H l ch l Z hu w L 2 ch l 1/2 Ẑhû ŵ L 2 ˆ ch l 1/2 û ŵ H 1 S ˆ. Here, S ˆ denotes the patch S transformed to the reference settng va the affne lnear mappng F 1. Note, that the patch S ˆ has dameter ds ˆ = O1 and contans a ball of radus ρs ˆ = O1. For the frst part of 4.13 the transformaton to the reference element and the embeddng H 1 ˆ L 2 ˆ yeld u w H l ch l 1/2 û ŵ H l ˆ ch l 1/2 û ŵ H 1 S ˆ. Usng also the embeddng W 2,6/5 S ˆ H 1 S ˆ, the estmate 4.13 smplfes to u Z h u H l ch l 1/2 û ŵ W 2,6/5 S ˆ Now, we employ a Deny-Lons type argument, e. g. the verson from heorem 3.2 n [17] where the estmate depends only on ds ˆ and ρs ˆ, as well as the transformaton back to S, and arrve at u Z h u H l ch l 1/2 û W 2,6/5 S ˆ ch2 l 1/2 5/6 u W 2,6/5 S, 4.15 where S was exploted n the last step. Henceforth, we have to dstngush among the cases whether S touches a corner or only a sngle edge. We frst consder the case that S touches the edge M k for some k X j, but s away from the corners. he Hölder nequalty wth q := 5p/6 and 1/q + 1/q = 1 yelds v 6/5 W 0,6/5 S = r k x 6k/5 vx 6/5 r k x 6k/5 dx S 6/5p 1/q r k x p k vx p dx r k x q 6 k /5 dx S S he second ntegral can be ntegrated exactly n cylnder coordnates r k, ϕ k, z k around M k and s bounded f 2 q 6 k /5 > 0. hs condton s equvalent to k < 5/3 2/p when nsertng the defnton of q and q. As S s contaned n a cylndrcal sector around M k havng length and radus proportonal h there exst constants c > 0, {1, 2, 3}, such that S r k x q 6 k /5 dx 5/6q c1 +c 2 h c3 h c c 1 0 5/6q r 1 q 6 k /5 k dr k dz k ch k 5/6 1/p,

15 where we used 1/q = 1 6/5p and h 3 n the last step. nsertng 4.16 wth 4.17 nto 4.15 leads to u Z h u H l ch 2 l k 1/2 1/p ρ k β j j, u W 2,p β, S, 4.18 where we already nserted the remanng weghts and exploted that S s away from the corner. Let now S contan also the corner c := x j. Analogous to 4.16 we derve an embeddng nto an approprate weghted Sobolev space and obtan usng the Hölder nequalty v 6/5 W 0,6/5 S = ρ j x 6β j/5 S k X j c ρ j x pβ j S k X j ρ j x q 6β j /5 S 6k /5 rk x vx 6/5 ρ j x 6β j/5 ρ j pk rk x vx p dx ρ j k X j q 6 rk k /5 x ρ j 6/5p k X j 6k /5 rk x dx ρ j 1/q dx We ntroduce sphercal coordnates ρ, ϕ k, ϑ k, whch are centred at c and concde wth the edge M k for ϑ k = 0. hs defnton mples that r k /ρ j = snϑ k. he ntegrals over ϑ k are bounded by a constant ndependent of h under the condton q 6 k /5 > 2 whch s mpled by 2/p < k < 5/3 2/p for all k X j. Hence, a constant c 1 > 0 exsts such that the second ntegral n 4.19 can be smplfed to ρ j x q 6β j /5 S k X j q 6 rk k /5 x ρ j dx 6/5q c1 h 6/5q c ρ 2 q 6β j /5 dρ 0 ch β j 5/6 1/p, 4.20 provded that q 6β j /5 > 3 whch s equvalent to the assumpton β j < 5/2 3/p. As a consequence, we get from 4.15 usng 4.19 and 4.20 the estmate u Z h u H l ch 2 l β j 1/2 1/p u W 2,p β, S, f ρ j,s = 0, and estmate 4.10 s proved completely. Let us now nvestgate how the estmate 4.11 can be deduced from he factors contanng r k, and ρ j, obvously depend on the poston of. herefore we ntroduce the followng defntons. We denote the nteror angle between the edges M k and M l, k, l X j, by α k,l and wrte α j := 1 4 mn k,l X j α k,l for the quarter of the mnmal angle between all edges havng an endpont n x j. We defne some cones C α j k, k X j, also llustrated n Fgure 2, by C α j k := {x U j Ω: r k x/ρ j x sn α j }. Outsde of ths cone, the angular dstance r k x/ρ j x s then bounded from below by a constant dependng only on the angles α j, but not on h. f C α j k = for all k X j, the 15

16 angular dstances to all edges are bounded from below,. e. k rk, c. k X j ρ j, ogether wth ρ β j j, r κ the estmate 4.12 leads to Otherwse, f C α j k for some k X j, we have r = r k,, and snce the angular dstances to the other edges are bounded from below we get from 4.12 the estmate u Z h u H l ch 2 l 1/2 1/p ρ k β j j, r k k, u W 2,p β, S Let us dscuss the terms dependng on r k, and ρ j,. n case of k β j we have ρ k β j j, c. Otherwse, we explot r k, ρ j, and arrve at ρ k β j. Hence, for both cases j, r k k, cr β j estmate 4.11 follows from o deduce estmate 4.11 from 4.10 for r S = 0, the same technque can be appled. n case of ρ j,s > 0 we merely explot that ρ j, ch. Remark 12. Let us recall the embeddngs 4.14 and 4.16 f ρ j,s > 0 or 4.19 f = 0 used n the proof of Lemma 11. Generally speakng one can say that we exploted ρ j,s W 2,p β, S W 2,6/5 S H 1 S, whch holds n partcular for k < 5/3 2/p. n the regularty result of heorem 4 we demanded k > 2 2/p λ e k, and we can always fnd a weght k satsfyng both condtons when λ e k > 1/3 whch s ndeed always the case. f we had used the Lagrange nterpolant h whch promses stablty only n L, we would have used the embeddngs W 2,p β, W 2,3/2 L, whch hold only for k < 4/3 2/p. One easly confrms that ths condton and the assumptons of heorem 4 can only be satsfed when λ e k > 2/3 whch s not the case for domans wth edges havng nteror angle larger or equal 270. Usng the local nterpolaton error estmates derved n Lemma 11 one can prove convergence rates for the fnte element approxmaton of problem 4.1 n H 1 Ω and L 2 Ω. heorem 13. Let Ω be decomposed nto a famly of trangulatons h satsfyng the condton 4.2. Let the mesh refnement parameter µ and the weghts β [0, 1 d and [0, 2/3 d satsfy the nequaltes hen, for arbtrary nput data f W 0,2 β, 1 λ e k < k 1 µ, k E, 1/2 λ c j < β j 1 µ, j C. y y h H l Ω ch 2 l y W ch2 l β, Ω holds for l = 0, 1. 1/ Ω, g W Γ the a pror error estmate β, f W 0,2 β, Ω + g W 1/ β, Ω

