Journal of Computational and Applied Mathematics. Finite element analysis for the axisymmetric Laplace operator on polygonal domains

Transcription

1 Jounal of Coputational and Applied Matheatics 235 (2011) Contents lists available at ScienceDiect Jounal of Coputational and Applied Matheatics jounal hoepage: Finite eleent analysis fo the axisyetic Laplace opeato on polygonal doains Hengguang Li Institute fo Matheatics and its Applications, Univesity of Minnesota, Minneapolis, MN 55455, USA aticle info abstact Aticle histoy: Received 10 Novebe 2010 Received in evised fo 8 Apil 2011 Keywods: Axisyetic Poisson s equation Weighted Sobolev space Finite eleent ethod Optial convegence ate Let L := 2 ( ) 2 2 z. We conside the equation Lu = f on a bounded polygonal doain with suitable bounday conditions, deived fo the thee-diensional axisyetic Poisson s equation. We establish the well-posedness, egulaity, and Fedhol esults in weighted Sobolev spaces, fo possible singula solutions caused by the singula coefficient of the opeato L, as 0, and by non-sooth points on the bounday of the doain. In paticula, ou estiates show that thee is no loss of egulaity of the solution in these weighted Sobolev spaces. Besides, by analyzing the convegence popety of the finite eleent solution, we povide a constuction of ipoved gaded eshes, such that the quasi-optial convegence ate can be ecoveed on piecewise linea functions fo singula solutions. The intoduction of a new pojection opeato fo the weighted space to the finite eleent subspace, cetain scaling aguents, and a calculation of the index of the Fedhol opeato, togethe with ou egulaity esults, ae the ingedients of the finite eleent estiates Elsevie B.V. All ights eseved. 1. Intoduction Let := [0, 2π) R 3 be a bounded doain, foed by the evolution of the polygon R 2 with espect to the z-axis (see Fig. 1). Conside the thee-diensional Poisson s equation in, with zeo Diichlet bounday conditions. In the pesence of axisyety in the data, the Laplace opeato in the thee-diensional doain becoes the two-diensional elliptic opeato L := 1 2 ( ) 2 2 z, > 0, whee and z ae the vaiables in the cylindical coodinates (,θ,z). Consequently, the thee-diensional axisyetic Poisson s equation can be educed to Lu = f in, u Γ0 = 0, whee Γ 0 :=. We ae inteested in studying the finite eleent ethod (FEM) fo the elliptic equation (1). The eduction of the diension (fo thee diensions to two diensions) leads to substantial savings on the coputation of the nueical solution fo the oiginal thee-diensional elliptic bounday value poble, and hence is of pactical inteest. Suppose the closue of the doain intesects the z-axis. Despite the benefit in nueical coputation, this pocess, howeve, intoduces singula coefficients in the elliptic opeato L and esults in Sobolev spaces H () ={v, 1/2 i j z v L2 (), i + j } with weights vanishing at = 0, which aises difficulties both in the analysis of the equation and in the estiates of the FEM. Fo the validation on the eduction of the diension, it is shown in [1,2] that, the thee-diensional Poisson s equation (1) E-ail addess: li_h@ia.un.edu /$ see font atte 2011 Elsevie B.V. All ights eseved. doi: /j.ca

2 5156 H. Li / Jounal of Coputational and Applied Matheatics 235 (2011) is equivalent to the two-diensional equation (1), by using Fouie analysis to pove cetain isoophiss between the usual Sobolev spaces H ( ) and the weighted spaces H (). An appoxiation popety of the finite eleent solution fo the axisyetic Stoes poble in the space H is discussed in [3]. We also ention [4,5], in which the Fouie FEM, a cobination of the appoxiating Fouie and the FEM, is studied fo the axisyetic Poisson s equation. In addition, estiates on the convegence of the ultigid ethod fo the axisyetic Laplace opeato and fo the Maxwell opeato can be found in [6,7], espectively. Assuing sufficient egulaity of the solution of Eq. (1), the existing esults (see [3,6 8] and efeences theein) suggest that the H 1 -no of the eo between the linea finite eleent solution and the eal solution is bounded by Ch on the tiangulation with quasi-unifo tiangles of size h. This povides the analogy of the quasi-optial convegence ate of the finite eleent solution fo elliptic bounday value pobles with egula coefficients in the usual Sobolev spaces and ensues good finite eleent appoxiations fo the thee-diensional axisyetic equation with a uch lighte coputational load than solving the oiginal thee-diensional poble. Futheoe, the solution of Eq. (1) ay have singulaities even in these weighted spaces H (), due to the non-sooth points on the bounday and to the singula coefficient when 0. The less egulaity in the solution slows down the convegence ate of the finite eleent solution, as well as aises well-posedness concens in these weighted spaces. Note that nea the vetices of that ae not on the z-axis, the coefficients of the opeato L ae bounded and theefoe, the singulaities in the solution have the sae chaacte as the cone singulaities of egula elliptic equations on polygonal doains. Thee exists a geat deal of liteatue egading diffeent aspects of cone singulaities of two-diensional elliptic equations. See fo exaple the onogaphs [9 14], eseach papes [15 24] on the analysis of the singula solution, and [25 28,16,29 31] and efeences theein on the nueical appoxiation fo singula solutions of this type. Fo vetices on the z-axis, the situation is diffeent, since the coefficient 1/. It tuns out that the possible singulaities nea these vetices ae closely elated to the thee-diensional vetex singulaities of elliptic equations. This is ou stating point fo the wo pesented in this pape. See [32 36] fo discussions on singula solutions of thee-diensional diffeential equations. Diffeent fo the existing esults entioned above [3,2,6,7,4,5], we shall focus hee on establishing well-posedness and egulaity esults fo singula solutions of Eq. (1) in suitable Sobolev spaces and on the constuction of siple, explicit finite eleent schees to appoxiate these solutions quasi-optially. Ou goal shall be achieved by intoducing the faewo in a odified weighted Sobolev space K a, () (Definition 2.7), which allows us to apply cetain usual finite eleent foulations to Eq. (1). In the convegence analysis of the finite eleent solution, we intoduce a new intepolation opeato fo a local egulaization pocess (Definition 4.4). Copaed with the usual nodal intepolation, this egulaization technique deonstates citical popeties of functions in the weighted spaces, which ae also useful to teat othe axisyetic pobles (see [6,37]). The est of the pape is oganized as follows. In Section 2, we fist biefly ecall soe existing esults in the liteatue fo the axisyetic equation. Then, we define two types of weighed Sobolev spaces fo futhe analysis in Sections 3 and 4, as well as notation that will be used thoughout this pape. In addition, seveal elevant popeties of the weighted Sobolev space will be discussed. In Section 3, we establish ou a pioi estiates (well-posedness, egulaity, and the Fedhol popety) fo the axisyetic equation in the weighted space K a, (). In paticula, we shall show the opeato L : K 2 a+1,+ () {v Γ 0 = 0} K 0 a 1, () defines an isoophis fo a > 0 sall and is Fedhol as long as a is away fo a countable set of values. This allows us to copute the ange of the index a, in which the isoophis above still holds. The finite eleent solution fo Eq. (1) is studied in Section 4. In the fist pat of this section, we biefly pesent the appoxiation popety of piecewise linea polynoials in the weighted space H 2 (). With a new intepolation opeato, we show that the quasi-optial convegence ate of the linea finite eleent solution is attained, assuing the solution is sufficiently egula. Based on these esults and on a scaling aguent, in the second pat of Section 4, we analyze the convegence ate of the nueical solution in the weighted space K a, (). Then, we descibe a constuction of a sequence of tiangulations suitably gaded to the vetices, such that the quasi-optial ate is ecoveed fo singula solutions. In Section 5, we pesent nueical tests fo Eq. (1) on two doains fo diffeent singulaities (on the z-axis o away fo the z-axis). The ates of convegence of the finite eleent solutions fo diffeent eshes ae copaed. These tests suggest that the quasi-optial convegence ates ae achieved on ou gaded eshes, which is in coplete ageeent with the theoy. 2. Weighted Sobolev spaces H and K a, In this section, we foally intoduce the axisyetic Poisson s equation and the definitions of soe weighted Sobolev spaces with elevant popeties The axisyetic Poisson s equation Let := [0, 2π) R 3 be a bounded doain, which is the evolution of about the z-axis. Suppose intesects the z-axis and its half section (the intesection of and a eidian half plane) R 2 is a polygon (see, fo exaple, Fig. 1).