17 α j Fgure 2: Defnton of the cones C α j k at a reentrant corner Proof. As a consequence of Cea s Lemma and the decomposton of the doman Ω we obtan y y h 2 H 1 Ω h y Z h y 2 H t remans to nsert the nterpolaton error estmates from Lemma 11 and to adjust the gradng parameter such that the desred convergence rate s obtaned. n case of r S = 0 we have h = h 1/µ and wth µ 1 κ we get y Z h y H 1 ch 1 κ/µ y W β, S ch y W β, S. Otherwse, f r S > 0 the mesh condton yelds h = hr 1 µ and consequently y Z h y H 1 chr 1 µ κ y W β, S ch y W β, S. nsertng these local estmates nto 4.23 leads to y y h H 1 Ω ch y W ch f β, Ω W 0,2 β, Ω + g W 1/ β, Ω Here, we also appled the regularty result of heorem 4 whose condtons are satsfed under our assumptons upon β and. Snce β 0 and 0, hence L 2 Ω W 0,2 β, Ω, we fnally obtan the estmate n the L 2 Ω-norm by the Aubn-Ntsche method. f we assume slghtly better regularty of the nput data n classcal Sobolev spaces we get by the embeddngs from Lemma 6 the followng smplfed verson of heorem 13. Corollary 14. Let be f L 2 Ω and g H 1/2 Γ. hen, the error estmate y y h H l Ω ch 2 l holds, provded that one of the followng assumptons holds: 1. he famly of trangulatons h s quas-unform.e. µ = 1 and the sngular exponents satsfy λ e k > 1, λc j > 1/2 for all k E and j C. 17

18 2. he famly of trangulatons h s refned accordng to 4.2 wth refnement parameter µ < mn{mn k E λe k, mn j C 1/2 + λc j}. n the remander of ths secton, the fnte element error on the boundary Γ s nvestgated. he ntal step of the convergence proof s an approprate decomposton of Γ. n order to extract those parts of the doman whch are under nfluence of sngulartes we defne the sets Ω R := {x: 0 < rx < 1} Ω, Γ R := Ω R Γ, ˆΩ R := {x: 0 < rx < 1 2 } Ω, ˆΓR := ˆΩ R Γ Remember that r := mn k E r k stands for the mnmum dstance to the sngular ponts. he boundary part whch s not nfluenced by sngulartes s denoted by ˆΓ 0 := Γ\ˆΓ R. ˆΓ 0 ˆΓ R Fgure 3: Decomposton of the boundary nto ˆΓ R and ˆΓ 0. For techncal reasons we ntroduce a dyadc decomposton of the doman Ω R. herefore, let d := 2, = 0,..., and let c 1 be a constant ndependent of h such that d = c h 1/µ and hence ln h. he constant c wll be specfed at the end of the proof of heorem 17 and wll play an mportant role. As llustrated n Fgure 4 we defne the sets { {x: d +1 < rx < d } Ω R, for = 0, 1,..., 1, Ω := {x: 0 < rx < d } Ω R, for =, whch form a decomposton of Ω R and ˆΩ R,. e. Ω R = nt Ω, =0 ˆΩR = nt Ω. =1 A decomposton of the boundary part Γ R s then gven by Γ := Ω Γ, = 0,...,

19 Ω 0 Ω 1.. Ω M k Fgure 4: Dyadc decomposton of Ω R along an edge. Note that the elements contaned n Ω or ntersectng Ω satsfy { hd 1 µ, for = 0, 1,..., 1, h h 1/µ, for =. Furthermore, we wll need the patches of Ω wth ts adjacent sets defned by Ω k := nt Ωmax{0, k}... Ω... Ω mn{,+k}, k N, and wrte Ω := Ω1, Ω := Ω2. he frst step of the proof s to derve nterpolaton error estmates on the subdomans Ω. We wll need estmates n the H l Ω -norm l = 0, 1 as well as n the L Ω -norm. n what follows, κ := max{max k E k, max j C β j } denotes agan the largest weght. Lemma 15. Let be gven some functon u H 1 Ω such that D α u W 1,p α = 1, wth p [2, ], non-negatve nteger k, and weghts satsfyng 0 k < 5/3 2/p, k E, 0 β j < 5/2 3/p, j C. hen, for l {0, 1}, there holds h 2 l d 2 l1 µ+1 2/p κ u Z h u H l Ω k c u W 2,p c Θ 1+1 2/p h 3 l 2/p κ/µ u W 2,p where Θ 1 := max{0, 7/2 l 3/p1 µ κ}. Moreover, f D α u W 1, the estmate u h u L Ω k β, Ωk+1 β, Ωk+1 β, Ωk+1 for all α = 1, wth weghts satsfyng 0 k < 5/3, k E, 0 β j < 2, j C, ch 2 d 21 µ κ c u W 2, β, c Θ 2 h 2 κ/µ u W 2, β, 19 Ωk+1 Ωk+1 β, Ωk+1 for all, for = 0, 1,..., k 2,, for = k 1,...,,, for = 0, 1,..., k 2,, for = k 1,...,,

20 holds, where Θ 2 := max{0, 21 µ κ}. Proof. Wthout loss of generalty we prove the asserton for k = 0. he same arguments can be appled n case of k > 0 ether. Let us frst derve the estmate n the H l Ω -norm for = 0, 1,..., 2. By the dscrete Hölder nequalty we get u Z h u 2 H l Ω 1 2/p 1 2/p u Z h u p H l Ω Ω he number of elements contaned n Ω can be estmated by Ω Ω 1 c mn Ω cd2 max Ω n the proof of Lemma 11 we have already shown that the local nterpolaton error for wth r > 0 can be estmated by u Z h u H l ch 2 l 1/2 1/p r κ u W 2,p β, S nsertng ths together wth the mesh condton h hd 1 µ 4.28 nto 4.27 leads to as well as r d +1 = 1 2 d and u Z h u 2 H l Ω ch22 l d 22 l1 µ+1 2/p κ u 2 W 2,p β, Ω. Extractng the root yelds the frst asserton. Let us now consder the case = 1,. he number of elements can be estmated by 1 cd 2 mn 1, Ω where mn s an arbtrary element touchng a sngular edge havng dameter h mn h 1/µ. Due to the mesh condton we have to dstngush between the cases whether touches the sngular ponts or not. f s away from the sngular edges/corners the estmate 4.29 can be appled agan and usng h 3, the mesh condton h hr 1 µ, as well as h 1/µ r d c h 1/µ we obtan u Z h u H l ch 2 l+3/2 3/p r 2 l+3/2 3/p1 µ κ u W 2,p β, S cc Θ 1 h 2 l κ/µ mn 1/2 1/p u W 2,p β, S, 4.30 where we used the property mn h 3/µ n the last step. Wth the local nterpolaton error estmate from Lemma 11 and the mesh crteron h h 1/µ we get almost the same estmate for elements touchng the sngular ponts, namely u Z h u H l ch 2 l κ/µ mn 1/2 1/p u W 2,p β, S he estmates 4.30 and 4.31 can be combned to u Z h u H l cc Θ 1 h 2 l κ/µ mn 1/2 1/p u W 2,p β, S h : Ω. 20

21 Next, we apply the Hölder nequalty 4.27 and obtan u Z h u H l Ω cc Θ 1 h 2 l κ/µ d 1 2/p u W 2,p β, Ω cc Θ 1+1 2/p h 3 l 2/p κ/µ u W 2,p β, Ω. n the remander of ths proof the L Ω error estmates are consdered. herefore, let be the element where the maxmum of ux h ux s attaned. Recall the coverng {U j } d j=1 of Ω ntroduced on page 6. Wthout loss of generalty we may assume that U j. f Ω for some = 0, 1,..., 2, we have u W 2,, and thus we may apply a standard nterpolaton error estmate and ntroduce the weghts afterwards. We obtan k u h u L Ω u h u L ch 2 ρ β k rk, j, u W 2, β, k X j ρ j, ch 2 r κ u W 2, ch2 d 21 µ κ u W 2, β, β, Ω n the thrd step we used the technque already appled n the proof of Lemma 11, where all factors dependng on ρ j, and r k, were smplfed to r κ. he last step follows from the mesh condton h hd 1 µ and from r d. For = 1, we have lower regularty on elements touchng the sngular ponts. hus, standard nterpolaton error estmates cannot be appled. Analogous to the proof of Lemma 11 we ntroduce a polynomal w P 1, splt the error nto two parts and consder the approxmaton error on a reference element. Wth the embeddng W 1,p ˆ L ˆ for some p > 3, we obtan the estmate u h u L û ŵ L ˆ + Îhû ŵ L ˆ c û ŵ W 1,p ˆ. Frst, we consder the case that s away from the corner ponts but possesses an edge whch s contaned n the edge M k, k E of Ω. We apply the embeddng W 2,p 1,1 ˆ W 1,p ˆ from Lemma 6 and the Bramble-Hlbert type argument n weghted Sobolev spaces presented n Lemma 10, and obtan u h u L û ŵ W 2,p 1,1 ˆ c û W 2,p 1,1 ˆ Under the assumpton k < 5/3 and p = 3 + ε wth suffcently small ε > 0 we get from Lemma 6 and the transformaton from ˆ to usng 4.4 the estmate û W 2,p 1,1 ˆ c û W 2, k, ˆ ch2 k k ρ k β j j, u W 2, β, n the last step we already nserted the remanng weghts and used the fact that ρ j, > 0. Furthermore, analogous to the proof of Lemma 11 we can show h 2 k ρ k β j j, ch 2 κ, and conclude from 4.34 and 4.33 u h u L ch 2 κ u W 2, β, f touches the edge M k only n a sngle pont we obtan exactly the same estmate by replacng n 4.33 and 4.34 the space W 2,p 1,1 ˆ by W 2,p 1,0 ˆ and W 2, k, k ˆ by W 2, k,0 ˆ, and by usng the property 4.6 nstead of

22 Let now touch addtonally the corner x j and let an edge of be contaned n M k, k X j. he other edges M l, l X j \ {k}, meetng n x j can be neglected, as touches them only n x j. From 4.6 and 4.6 we conclude for these edges 1 = ˆρ/ˆρ r l /ρ j. We consder agan 4.33, employ the embeddng from Lemma 6 wth β j < 2 and k < 5/3, and obtan usng 4.4 and 4.5 the estmate û W 2,p 1,1 ˆ c û W 2, β j, k ˆ ch2 β j u W 2, β, n the last step we merely nserted the remanng weghts whch are bounded on. After nserton nto 4.33 we arrve agan at n concluson, we have shown that f touches the sngular ponts the estmate u h u L ch 2 κ/µ u W 2, β, 4.36 holds, and otherwse, we obtan from 4.32 wth h 1/µ r d c h 1/µ, = 1,, and some computatons u h u L ch 2 r 21 µ κ u W 2, β, ccθ 2 h 2 κ/µ u W 2, β, he estmates 4.36 and 4.37 mply the last asserton snce Ω. We frst show an ntal error estmate on a sngle boundary strp Γ and wll combne t to a global estmate n heorem 17. Lemma 16. Let y H 1 Ω R L Ω R and denote by y h ts Rtz projecton,. e. Ω R y y h v h + y y h v h = 0 v h V h. hen, for all {1,..., } the local estmate y y h L 2 Γ c d 1/2 ln h nf y χ L χ V Ω + d 1/2 y y h L 2 Ω h 4.38 holds. Proof. Let us frst show the asserton for 1,..., 2. For techncal reasons we decompose the doman Ω as follows. Let x k, y k, z k, k E, denote Cartesan coordnate systems havng orgn n some corner c = x j, j C wth k X j, such that the z k -axes concde wth the edges M k. Moreover, defne α j mn := mn k,l X j α k,l where α k,l s the angle between the edges M k and M l, and ntroduce the set Ω c := {x Ω : z k x < 1 + A d }, where A := 2 cot α j 2 k X j wth outer boundary Γ c := Ωc Γ compare Fgure 5a. t s easy to confrm that Γc d2. o obtan the desred estmate on Γ c we apply the Hölder nequalty and a trace theorem and obtan mn 1, y y h L 2 Γ c d y y h L Γ c d y y h L Ω c