3 H. Li / Jounal of Coputational and Applied Matheatics 235 (2011) Fig. 1. A thee-diensional axisyetic doain (left); the coesponding two-diensional half section (ight). We then conside the thee-diensional Poisson s equation in with the Diichlet bounday condition, ũ = ( 2 x + 2 y + 2 z )ũ = f in, ũ = 0 on. (2) Recall the Sobolev space H ( ) ={v, α v L 2 ( ), α } and H 1( 0 ) := H 1 ( ) {v = 0} in the tace sense, whee α = (α 1,α 2,α 3 ) is a ulti-index. In the pesence of axisyety in the data and in the solution ( θ ũ = θ f = 0) with espect to the cylindical coodinates (,θ,z), we define u(, z) = ũ(,θ,z) and f (, z) = f (,θ,z). (3) Recall is the intesection of and the z-plane fo > 0. We let Γ 0 := be the pat of the bounday iposed with the Diichlet condition and Γ 1 := Γ 0 be its copleent set on the z-axis (Fig. 1). Thus, it is well nown that Eq. (2) can be witten as the following elliptic equation (see also [1,7]), Lu = f in (4) u = 0 on Γ 0, whee L := 1 2 ( ) 2 2 z = z. Note that the deivation of the wea solution of Eq. (4) also equies bounday conditions on Γ 1, which we will discuss in Rea 2.5. Fo now on, we shall concentate on the analysis of Eq. (4) and its finite eleent appoxiations. Ou techniques, nevetheless, ay be also useful fo dealing with diffeent bounday conditions and othe types of axisyetic pobles Weighted Sobolev spaces and the wea solution We hee define diffeent weighted Sobolev spaces on and the wea solution of Eq. (4), with which futhe analysis can be caied out in Sections 3 and 4. Definition 2.1. We fist define the following weighted Sobolev spaces L 2 () := v : R, v 2 ddz <, H () := {v : R, i j v z L2 (), 0 i + j }, = 0, 1, 2. Clealy, H 0() = L2 (). Then, the nos and the sei-nos fo any v H (), = 0, 1, 2, ae defined by v 2 H () := ( i j z v)2 ddz, v 2 H () := ( i j z v)2 ddz. i+j=0 i+j= Note that H () is closely elated to the usual Sobolev space H ( ), = 0, 1, 2. In paticula, we suaize a nube of esults fo the liteatue fo a bette undestanding of the space H (). Let H ( ) H ( ), = 0, 1, 2, be the subspace of all axisyetic functions in H ( ). The following two popositions can be found in [1,2]. Poposition 2.2. Fo = 0, 1, the tace apping ṽ(,θ,z) v(, z) in (3), is well defined fo sooth functions and extends to an isoety fo H ( ) onto H (). The ecipocal lifting, v H () ṽ H ( ) also defines an isoety. 2πv 2 H () =ṽ2 H ( ).