23 Γ +1 Γ c Γ Γ e d d d α k,l /2 Ad Γ e,j Ω e,j Γ 1 e Ω e,j a Defnton of Ω e and Ω c b Defnton of Ω e,j and ts patch Fgure 5: llustraton of the domans defned n the proof of Lemma 16 Now we can apply the local maxmum norm estmate of heorem 10.1 and Example 10.1 n [37], whch reads n our stuaton y y h L Ω c c ln h nf y χ L χ V Ω + d 3/2 y y h L 2 Ω, 4.40 h wth d := dst Ω c \ Γ, Ω \ Γ. Due to our constructon we fnd that d d and nsertng 4.40 nto 4.39 yelds y y h L 2 Γ c c d ln h nf y χ L χ V Ω + d 1/2 y y h L 2 Ω h t remans to derve ths estmate on that part of Ω whch excludes neghbourhoods of the corners. We fx an edge e := M k havng length L and endponts x j, x j, ntroduce the nterval Z := 1 + A d, L 1 + A d and defne the doman Ω e := {x Ω : z k x Z}, havng outer boundary Γ e := Ωe Γ whch s llustrated n Fgure 5a. However, the measure of Γ e s only of order d and a drect applcaton of would yeld a worse estmate. Hence we ntroduce a further decomposton and splt the nterval Z nto N d 1 subntervals Z j such that Z j d. hen, Ω e can also be expressed as the unon of the sets Ω e,j := {x Ω : z k x Z j }, = 1,..., N Agan, we wrte Γ e,j := Ωe,j Γ and confrm that the desred property Γe,j d2 holds. o apply the local maxmum norm estmate 4.40 n a reasonable way we also have to defne patches Ω e,j := { x Ω : z k x nt } Z j 1 Z j Z j+1, 23

24 where we set Z 0 := Ad, 1 + A d, Z N+1 := L 1 + A d, L Ad. Due to ths constructon we have the property dst Ω e,j \ Γ, Ω e,j \ Γ d Explotng the decomposton 4.42, the Hölder nequalty, a trace theorem and the local maxmum norm estmate 4.40 wth property 4.43 leads to N y y h 2 L 2 Γ e d 2 y y h 2 L Ω e,j c c j=1 N j=1 n the last step we exploted that N =1 1 d 1 that Γ = Γ xj j C d 2 ln h 2 y χ 2 L Ω e,j + d 1 y y h 2 L 2 Ω e,j d ln h 2 y χ 2 L Ω + d 1 y y h 2 L 2 Ω and that N j=1 Ω,j Ω. Fnally we observe and together wth 4.41 and 4.44 we arrve at the asserton. t remans to show 4.38 also for = 1, whch cannot be obtaned wth the same technque, snce the local maxmum norm estmate 4.40 s not applcable f Ω contans the sngular ponts. herefore, we frst nsert an arbtrary element χ V h as ntermedate functon and apply the trangle nequalty whch leads to k M Γ M k y y h L 2 Γ c y χ L 2 Γ + χ y h L 2 Γ Next, we apply the Hölder nequalty and a trace theorem to get y χ L 2 Γ cd 1/2 y χ L Ω For the second part of 4.45 we explot that χ y h s a functon from a fnte-dmensonal space. Let F denote the set of faces F of elements h wth F Γ. he faces have dameter h F h f F. Applyng a trace theorem on a reference settng as well as a norm equvalence leads to χ y h 2 L 2 Γ F F χ y h 2 L 2 F c h 2 F ˆχ ŷ h 2 L 2 ˆF c h 2 ˆχ ŷ h 2 L 2 ˆ F F Ω c h 1 χ y h 2 L 2 ch 1/µ χ y h 2 L 2 Ω Ω 24

25 Extractng the root and takng d h 1/µ nto account yelds χ y h L 2 Γ cd 1/2 χ y h L 2 Ω. Wth the trangle nequalty and the Hölder nequalty explotng that Ω d2, we obtan χ y h L 2 Γ c d 1/2 y χ L Ω + d 1/2 y y h L 2 Ω nsertng 4.46 and 4.48 nto 4.45 mples the estmate 4.38 for = 1,. he next step of the proof s to derve a fnte element error estmate on the boundary ˆΓ R whch s under nfluence of corner and edge sngulartes. herefore, we localse the soluton y wth a smooth cut-off functon ω C Ω satsfyng and defne ỹ := ωy. Let ω ˆΩR 1 and supp ω Ω R, 4.49 V h Ω R := {v h V h : v h 0 n Ω \ Ω R } denote the space of ansatz functons vanshng outsde of Ω R. hs defnton mples that supp v h Ω R. n what follows, ỹ h V h Ω R denotes the Rtz-projecton of ỹ,.e. aỹ ỹ h, v h = 0, for all v h V h Ω R An error estmate for ths Rtz-projecton s consdered n the followng theorem: heorem 17. Let D α ỹ W 1,2 α, γ Ω R W 1, β, Ω R for all α = 1, wth weghts satsfyng 0 α j < 1, 0 β j < 2, j C, 0 γ k < 2/3, 0 k < 5/3, k E. Assume that the refnement condton 4.2 wth parameter µ 1 κ 2, wth κ 2 := max{max j C α j, max k E γ k}, µ 5 4 κ 2, wth κ := max{max j C β j, max k E k}. hen, for ỹ and ỹ h defned n 4.50 the estmate ỹ ỹ h L 2 ˆΓ R ch2 ln h 3/2 ỹ W α, γ Ω R + ỹ W 2, β, Ω R holds. Proof. We consder the decomposton of the boundary ˆΓ R nto the segments Γ := Ω Γ ntroduced n n Lemma 16 we have already derved the local estmate ỹ ỹ h L 2 Γ c d 1/2 ln h ỹ h ỹ L Ω + d 1/2 ỹ ỹ h L 2 Ω

26 on Γ for all = 1,...,. Now, we sum up all boundary parts Γ and ncorporate the property d = 1 2 d d + r := γr on Γ whch leads to ỹ ỹ h 2 L 2 ˆΓ R c ln h 2 d ỹ h ỹ 2 L Ω + γ 1/2 ỹ ỹ h 2 L 2 Ω R =1 For the frst term on the rght-hand sde of 4.52 we explot ln h and the local nterpolaton error estmate n L Ω -norm from Lemma 15 whch leads to =1 d ỹ h ỹ 2 L Ω c ln h h4 ỹ 2 W 2, β, Ω R Here, we already nserted the assumpton µ 5/4 κ /2. he second part requres an estmate for a weghted L 2 Ω R error. herefore, we adopt the technque that was appled n the proof of Lemma 6.2 n [33] and use a dualty argument. Frst, we decompose the error nto γ 1/2 ỹ ỹ h L 2 Ω R γ 1/2 ỹ ỹ h L 2 Ω R \Ω 0 Ω 1 + γ 1/2 ỹ ỹ h L 2 Ω 0 Ω 1. On the outermost rngs Ω 0 Ω 1 we explot that γ 1 and use heorem 13 whch leads to γ 1/2 ỹ ỹ h L 2 Ω 0 Ω 1 c ỹ ỹ h L 2 Ω R ch 2 ỹ W α, γ Ω R For the error on Ω R := Ω R \Ω 0 Ω 1 we apply the representaton and consder the auxlary problem γ 1/2 ỹ ỹ h L 2 Ω R = sup γ 1/2 ỹ ỹ h, g 4.55 g C 0 Ω R g L 2 ΩR =1 w + w = γ 1/2 g n Ω R, n w = 0 on Ω R From the weak formulaton of 4.56 we can deduce γ 1/2 ỹ ỹ h, g = ỹ ỹ h, γ 1/2 g = aỹ ỹ h, w We ntroduce a further cut-off functon η C 0 Ω R such that η 1 on Ω R := Ω R \ Ω 0 Ω 1, and supp η ˆΩ R, and we make use of the decomposton w = w 1 + w 2 wth w 1 := ηw and w 2 := 1 ηw. he defnton of w 1 mples that Z h w 1 V h Ω R whch allows us to apply the Galerkn orthogonalty hs yelds c aỹ ỹ h, w 1 = aỹ ỹ h, w 1 Z h w 1 3 ỹ ỹ h H 1 Ω w 1 Z h w 1 H 1 Ω + =3 = 2 ỹ ỹ h H 1 Ω w 1 Z h w 1 H 1 Ω + ỹ ỹ h H 1 Ω 1 w 1 Z h w 1 H 1 Ω

27 t remans to estmate the three terms on the rght-hand sde. he cases = 3,..., 3 and = 2,..., such as = 0, 1, 2 are consdered separately. For the frst part of the rght-hand sde of 4.58,. e. = 3,..., 3, we apply the local H 1 Ω error estmate of Corollary 9.1 n [37] whch reads n our stuaton ỹ ỹ h H 1 Ω c ỹ Z h ỹ H 1 Ω + d 1 ỹ Z h ỹ L 2 Ω + d 1 ỹ ỹ h L 2 Ω nsertng the nterpolaton error estmates of Lemma 15 leads to the nequalty ỹ ỹ h H 1 Ω c hd 2 µ κ 1 + hd µ ỹ W 2, ỹ ỹ h L 2 Ω β, Ω + d 1 Further, note that hd µ hd µ = c µ 1 n case of c > 1. o estmate the term w 1 Z h w 1 H 1 Ω for = 3,..., 3 n 4.58 we drectly apply the local H 1 Ω nterpolaton error estmates of Lemma 15 wth p = 2. Note that w 1 s an element of W 1/2, 1/2 Ω R for arbtrary polyhedra, snce the assumptons of heorem 4 are always satsfed as λ e k > 1/2 k E and λ c j > 0 j C. Wth κ = 1/2, Lemma 15 now mples w 1 Z h w 1 H 1 Ω chd 1/2 µ w 1 W 1/2, 1/2 Ω = chd1/2 µ w W 1/2, 1/2 Ω he last step s a consequence of the fact that η 1 on Ω R and hence w 1 w on all Ω wth = 3,..., 3. Combnng 4.60 and 4.61 yelds for = 3,..., 3 the estmate ỹ ỹ h H 1 Ω w 1 Z h w 1 H 1 Ω c h 2 d 5/2 2µ κ ỹ W 2, β, Ω + hd 1/2 µ ỹ ỹ h L 2 Ω w W 1/2, 1/2 Ω c h 2 ỹ W 2, β, Ω + c µ γ 1/2 ỹ ỹ h L 2 Ω w W 1/2, 1/2 Ω he last step follows from the assumpton upon µ and the defnton of the domans Ω, more precsely we exploted d µ < c µ h 1. o obtan a smlar estmate n case of = 2,..., we have to nsert the approprate nterpolaton error estmates of Lemma 15 nto 4.59 whch mples ỹ ỹ h H 1 Ω c c Θ 1+1 h 2 κ /µ ỹ W 2, ỹ ỹ h L 2 Ω, β, Ω + d 1 w 1 Z h w 1 H 1 Ω cc max{0,1/2 µ} h 1/2µ w 1 W 1/2, 1/2 Ω. n the followng we neglect the factor c where not needed snce t s ndependent of h. Combnng both estmates leads to ỹ ỹ h H 1 Ω w 1 Z h w 1 H 1 Ω c h 5/2 κ /µ ỹ W 2, β, Ω + cmax{0,1/2 µ} h 1/2µ d 1 ỹ ỹ h L 2 Ω w 1 W 1/2, 1/2 Ω c h 2 ỹ W 2, β, Ω + cmax{ 1/2, µ} γ 1/2 ỹ ỹ h L 2 Ω w W 1/2, 1/2 Ω