4 5158 H. Li / Jounal of Coputational and Applied Matheatics 235 (2011) On the othe hand, fo = 2, we have the following poposition. Poposition 2.3. Let H+ 2 () := {v H2(), v/ L 2 ()}, with v H 2 + () = v 2 + ( v) 2 1/2. ddz H 2 () (5) The tace opeato in Poposition 2.2 defines an isoophis fo H2 ( ) to H 2 + (). Besides, the following density popety can be found in [38] (Poposition 7.6). Poposition 2.4. Fo = 0, 1, 2, the space of sooth functions C ( ) is dense in H (). Rea 2.5. It is shown in [1], that fo any u H+ 2 (), the tace u u Γ1 is well defined, and u Γ1 = 0 in L 2. Theefoe, if the solution u belongs to H+ 2 () {v Γ 0 {v Γ0 = 0} of Eq. (4) by = 0}, with integation by pats, we define the wea solution u H 1() a w (u,v):= ( u v + z u z v)ddz = f vddz, (6) fo any v H 1() {v Γ 0 = 0}. By the Poincaé inequality on the thee-diensional doain and the Lax Milga Lea, it is seen that the wea fo (6) deteines a unique solution u H 1() {v Γ 0 = 0}, fo any f L 2 (). In fact, one can futhe show u H+ 2 (G), fo any G away fo the vetices, based on the standad egulaity estiates fo Eq. (2). In addition, Poposition 2.2 iplies that ũ(,θ,z) = u(, z) is the wea solution of the oiginal Poisson s Eq. (2), with f (,θ,z) = f (, z) L 2 ( ) (see also [1,7] and efeences within). We wite a few wods about the tace on Γ 0 fo any function in H 1 (). Note that Γ 0 is coposed of line segents γ i. Then, on a segent γ i Γ 0, γ i { = 0} =, the tace of v H 1 () is well defined in L2, because in the neighbohood of γ i, H 1 is equivalent to H 1. Fo a segent γ i whose closue intesects the z-axis, we ecall the following esult fo [6]. Poposition 2.6. Let T be a tiangle with diaete h, such that T { = 0} =. Let e be an edge of T, not sitting on the z-axis, but ē { = 0} =. Then, fo any v C ( T), Case 1: if T { = 0} is only a point, then 2 v 2 ds C e h(h 2 v 2 L 2 (T) + v 2 ); H 1 (T) e Case 2: if T { = 0} is an edge, then v 2 L 2 (e) C e (h 2 v 2 L 2 (T) + v 2 H 1 (T) ). The constant C above depends on the shape egulaity of the tiangle, not on v. The extension of these inequalities fo v H 1 () follows fo the density aguent in Poposition 2.4. Thus, fo any v H 1 (), if γ i { = 0} is an isolated point in { = 0}, γ i can be included in a tiangle of Case 1, and hence γ i 2 v 2 ddz is well defined; if γ i { = 0}, howeve, is an end point of a segent in { = 0}, then v has a L 2 -tace on γ i, since γ i can be a edge of a tiangle of Case 2. This tace esult will also be useful in ou finite eleent analysis in Section 4. Recall that the solution of Eq. (4) ay have singulaities in H+ 2 (), due to the non-sooth points on the bounday and to the singula coefficient in L, even if the ight hand side f is sooth. To handle these possible singula solutions, we need the following weighted Sobolev space. Let Q i be the ith vetex of and S ={Q i } be its vetex set. Denote by l the iniu of the non-zeo distances fo a point Q i to a bounday edge of. Let l := in(1/2, l/4) and Vi := B(Q i, l), (7) whee B(Q i, l) denotes the ball centeed at Qi S with adius l. Note that the sets Vi ae disjoint. Then, we define the function ϑ C ( S) = x Qi in V i ϑ(x) (8) l/2 in ( Vi ). Thus, the space K a, () is given by the definition below.

5 H. Li / Jounal of Coputational and Applied Matheatics 235 (2011) Definition 2.7. We define K () := {v : R, a, ϑi+j a i j v z L2 (), i + j }, a R, = 0, 1, 2. Fo any open set G and any v : G R, v 2 Ka, (G) := ϑ i+j a i j z v2, L 2 (G) i+j=0 The inne poduct on the Hilbet space K a, () is (u,v) K a, () = ϑ 2(i+j a) ( i j u)( i j v)ddz. z z i+j v 2 Ka, (G) := ϑ a i j z v2. L 2 (G) i+j= A subspace of K 2 () a, that can be egaded as the countepat of H2 + () is Ka,+ 2 () := K2 () a, v, ϑ ( v) 2 4 2a ddz <, with the no on any open set G, whee v 2 K 2 a,+ (G) =v2 K 1 a, (G) + v 2 K 2 a,+ (G), v 2 K 2 a,+ (G) = G ϑ ( 4 2a 2 v)2 + ( 2 z v)2 + ( z v) 2 v 2 + ddz. In addition, we denote by K 1 a, () := (K1 a, () {v Γ 0 = 0}) the dual space of K 1 a, () {v Γ 0 = 0} with espect to the pivot space L 2 (), w K 1 a, () := sup v K 1 a, () {v Γ0 =0} vwddz, v = 0. v K 1 a, () Rea 2.8. The space K () a, fo Eq. (4) is the analogue of the weighted space K a () fo cone singulaities of elliptic equations with bounded coefficients (see fo exaple [16,21,39]). In the definitions of weighted spaces H () and K a, (), we conside only fo = 0, 1, 2. This is sufficient fo ou FEM using linea appoxiation functions. An extension fo > 2 is also possible [2]. By Poposition 2.2, it is natual to intoduce H () if the solution is egula enough. On the othe hand, we shall show that K a, () is an appopiate space to study singula solutions fo the singula coefficients and fo the non-sooth bounday fo Eq. (4) Soe leas We give seveal popeties of K a, () that ae useful fo futhe analysis. To avoid any confusion on notation, in the text below, we use ρ and φ as the vaiables in the pola coodinates (ρ, φ), since the vaiable is used in the equation, whee ρ denotes the distance to the oigin and φ is the angle. In addition, by A B, we ean that thee exist constants C 1, C 2 > 0, such that C 1 A B C 2 A. Fo siplicity, we wite K := a, K () a, and H := H (). In the following leas, we will oit the poof if it is ainly based on definitions of the nos and diect calculation. We fist have the following altenative expessions fo the opeatos and z. Lea 2.9. On evey V i := B(Q i, l), we set a local pola coodinate syste (ρ, φ), whee Qi = ( i, z i ) is the new oigin, and i = ρ sin φ, z z i = ρ cos φ. Then, on V i, = (sin φ) ρ + cos φ ρ φ, z = (cos φ) ρ sin φ ρ φ. Meanwhile, the elation between the Catesian coodinates (x, y, z) and the cylindical coodinates (,θ,z) eads x = (cos θ) sin θ θ, y = (sin θ) + cos θ θ. Recall the function ϑ in (8). Then, we have an uppe bound fo the following function. Lea The function ϑ j+ a j z ϑ a is bounded on. This lea leads to the following isoophis between weighted Sobolev spaces. Lea We have ϑ b K a, = K a+b,, whee ϑ b K a, ={ϑ b v, v K a, }.

6 5160 H. Li / Jounal of Coputational and Applied Matheatics 235 (2011) Poof. Let v K a, and w = ϑ b v. Then ϑ i+j a i j zv L 2, fo i + j. Thus, we veify w K a+b, by checing the inequalities below, ϑ ϑ i+j a b i j w = i+j a b i j z s s t t ϑ b z i s j t z v s i,t j C ϑ (i+j s t) a i s j t z v L 2, s i,t j whee the last inequality follows fo Lea Theefoe, ϑ b K a, is continuously ebedded in K a+b,. Naely, the ap ϑ b : K a, K a+b, is continuous. On the othe hand, because this ebedding holds fo any eal nube b, we have K a+b, = ϑ b ϑ b K a+b, ϑ b K a,. To coplete the poof, we also notice that the invese of ultiplication by ϑ b is ultiplication by ϑ b, which is also continuous. Recall that V i := B(Q i, l) in (7). Theefoe, ϑ(, z) l on Vi, and we have the following lea. Lea Let G V i be an open subset of V i, such that ϑ ξ l on G. Then, fo and a a, we have K a, K a, and v K a a ξ vk a (G) a, (G), v K. a,, The following lea assets that the H -no and the K a,-no ae equivalent on a subset of, whose closue is away fo the vetex set S. Lea Let G be an open subset, such that inf x G ϑ(x) > 0. Then, v H (G) M 1 v K a, (G) and v K a, (G) M 2 v H (G), v H (G). In addition, v H 2 + (G) M 1u K 2 a,+ (G) and v K 2 a,+ (G) M 2v H 2 + (G), v H2 + (G), whee M 1 and M 2 depend on the infiu of ϑ(x) on G and, but not on v. Using Lea 2.12, we have the following copaison fo K a, (V i) and H (V i). Lea Let G V i be an open subset, on which ϑ ξ l. Fo = 0, 1, 2, v H (G) ξ a v K a, (G), a ; v K a, (G) ξ a v H (G), a 0. By Lea 2.13, we have the extension of Lea 2.14 to the entie doain. Coollay Fo a function v, we have v H M 1 v K, and v K a, M 2 v H fo a 0, whee M 1 and M 2 depend on and a. Recall the opeato L := z. Lea The ap L : Ka+1,+ 2 K0 a 1, is well defined and continuous. Poof. We show that thee is C > 0 such that Lv K 0 a 1, the calculation below. Lv 2 K 0 a 1, = ϑ 2 2a (v + 1 v + v zz ) 2 ddz C ϑ 2 2a (v v 2 + v2 + zz v2 z )ddz Cv 2 K 2 a+1,+. Cv K 2 fo all v a+1,+ K2 a+1,+. The poof is then copleted by 3. Well-posedness and egulaity in weighted Sobolev spaces Based on the elation between Eq. (4) and the thee-diensional Poisson s equation (2), it can be shown that the solution of (4) u H+ 2 H2, povided that f L2 and has only good cones. See [7] fo the case = (0, 1) (0, 1). On an abitay polygonal doain, howeve, the non-sooth bounday and the singula coefficients of the elliptic opeato ay affect the well-posedness and egulaity of the solution in H, and the stateents above ae in geneal not tue.

7 H. Li / Jounal of Coputational and Applied Matheatics 235 (2011) In this section, we analyze the solution of Eq. (4) in the weighted space K a, := K a, () on polygonal doains. To be oe pecise, we loo fo a inial egulaity solution u K 1 {v 1, Γ 0 = 0} that satisfies the vaiational foulation a(u,v)= ( u v + z u z v)ddz = f vddz, (9) fo any v K 1 {v 1, Γ 0 = 0}. We fist pove the well-posedness of the solution in K 1 {v a+1, Γ 0 = 0}, fo a > 0 sall. Then, we povide a egulaity esult in the weighed Sobolev space fo the solution. The Fedhol popety of the opeato L will be discussed in the last subsection. Thoughout Section 3, we denote by =( x, y, z ), the gadient in the conventional Catesian coodinates Well-posedness We fist need the following lea fo ou well-posedness esult in Theoe 3.2. Lea 3.1. Fo any u K 1 {v 1, Γ 0 = 0}, we have a(u, u) = ( u) 2 + ( z u) 2 u 2 ddz C ϑ ddz, 2 and theefoe the bilinea fo a(, ) in (9) is stictly coecive on K 1 1, {v Γ 0 = 0}. Poof. Recall the neighbohood V i of Q i S. Define V i /α = B(Q i, l/α) fo α N and let O = Vi /2. By the definitions of the nos involved, we need to veify the following Poincaé-type inequality on evey V i, and also on O, ( u) 2 + ( z u) 2 u 2 ddz C ddz, (10) D ϑ 2 whee D is eithe V i o O and C is independent of u. Noting u K 1 1, H 1, we let ũ(,θ,z) = u(, z) be the axisyetic function in the thee-diensional doain as in Poposition 2.2. Fo D = V i, the neighbohood of Q i that is away fo the z-axis, note that the desied estiate is well nown in [32,12,13] fo weighted spaces without the paaete. Thus, the justification of (10) fo this sub-doain follows, since is bounded on D and K 1 1, is equivalent to the space in the above efeences. We now veify (10) fo D = V i, the neighbohood of the vetex Q i sitting on the z-axis. Recall ũ(,θ,z) = u(, z). Let Vi = V i [0, 2π) be the doain fo the evolution of V i about the z-axis. Thus, Vi can be chaacteized in the spheical coodinates (ρ, θ, φ) centeed at Q i by Vi ={(ρ, ω), 0 <ρ< l, ω ωqi }, whee ω Qi S 2 is the polygonal doain on the unit sphee S 2. Then, by Lea 2.9, we have and ũ 2 = ũ 2 + x ũ2 + y ũ2 = z ũ2 ρ + ũ2 2 φ ρ + ũθ 2 ρ 2 sin 2 φ, ω Qi ũ 2 ds C ω Qi ũ 2 φ + ũ2 θ sin φdφdθ, sin 2 φ which is just the Poincaé inequality on ω Qi and ds = sin φdφdθ is the volue eleent on ω Qi (see also [32,33]). Thus, we obtain 2 2 u 2π V i ϑ ddz = ũ l 2 Vi ρ dxdydz = ũ 2 dsdρ 2 0 ω Qi l C ũ 2 ρ + ũ2 2 φ 0 ω Qi ρ + ũ θ ρ 2 dsdρ 2 ρ 2 sin 2 φ = C ũ 2 dxdydz = 2πC (u 2 + u2)ddz. z Vi V i We now veify fo D = O. Recall = [0, 2π). Then, we have 2π ( u) 2 + ( z u) 2 2π ddz = ( ũ) 2 + ( z ũ) 2 ddzdθ 0 = ũ 2 dxdydz C ũ 2 dxdydz = 2πC u 2 ddz 2πC ϑ 2 u 2 ddz, D