28 he last step follows from the assumpton upon µ and the fact that d d = c h 1/µ for = 2,...,. Moreover, we exploted w w 1 on Ω for = 2,...,. For the last part of 4.58 we apply heorem 13 for the fnte element error compare also 4.54 and Lemma 15 for the nterpolaton error. he factors d 0, d 1 and d 2 are of order one and can thus be neglected. We then obtan ỹ ỹ h H 1 Ω 1 w 1 Z h w 1 H 1 Ω 1 ch 2 ỹ W 2, β, Ω R w 1 W ch 2 ỹ W 2, β, Ω R 1/2, 1/2 Ω 1 w W 1/2, 1/2 Ω 1 + w H 1 Ω he last step s a consequence of the Lebnz rule and the fact that D α η c for all α 2, as well as the property 1/16 rx 1 on Ω 1 whch allows us to neglect the weghts hdden n the norm defnton. We may now nsert the estmates 4.62, 4.63 and 4.64 nto 4.58 whch leads to aỹ ỹ h, w 1 c h 2 ỹ W 2, β, =3 Ω + cmax{ 1/2, µ} γ 1/2 ỹ ỹ h L 2 Ω w W 1/2, 1/2 Ω ch ỹ 2 W α, γ Ω R + ỹ W 2, β, Ω w R W 1/2, 1/2 Ω R + w H 1 Ω R c h 2 ln h 1/2 ỹ W α, γ Ω R + ỹ W 2, β, Ω + c max{ 1/2, µ} R γ 1/2 ỹ ỹ h L 2 Ω R w W 1/2, 1/2 Ω R + w H 1 Ω R Next, we show that w possesses the regularty demanded by the rght-hand sde, whch follows from heorem 4 and the Lax-Mlgram Lemma once we have shown that γ 1/2 g W 0,2 1/2, 1/2 Ω R H 1 Ω R For some fxed x U j defne k X j such that r kx = rx. he angular dstance to the edges M k wth k X j \ { k} s bounded from below,.e. r k /ρ j c compare also Fgure 2. Consequently, we obtan and drectly conclude γ 1 x = d + rx 1 rx 1 = r kx 1 = ρ j x 1 r k cρ j x 1 k X j 1 x ρ j 1 rk x, 4.68 ρ j γ 1/2 g W 0,2 1/2, 1/2 Ω R c g L 2 Ω R c o show the boundedness n H 1 Ω R we use the operator norm representaton, the Cauchy- Schwarz nequalty as well as the boundedness of g n L 2 Ω R, and arrve at γ 1/2 g H 1 Ω R = sup g, γ 1/2 ϕ ΩR ϕ H 1 Ω R ϕ H 1 Ω R 28 c sup ϕ H 1 Ω R γ 1/2 ϕ L 2 Ω R ϕ H 1 Ω R

29 akng agan 4.68 nto account leads to γ 1/2 ϕ L 2 Ω R c ϕ W 0,2 1/2, 1/2 Ω R c ϕ H 1 Ω R, 4.71 where the second step s a consequence of the embeddng W 1,2 0, 0 Ω R W 0,2 1/2, 1/2 Ω R see Lemma 6 and the fact that the spaces W 1,2 0, 0 Ω R and H 1 Ω R are equvalent. nsertng 4.71 nto 4.70 and takng also 4.69 nto account yelds 4.67, and consequently he estmate 4.65 then becomes aỹ ỹ h, w 1 c h 2 ln h 1/2 ỹ W α, γ Ω R + ỹ W 2, β, Ω R w W 1/2, 1/2 Ω R + w H 1 Ω R c c max{ 1/2, µ} γ 1/2 ỹ ỹ h L 2 Ω R t remans to derve a smlar estmate wth w 2 nstead of w 1. herefore, we explot that w 2 0 on Ω R, and n w 2 0 on Ω R. Partal ntegraton then yelds aỹ ỹ h, w 2 = ỹ ỹ h, w 2 + ỹ ỹ h, w 2 + ỹ ỹ h, n w 2 ΩR ỹ ỹ h L 2 Ω R w 2 H 2 Ω R \ Ω R We explot the property D α η c for all α 2 and the fact that Ω R \ Ω R has postve dstance to the sngular ponts, and arrve at w 2 H 2 Ω R \ Ω R c w H 2 Ω R \ Ω R w c H 1 Ω R + w W 1/2, 1/2 Ω c, R where the last estmate s another applcaton of estmate from heorem 13 nto 4.74 and get Moreover, we nsert the global aỹ ỹ h, w 2 ch 2 ỹ W α, γ Ω R Wth the equatons 4.55 and 4.57, the decomposton w = w 1 + w 2, and the estmates 4.73 and 4.75, we get γ 1/2 ỹ ỹ h L 2 Ω R c c Θ 1+1 h 2 ln h ỹ 1/2 W α, γ Ω R + ỹ W 2, β, Ω R + c max{ 1/2, µ} γ 1/2 ỹ ỹ h L 2 Ω R We fx the generc constant c and choose c suffcently large such that cc max{ 1/2, µ} 1/2. hs allows us to apply a kck-back argument and we consequently arrve at γ 1/2 ỹ ỹ h L 2 Ω R ch2 ln h ỹ 1/2 W α, γ Ω R + ỹ W 2, β, Ω. R akng also 4.54 nto account leads to γ 1/2 ỹ ỹ h L 2 Ω R ch 2 ln h ỹ 1/2 W α, γ Ω R + ỹ W 2, β, Ω. R nsertng ths together wth 4.53 nto 4.52 yelds the asserton. 29