8 5162 H. Li / Jounal of Coputational and Applied Matheatics 235 (2011) whee we applied the equation ũ 2 = ũ 2 +(ũ θ/) 2 +ũ 2 z, the usual Poincaé inequality on, and the fact that ϑ is bounded fo 0 on O. Adding all the inequalities, we actually show that inequality (10) holds on, and hence coplete the poof. Based on Lea 3.1, we note that the two spaces K 1 1, {v Γ 0 = 0} and H 1 {v Γ 0 = 0} ae essentially the sae, since they can be equipped with the sae no a(, ) 1/2. Theefoe, the K 1 1 -wea solution fo (9) is the sae solution defined by (6) in Section 2. We now have the solvability esult in the space K 1 a+1,. Theoe 3.2. Thee exists η>0, such that fo 0 a <ηand f K 0 a 1,, the vaiational foulation (9) defines a unique solution u K 1 a+1, of Eq. (4). Poof. We fist veify, fo a = 0, the uniqueness of the solution u K 1 {v 1, Γ 0 = 0}. Note that the Cauchy Schwaz inequality gives a(u,v) u K 1 v 1, K 1 u 1, K 1 v 1, K 1. 1, Based on this continuity popety and Lea 3.1, the Lax Milga Lea then poves that L : K 1 1, {v Γ 0 = 0} K 1 1, := (K1 1, {v Γ 0 = 0}) is an isoophis. Theefoe, we conclude that thee exists a unique solution u K 1 1, {v Γ 0 = 0} fo f K 0 a 1, K 1 1,. Fo a 0, note that the faily of opeatos ϑ a Lϑ a : K 1 1, {v Γ 0 = 0} K 1 1, depends on a continuously in no. Theefoe, thee exists η>0, depending on the doain and the opeato L, such that fo 0 a <η, the opeato ϑ a Lϑ a : K 1 1, {v Γ 0 = 0} K 1 1, is invetible. Hence, by Lea 2.11, fo 0 a <η, since K 0 1, K 1 1,, the invetibility of ϑ a Lϑ a : K 1 {v 1, Γ 0 = 0} K 1 1, poves the solution u K 1 {v 1, Γ 0 = 0} is in fact a solution in K 1 {v a+1, Γ 0 = 0}, fo any f K 0 = ϑ a a 1, K 0 1,. See also [16,39 41] fo elated discussions. The paaete η plays an ipotant ole in ou analysis of the FEM in Section 4. By coputing the index of L in the weighted Sobolev space K a+1,, we shall evaluate η explicitly in Theoe 3.5 using the Fedhol popety of the opeato Regulaity Based on the egulaity estiates in [32,16,33,21,39] fo the Laplace opeato in two-diensional polygonal and theediensional polyhedal doains, we now have the following egulaity esult fo the solution of the axisyetic bounday value poble (4), in the weighted Sobolev space K a,. Theoe 3.3. Let 0 a < 1 and f L 2. Suppose u K1 a+1, is the unique solution of Eq. (4). Then, we have u K 2 a+1,+ Cf L 2, whee the constant C = C(a, ) >0 is independent of f and u. Poof. Note that fo 0 a < 1, f L 2 K0 a 1,. By Lea 3.1 and the definitions of weighted spaces, u K1 a+1, H1. We let ũ(,θ,z) := u(, z) and f (,θ,z) := f (, z) as in Poposition 2.2. Then, ũ H1 ( ) solves the thee-diensional Poisson s equation (2) with the ight hand side f L2 ( ). Accoding to Theoe 3.2, ϑ a Lϑ a : K 1 {v 1, Γ 0 = 0} K 1 1, is invetible. Thus, ϑ a u K 1 Cϑ a f 1, K 1 = ϑ a f wddz C sup. 1, 0=w K 1 1, {v Γ 0 =0} w K 1 1, Fo f L 2 K0 a 1,, based on Lea 2.11, the Cauchy Schwaz inequality, and the estiate above, we have u K 1 a+1, C sup 0=w K 1 1, {v Γ 0 =0} f K 0 w a 1, K 0 1, w K 1 1, Theefoe, we only need to veify u K 2 a+1,+ Cf L 2. Cf K 0 Cf a 1, L 2. (11)

9 H. Li / Jounal of Coputational and Applied Matheatics 235 (2011) Recall V i = B(Q i, l) in (7) and Vi /α = B(Q i, l/α) fo α N. Regulaity is a local popety. We pove this estiate on each V i /2 and on O := ( V i /2), espectively. Let O := ( O) [0, 2π) (esp. P := ( V i /4) [0, 2π) ) be obtained by the evolution of O (esp. ( V i /4)) about the z-axis. By the standad egulaity esult fo Eq. (2), we have ũ H 2 ( O ) C( f L 2 ( P ) +ũ H 1 ( ) ), (12) since O is away fo the singula points. Then, using the fact that ϑ is bounded above and below fo 0 on O and the elations in Lea 2.9, we have 2π u 2 2π ϑ ( 2 2a 2 K a+1,+ 2 (O) u)2 + ( 2 z u)2 + 2( z u) 2 u 2 + ddz = C O 2π 0 ϑ ( 2 2a 2 ũ)2 + ( 2 z ũ)2 + 2( z ũ) 2 ũ 2 + ddzdθ O ( 2 x ũ)2 + ( 2 y ũ)2 + ( 2 z ũ)2 + 2( x y ũ) 2 + 2( x z ũ) 2 + 2( y z ũ) dxdydz 2 O C ũ 2 H 2 ( O ). Theefoe, based on (11) (13), and Poposition 2.2, we obtain u K 2 a+1,+ (O) C ũ H 2 ( O ) C( f L 2 ( P ) +ũ H 1 ( ) ) C(f L 2 (( V i /4)) +u H 1 () ) C(f L 2 (( V i /4)) +u K 1 a+1, ()) Cf K 0 a 1, () Cf L 2 (). Fo the estiates nea the vetices, we need to distinguish the vetices away fo the z-axis and those on the z-axis. Fo a vetex Q i that is not on the z-axis, in its neighbohood V i /2, is bounded below fo 0. Theefoe, Eq. (4) is elliptic with sooth coefficients and the zeo bounday condition. The egulaity esult is well nown in weighted Sobolev spaces equivalent to Ka+1,+ 2 (V i/2). Naely, fo 0 a < 1, u 2 K a+1,+ 2 (V i/2) C ϑ 2 2a ( 2 u)2 + ( 2 z u)2 + ( z u) 2 + ( u) 2 ddz C V i /2 V i ϑ 2 2a f 2 ddz Cf 2 K 0 a 1, (V i) Cf 2 L 2 (V i ). Let Q i be the oigin in the local pola coodinates (ρ, φ). A siple poof of the estiate above is obtained by using the Mellin tansfo, a patition of unity of the fo φ n (ρ) := φ(ρ n), and applying the standad egulaity esults fo sooth doains to the function φ n u. See fo exaple [16,21,39,41] fo details. Fo a vetex Q i on the z-axis, the desied esult on V i /2 can be deived fo the egulaity of ũ as follows (identifying u with ũ as in Poposition 2.2). Let Vi := V i [0, 2π) (esp. Vi /2 := V i /2 [0, 2π) ). Then, we have the following weighted estiate on Vi fo [32,33] fo the thee-diensional vetex, ϑ 2 2a ( 2 x ũ)2 + ( 2 y ũ)2 + ( 2 z ũ)2 + ( x y ũ) 2 + ( x z ũ) 2 + ( y z ũ) 2 dxdydz C ϑ 2 2a f 2 dxdydz. (14) Vi Vi /2 Note that (14) is in fact the estiate in the weighted space K a fo singula vetices in [32,33], whee a siila poof to two-diensional vetices was caied out, with a patition of unity in a thee-diensional doain. Thus, fo 0 a < 1, 2π u 2 2π ϑ ( 2 2a 2 K a+1,+ 2 (V i/2) u)2 + ( 2 z u)2 + 2( z u) 2 u 2 + ddz V i /2 = ϑ 2 2a ( 2 x ũ)2 + ( 2 y ũ)2 + ( 2 z ũ)2 + 2( x y ũ) 2 + 2( x z ũ) 2 + 2( y z ũ) 2 dxdydz C Vi /2 Vi ϑ 2 2a f 2 dxdydz = 2πC Adding up all estiates copletes the poof of this theoe. V i ϑ 2 2a f 2 ddz 2πCf 2 L 2 (V i ). Note that we can extend the above poof to f K 0 a 1, fo 0 a < 1 and obtain u K a+1,+ 2 Cf K 0. a 1, (13)

10 5164 H. Li / Jounal of Coputational and Applied Matheatics 235 (2011) The Fedhol popety Based on the well-posedness and egulaity esults established in the pevious two subsections, we shall futhe study the Fedhol popety of the opeato L in the weighted Sobolev space K a,. An application of this study on the FEM will be shown in Section 4. Recall that a continuous opeato A : X Y between Banach spaces is Fedhol if the enel of A (that is, the space e(a) := {Ax = 0}) and Y/AX ae finite diensional spaces. We also define its index by the foula ind(a) = di e(a) di(y/ax). Then, it is possible to deteine η in Theoe 3.2 by the Fedhol popety of the axisyetic opeato L. Befoe we poceed with the discussion on this popety, we ealize the following lea. Lea 3.4. Let G be an open subset of the doain that is away fo the vetices. Then, the space H+ 2 (G) is copactly ebedded in H 1 (G). Poof. This lea can be justified by the Rellich Kondachov Theoe fo usual Sobolev spaces and the isoophiss in Popositions 2.2 and 2.3, between these weighted Sobolev spaces and the usual Sobolev spaces on the coesponding thee-diensional doain. Let Q i S be a vetex of away fo the z-axis. In the neighbohood V i, by feezing the coefficient at Q i, the local behavio of the solution is deteined by the pincipal pat 2 2 z of the opeato L, since is bounded away fo 0. Let ı := 1. Then, in the pola coodinates (ρ, φ), α i φ β i on V i, the opeato pencil P i (τ) associated to L in V i is defined by ( z )(ρıτ+ ζ (φ)) = ρ ıτ+ 2 P i (τ)ζ (φ), whee ζ (φ) is any sooth function with the zeo Diichlet bounday condition fo φ = α i and φ = β i. Thus, based on the foula z = ρ 2 ((ρ ρ ) φ ) and (ρ ρ) 2 ρ ıτ+ = ρ ıτ+ (ıτ + ) 2, we obtain P i (τ) = (τ ı) 2 2 φ. Let θ i = β i α i be the inteio angle of the cone with vetex Q i. It is well nown that the spectu of the opeato φ 2, with zeo bounday conditions, is π 2 Σ i =, = 1, 2, 3,..., (15) θ i and hence P i (τ) is invetible fo all τ R, given = ±π/θ i. On the othe hand, fo a vetex Q i on the z-axis, we chaacteize the evolution Vi = V i [0, 2π) of V i by Vi ={(ρ, ω), 0 <ρ< l, ω ωqi }, in spheical coodinates, whee ω Qi S 2 is the pojection of Vi on the unit sphee S 2. Then, on Vi, we have the foula Lu = ũ = ρ 2 ((ρ ρ ) 2 + ρ ρ + )ũ, fo the axisyetic function ũ, whee = (cot φ) φ + 2 φ + (sin2 φ) 2 2 θ denotes the Laplace Beltai opeato on ω Qi. The opeato pencil fo on Vi is thus given by P i (τ) = ((ıτ + 1/2)(ıτ + + 1/2) + ). Inheiting the bounday condition fo the oiginal equation (2), that is, the zeo bounday condition, and taing θ ũ = 0 into account, the sallest eal eigenvalue of the opeato on ω Qi is stictly positive λ i,1 > 0, (see [11]), and can be coputed nueically. Theefoe, P i (τ) is invetible fo all τ R, when < λ i,1 + 1/4. Recall the isoeties between diffeent spaces fo Poposition 2.2. Note that K (G) a, is equivalent to H (G) fo G away fo the vetices, and hence Ka,+ 2 (G) is copactly ebedded in K1 a, (G) by Lea 3.4. Define η 1 := in( λ i,1 + 1/4). i Thus, fo <η 1 and = ±π/θ i, following Kondatiev s ethod [21], we obtain the Fedhol conditions on the opeato ϑ Lϑ : K1,+ 2 {v Γ 0 = 0} K 0 1,, which iplies that L : K2 +1,+ {v Γ 0 = 0} K 0 1, is Fedhol by Lea See [21,39] and efeences theein fo oe details on the Kondatiev s ethod. Fo out of the ange above, the opeato L ay not be Fedhol, o is Fedhol but has a non-zeo index, and hence is not invetible. Fo the coputation of non-zeo indices of Fedhol opeatos, we efe to [42,43]. Now, we ae in the position to specify the index a and ipove ou well-posedness esult.