30 Now we are able to prove the man result of ths secton. heorem 18. Assume that f C 0,σ Ω and g 0 and that the trangulaton h satsfes the condton 4.2. Moreover, let be gven weghts α [0, 1 d, β [0, 2 d and γ [0, 2/3 d, [0, 5/3 d, whch satsfy 1 2 λc j < α j 1 µ, 2 λ c j < β j 5 2µ, j C, 2 1 λ e k < γ k 1 µ, 2 λ e k < k µ, k E. Let y denote the soluton of the boundary value problem 4.1 and y h ts fnte element approxmaton. hen some c > 0 exsts such that y y h L 2 Γ ch 2 ln h 3/2 D α y W 1,2 α, γ Ω + D α y W 1, β, Ω + y L Ω α =1 α =1 Proof. For techncal reasons we ntroduce further subsets Ω R := nt Ω, =2 ΩR := nt Ω, ΓR := Ω R Γ, ΓR := Ω R Γ. =3 Note that we have the relaton Ω R Ω R ˆΩ R Ω R Ω n the sense of Wahlbn [37, Chapter 10]. Let ω be the cut-off functon defned n n order to apply heorem 17 we nsert the ntermedate functon ỹ h and explot that ỹ := ωy concdes wth y n ˆΩ R. hs leads to y y h L 2 Γ R ỹ ỹ h L 2 ˆΓ R + ỹ h y h L 2 Γ R For the frst part we may now apply the result of heorem 17 and obtan ỹ ỹ h L 2 ˆΓ R ch2 ln h ỹ 3/2 W α, γ Ω R + ỹ W 2, β, Ω R Note that t s possble to construct a cut-off functon ω satsfyng 4.49 and D α ω L Ω R c wth some c > 0 dependng only on α and dst ˆΩ R \ Γ, Ω R \ Γ = 1/2. Moreover, we have D α ω = 0 on ˆΩ R for α 1. he Lebnz rule then yelds ỹ W 2,p β, Ω R = ωy W 2,p β, Ω R c y W 2,p β, Ω R + y W 1,p Ω\ˆΩ R c D α y W 1,p β, Ω + y L p Ω, p [1, ], 4.81 α =1 where we also exploted that the weghts are bounded wthn Ω \ ˆΩ R. Let us dscuss the second part of For the error of ỹ h y h we frst apply a trace theorem and explot that ỹ h y h s dscrete harmonc on ˆΩ R. Hence, the dscrete Caccoppoltype estmate of Lemma 3.3 n [15] can be appled and we obtan ỹ h y h L 2 Γ R c ỹ h y h H 1 Ω R c ỹ h y h L 2 ˆΩ R c ỹ ỹ h L 2 Ω R + y y h L 2 Ω. 30

31 Here we also used the propertes dst ˆΩ\Γ, Ω\Γ = 1/4 c and y = ỹ on ˆΩ R. An applcaton of heorem 13 yelds then ỹ h y h L 2 Γ R ch2 ỹ W α, γ Ω + y W α, γ Ω ch 2 D α y W 1,2 α, γ Ω + y L 2 Ω, 4.82 where we reused the technque appled n 4.81 n the last step. Consequently we get from 4.79 the estmate α =1 y y h L 2 Γ R ch2 ln h 3/2 D α y W 1,2 α, γ Ω R + D α y W 1, β, α=1 α=1 Ω + y L Ω Let us consder the error on the remanng part Γ\ Γ R where we have no nfluence of the sngulartes. One can drectly apply the trace theorem n L -norm whch yelds y y h L 2 Γ\ Γ R c y y h L Γ\ Γ R c y y h L Ω\ Ω R Now we may apply the local maxmum norm estmate of heorem 10.1 n [37] compare also 4.40 whch yelds y y h L Ω\ Ω R c ln h y h y L Ω\ Ω R + y y h L 2 Ω\ Ω R he second part on the rght-hand sde of 4.85 has been estmated n heorem 13 whose assumptons are obvously satsfed. he element where the maxmum of y h y s acqured s denoted by. An applcaton of a standard L nterpolaton error estmate mples y h y L Ω\ Ω R y hy L ch 2 y W 2, and we may nsert the weghts whch are bounded by a constant wthn Ω\ Ω R. hs estmate together wth 4.85 and 4.84 yelds y y h L 2 Γ\ Γ R ch2 ln h y W 2, Ω, β, and together wth 4.83 the desred estmate 4.78 follows. From the above theorem we drectly conclude the followng observatons. Corollary 19. Assume that f C 0,σ Ω for arbtrary σ 0, 1, and g 0. he error estmate y y h L 2 Γ ch 2 ln h 3/2 holds, provded that one of the followng assumptons s satsfed: 1. he trangulaton h s quas-unform. e. µ = 1, and the sngular exponents satsfy λ e k > 3/2, λc j > 3/2 for all k E and j C. 2. he trangulaton h s refned accordng to 4.2 wth parameter µ < } {mn 2 mn k E λe k, mn j C λc k

32 Remark 20. One observes that the refnement condton necessary for an optmal convergence rate n L 2 Γ-norm s a dfferent one than for error estmates n the L 2 Ω-norm see Corollary 14. However, on the one hand we have µ < λe k < λe k, snce λ e k > 1/2, and, on the other hand we have µ < λc j < λc j, snce λ c j > 0. hus, the mesh gradng condton requred for optmal error estmates on the boundary from Corollary 19 mples the condton requred for the estmates n the doman from Corollary 14. Remark 21. Let us recall the fact that λ e k > 1/2 k E and λc j > 0 j C. f there are only sngular edges but no corners, one can always fnd a refnement parameter µ > 1/2 satsfyng However, t may happen that the corner sngular exponent s very close to 0 and hence, we have to choose µ close to 1/4. Wth our refnement strategy the number of elements contaned n h s then not of order h 3, and as a consequence we would not get optmal convergence wth respect to the number of degrees of freedom #DOF. As a remedy, we can use another refnement strategy, for nstance the one from [25], whch allows a stronger refnement only towards the corner wth parameter ν 0, 1]. n a vcnty of a corner we can choose ν arbtrarly small and stll have #DOF = Oh 3. Unfortunately, the extenson of the proof of heorem 18 to ths advanced refnement strategy s not obvous. 5 Error estmates for the optmal control problem We are now n the poston to formulate the second man result of ths paper. Replacng the error norms n Lemma 1 wth the estmates obtaned n Corollary 14 and Corollary 19 yelds: heorem 22. Let y d C 0,σ Ω wth some σ 0, 1, and f L 2 Ω. By ȳ, ū and ȳ h, ū h we denote the solutons of 1.1 and 2.5, respectvely. Assume that the refnement crteron 4.2 s satsfed wth parameter hen, the error estmate holds. µ < mn {mn k E λe k, mn j C λc j ū ū h L 2 Γ + ȳ ȳ h L 2 Ω ch 2 ln h 3/2 }. 5.1 Proof. o show the desred estmate we nsert the results from Corollary 14 and Corollary 19 nto Lemma 1 and get whch holds true once we have shown ū ū h L 2 Γ + ȳ ȳ h L 2 Ω ch 2 ln h 3/2 5.2 ū H 1/2 Γ + ȳ ȳ d C 0,σ Ω c

33 As a consequence of the projecton formula 1.5, we get ū H 1/2 Γ ū c L 2 Γ + Π ad α 1 p H 1/2 Γ. 5.4 For the second term on the rght-hand sde we explot the defnton of the H 1/2 Γ-semnorm, dstngush among the subsets of Γ where ū s actve and nactve, and fnd analogous to [6, heorem 4.1] that Π ad α 1 p H 1/2 Γ c α 1 p H 1/2 Γ. ogether wth a trace theorem and the Lax-Mlgram Lemma we get from 5.4 ū H 1/2 Γ c ū L 2 Γ + ȳ y d L 2 Ω c f L 2 Ω + ū L 2 Γ + y d L 2 Ω c. 5.5 n order to show the regularty of ȳ n 5.3 we explot the embeddng H 3/2+εreg Ω C 0,σ Ω whch holds for σ 0, ε reg. he regularty ȳ H 3/2+εreg Ω wth some ε reg 0, 1/2] s stated n [14, Corollary 23.5] and holds under the assumpton f L 2 Ω and ū H 1/2 Γ, whch has been dscussed already n 5.5. Collectng up all regularty results leads to the asserton. Corollary 23. f the condtons of heorem 22 are satsfed the estmate holds. Proof. Usng the representaton 2.4 we get p p h L 2 Γ ch 2 ln h 3/2. p p h L 2 Γ = S ȳ y d S h ȳ h y d L 2 Γ c S S h ȳ y d L 2 Γ + S h ȳ ȳ h L 2 Γ. Applyng the boundedness of S h from L2 Ω to L 2 Γ, and the estmates from heorem 18 and heorem 13, leads to the asserton. Example 24 Pure edge sngulartes. he case of pure edge sngulartes s the easer case snce the sngular exponent λ e k := π/ω k s explctly known. Consequently, the assumpton of heorem 22 reads µ < π 2ω k, whch guarantees optmal convergence rates. f the angle tends to 360 the refnement parameter has to be chosen close to 1/2. he L-shaped doman wth nteror angle of 270 has edge sngular exponent λ e k = 2/3 and thus, the assumptons of heorem 22 requre µ < = he sngular exponents for the endponts of the sngular edge are gven by λ c j = 5/3 whch s proven n [35]. Due to λ c j > λe k no addtonal refnement towards the corner s necessary here. he three-dmensonal L-shape wll be consdered n our experments n Secton 6. 33

34 Example 25 Fchera corner. n Secton 3 we have already dscussed the sngular behavour at a Fchera corner and found that the corner sngular exponent s approxmately gven by λ c j For the edges havng an endpont n ths corner we have λ e k = 2/3. he assumptons of heorem 22 then yeld the condton µ < } {λ 2 mn c j, mn λ e k = 1 k X j 4 + λe k 2 = Because of λ c j > λe k, the Fchera corner does not requre an addtonal refnement n a vcnty of the corner. he crteron for the sngular edges s stronger n ths case. 6 Numercal experments o confrm our theoretcal nvestgatons we constructed a smple benchmark example. We consder the problem 1 mn y,u L 2 Ω U ad 2 y y d 2 L 2 Ω + α 2 u 2 L 2 Γ + yxg 2 xds x subject to the constrants y + y = f n Ω, n y = u + g 1 on Γ, and u 0.4. Due to the addtonal term n the target functonal the adjont problem now reads p + p = y y d n Ω, n p = g 2 on Γ. hs allows us to construct an example whch has an analytc soluton. Note that our theory can be smply extended to ths modfed problem. n ths example, Ω s the three-dmensonal L-shape, see Fgure 6. he exact soluton gven n cylnder coordnates r, ϕ, z wth z-axs concdng wth the sngular edge s set to ȳ = 16z 4 32z z 2 + 1r λe cosλ e ϕ. Obvously, t contans the domnant part of the edge sngularty f λ e := π/ω = 2/3. Furthermore, let p = ȳ be the optmal adjont state and let ū be defned by the projecton formula. he nput data f and y d can be computed by means of state and adjont equaton and the functons g 1, g 2 L 2 Γ are ntroduced such that the boundary condtons of state and adjont equaton are satsfed. he sngular behavour on ths doman has been dscussed n Example 24. We computed the soluton of our model problem on the unform mesh µ = 1, on a slghtly refned mesh µ = , on a mesh whch guarantees optmal convergence for the fnte element error n L 2 Ω-norm but not n L 2 Γ-norm µ = , and on a mesh satsfyng the assumptons of the man result heorem 22 µ = 0.5. he refnement routne works as follows. We marked all cells for those 1 µ r h > h wth r := sup x 2 1 R + x x holds. he parameter R s the refnement radus,. e. all elements whch have dstance less than R to the sngular edge are at least refned once when µ < 1, provded that all elements 34 Γ

35 a µ = 1 b µ = 0.5 Fgure 6: Local refnement of a very coarse grd wthn ths radus have equal dameter see e. g. Fgure 6a. n the present experment we used R = 0.2. All marked cells are then decomposed regularly,.e. each tetrahedron s dvded nto eght peces. o construct a conformng closure we adapt the strategy that has been studed n [11]. he markng and refnement procedure s repeated untl there s no element satsfyng 6.1 any more. hs refnement strategy appled to a coarse mesh can be seen n Fgure 6b ū ūh L 2 Γ 10 3 µ = 1 µ = µ = µ = 0.5 ON 1.16/3 ON 1.5/3 ON 1.71/3 ON 2/ Number of DOFs Fgure 7: Error ū ū h L 2 Γ, sold lnes: measured error, dotted lnes: expected behavour Fgure 7 confrms that the measured error concdes wth the theoretcally predcted behavour whch s llustrated by the dotted lnes. he results wdely concde wth experments for two-dmensonal problems wth the post-processng strategy n [6]. 35