11 H. Li / Jounal of Coputational and Applied Matheatics 235 (2011) Theoe 3.5. Define η := in(1, π/θ i, λ i,1 + 1/4). Then, fo any 0 a <η, thee is a unique solution u K 2 a+1,+ {v Γ 0 = 0} fo Eq. (4), povided that f K 0 a 1,. Poof. Theoes 3.2 and 3.3, Lea 2.16 and the discussion above show that, fo a = 0, the opeato L : Ka+1,+ 2 {v Γ 0 = 0} K 0 a 1, is Fedhol with index zeo, since it is invetible. By the hootopy invaiance of the index, L is Fedhol with index zeo fo 0 a <η. Note that the enel of L is non-inceasing as a inceases. Theefoe, L is injective between these spaces. Since the index is zeo, we conclude it is in fact a bijection fo 0 a <η. 4. The finite eleent estiates in weighted spaces In this section, we analyze the finite eleent appoxiation fo the solution of Eq. (4), especially fo singula solutions, in weighted Sobolev spaces. Pecisely, let T := {T i } be a tiangulation of with tiangles T i. Denote by S := S(T, 1) H 1 {v Γ 0 = 0} =K 1 1, {v Γ 0 = 0} the finite eleent space associated to the linea Lagange tiangle. Then, the finite eleent solution u S S is defined by a(u S,v S ) = ( u S v S + z u S z v S )ddz = f v S ddz, (16) fo any v S S. To obtain an eo estiate, we shall fist establish an appoxiation esult assuing the solution is sufficient egula in H 2. Then we descibe a siple and explicit constuction of a sequence of tiangulations T n, suitably gaded to points whee singulaities in the solution occu, such that the following quasi-optial ate of convegence can be achieved u u n H 1 C di(s n ) 1/2 f L 2, f L 2, whee S n = S(T n, 1) is the finite eleent space on the esh T n and u n := u Sn S n is the finite eleent solution. We fist need the following estiate fo Céa s Lea fo futhe analysis. Lea 4.1. Given the finite eleent solution u S defined above, then thee exists a constant C > 0, independent of u, such that u u S K 1 1, C inf u χ K 1. χ S 1, Poof. The poof is standad. Let u 2 a := a(u, u). Indeed, we have u u S a = inf u χ a, χ S because u S is the pojection of u onto S in the a-inne poduct. The esult then follows fo Céa s Lea and the equivalence of the a-no and the K 1 1, -no, given the Diichlet bounday condition on Γ 0 (Lea 3.1) Appoxiation in the space H In the est of the pape, we equie that all tiangles of the tiangulation T ae shape-egula, and adjacent tiangles have copaable size. Naely, let T i, T j T be two tiangles, such that Ti Tj =, then thee exists a constant C 0, dia T i ax C 0. T i,t j T n dia T j The Lagange intepolation opeato I : C 0 S is such that fo any v C 0 ( ), Iv(x i ) = v(x i ) at the nodes x i of each tiangle. In addition, fo a sub-doain G, we denote by P (G) the set of polynoials of degee on G. In this section, the constant C > 0 in ou estiates will in geneal depend on the shape egulaity of the tiangles in T, but not on the solution u o the given data f. We fist state a lea fo [3], egading the polynoial appoxiation popety in the weighted Sobolev space. It is an extension of the well-nown appoxiation esult in the usual Sobolev space. Lea 4.2. Fo a copact set K, let h K = dia K < 1 be its diaete. Suppose K is sta-shaped with espect to a ball of adius δh K. If K { = 0} =, inf p P 1 (K) (h 1 K v p L 2 (K) + v p H 1 (K) ) Ch K v H 2 (K), v H2 (K), whee the constant C depends on δ, but not on v o h K. Poof. Since the poof is athe long and siila to the pocess in [44,45] fo usual Sobolev spaces, we only give a setch. One can constuct a linea function p by using Taylo s Theoe fo sooth functions. Then, based on the estiates on the weight in the space H and the estiates on the esidue v p, the desied esult can be obtained fo v H 2 (K) using the density aguent. The coplete poof of this lea can be found in the long vesion of this pape [46]. See also the poof in [3] fo the uppe bound of inf p P1 (K) v p L 2 (K). (17)

12 5166 H. Li / Jounal of Coputational and Applied Matheatics 235 (2011) Fig. 2. A selection of (e(x i ), T i ) and (e(x j ), T j ) associated to T (left); the patch U based on this selection (ight). Rea 4.3. A weighted Sobolev ebedding esult v L ( ) Cv H 2 () is obtained in [8]. Thus, using the appoxiation popety (Lea 4.2) and the nodal intepolation opeato, it is possible to analyze the convegence ate fo the finite eleent solution, povided the eal solution u H 2 (). We hee, howeve, pesent a diffeent appoach by intoducing a new intepolation opeato Π : H 2 () S(T, 1) based on a local egulaization pocess (Definition 4.4). Exploiting citical popeties of functions in the weighted space fo the finite eleent analysis, this technique shall allow us to get shap eo analysis in this pape fo singula solutions, as well as povide useful tools in the futue wo fo oe coplex axisyetic pobles with low-egulaity data (e.g., [37]). Definition 4.4. Fo each node x i on the z-axis, we associate x i with an edge e(x i ) and a tiangle T i, such that x i is an endpoint of e(x i ), e(x i ) does not lie on the z-axis, and e(x i ) is an edge of T i. Denote by x i the othe endpoint of e(x i ) (see Fig. 2). In addition, if x i is the endpoint of Γ 0, we equie the associated edge e(x i ) lies on Γ 0 to peseve the bounday condition. Define the opeato π i : H 2 (T i) R e(x π i v = i )(t v)ds e(xi ) ds e(x i ), whee t denotes the unit vecto paallel to e(x i ), pointing fo x i intepolation opeato Π : H 2 () S(T, 1), Πv := v(x i )ψ i + (v(x i ) + π i v)ψ i, i,x i {=0} i,x i {=0} to x i, and = (, z ). We then define the new whee ψ i S(T, 1) is the usual linea basis function associated with x i. Note that the associations between x i and e(x i ) and between e(x i ) and T i, ae not unique. One can select any edge connected to x i as e(x i ) and any tiangle including e(x i ) as T i, as long as they satisfy the conditions in Definition 4.4. It is also clea that Πv n = v n fo any v n S(T, 1). We then have the following appoxiation popety of Πv away fo the z-axis. Lea 4.5. Let G be a sub-doain such that Mh on G, fo 0 < h < 1. Let T ={T i } be the tiangulation of G with quasi-unifo tiangles of size h. Then, v Πv H 1 (G) Ch v H 2 (G), v H2 (G), whee the constant C depends on G and the shape egulaity of the tiangles. Poof. Note that on G, Πv = Iv. Theefoe, by the usual estiate in Sobolev spaces, fo any tiangle T i T, we obtain v Πv H 1 (T i ) =v Iv H 1 (T i ) Ch v H 2 (T i ). Let i,in and i,ax be the sallest and the lagest distance fo any point in Ti to the z-axis, espectively. Then, thee exists a constant M 1, such that 1 < ax i ( i,ax / i,in ) M 1, since Mh on G. Theefoe, v Πv H 1 (T i ) 1/2 i,ax v Πv H 1 (T i ) C 1/2 i,ax h v H 2 (T i ) CM 1 1/2 i,in h v H 2 (T i ) CM 1 h v H 2 (T i ). The poof is thus copleted by adding up the estiates fo all tiangles. We now define soe special tes that we will use often in the text below. By an a-node, we ean a node of the tiangulation that does not lie on the z-axis; by a z-node, we ean a node on the z-axis.

13 H. Li / Jounal of Coputational and Applied Matheatics 235 (2011) Recall the associated edge e(x i ) to each z-node x i fo Definition 4.4. Fo any tiangle T T whose closue intesects the z-axis, let Z ={x i } be the union of its z-nodes. We associate to T the following patch U := inteio( xi Z { Tj, Tj ē(x i ) = } { Tj, Tj T = }), T j T. (18) Naely, the open set U fos a neighbohood of xi Z {ē(x i )} T. Theefoe, by (17), U is the union of finite ovelapped doains D i. Each D i is sta-shaped with espect to a ball of adius C 1 h = C 1 dia T, fo C 1 depending on the shape egulaity of the tiangles. Fo exaple, evey D i can be the union of two tiangles in U, shaing a coon edge (see Fig. 2). Then, based on Lea 4.2 and the standad appoxiation theoy in usual Sobolev spaces [44,47], on evey D i, we have inf p P1 (D i )(h 1 v p L 2 (D i ) + v p H 1 (D i ) ) Ch v H 2 (D i ). Hence, by a theoe developed in [45], inf p P 1 (U ) (h 1 v p L 2 (U ) + v p H 1 (U ) ) Ch v H 2 (U ) (19) fo C depending on the doains D i and the tiangulation. Thus, we have the following estiates in the neighbohood of the z-axis. Lea 4.6. Fo any T T, with T { = 0} =, let h = dia T < 1. Then, v Πv H 1 (T ) Ch v H 2 (U ), v H2 (U ), whee U epesents the patch defined in (18) and the constant C depends on the shape egulaity of the tiangulation. Poof. Denote by ψ j the linea basis function associated to the node x j. Then, fo any ψ j whose suppot intesects T, noting ax(( T )) is copaable with h, we fist deive the following estiate, 1/2 ψ j H 1 (T ) = C T ψ 2 j ddz + T ( ψ j 2 + z ψ j 2 )ddz (1 + h 2 )ddz T 1/2 C(h 2 h 2 h ) 1/2 = Ch 1/2, whee C depends on the tiangulation. Note fo any p P 1 (T ), v Πv H 1 (T ) v p H 1 (T ) +Π(v p) H 1 (T ). Setting w = v p, we shall veify the estiate fo w H 1 (T ) and Πw H 1 (T ). Recall that Z is the union of the z-nodes of T and fo any node x i Z, thee is an associated edge e(x i ). The othe endpoint x i of e(x i ) is an a-node (Definition 4.4). Meanwhile, we define the union of a-nodes associated to T, A ={x i, x i Z } {a-nodes T }. Note that, fo any x l A, we can associate it to a tiangle T l U, away fo the z-axis, such that x l T l is one of its vetices. Denote by ˆT the standad efeence tiangle with dia ˆT = 1. Then, the affine apping F between Tl and ˆT is defined by F(T l ) = ˆT and w(x) = ŵ(ˆx) = ŵ(f(x)). Theefoe, by the usual Sobolev ebedding Theoe, a scaling aguent, and the definition of the nos, w(x l ) ŵ L (ˆT) Cŵ H 2 (ˆT) C(h 1 w L 2 (T l ) + w H 1 (T l ) + h w H 2 (T l )) C(h 3/2 w L 2 (T l ) + h 1/2 w H 1 (T l ) + h1/2 w H 2 (T l ) ). In the last inequality, we used the fact that the atio ax(( Tl ))/ in(( Tl )) M 1 and in(( Tl )) is copaable with h, since Tl does not intesect the z-axis. Theefoe, by Definition 4.4 and the estiates above, we have Πw H 1 (T ) w(x i ) + π i w ψ i H 1 (T ) + w(x l ) ψ l H 1 (T ) i,x i Z C(h 1/2 i,x i Z π i w + l,x l A {x i } (h 1 w L 2 (T l ) + w H 1 (T l ) + h w H 2 (T l ))). (20) l,x l A Futheoe, by the shape egulaity of the tiangulation and (17), we notice that Ah 2 e(x i ) ds Bh2 and Ah e(x i ) Bh, fo A, B > 0 depending on the tiangulation. We then focus on the estiate fo π i w. Let T i U be a tiangle with e(x i ) as an edge.

14 5168 H. Li / Jounal of Coputational and Applied Matheatics 235 (2011) Case 1: Ti { = 0} is a single node x i. By Definition 4.4, the tace estiate in Poposition 2.6, and the Hölde inequality, we obtain e(x π i w = i )(t w)ds e(xi ) ds e(x i ) Ch 1 (t w)ds e(x i ) 1/2 Ch 1 w 2 2 ds 1/2 ds e(x i ) e(x i ) Ch 1/2 (h 2 w 2 H 1 (T i ) + w 2 H 2 (T i ) )1/2 Ch 1/2 (h 1 w H 1 (T i ) + w H 2 (T i ) ). Case 2: Ti { = 0} is a line segent. Siilaly, by Poposition 2.6, we have π i w Ch 1 (t w)ds Ch 1 w 2 ds 1/2 ds e(x i ) e(x i ) Ch 1/2 (h 2 w 2 H 1 (T i ) + w 2 H 2 (T i ) )1/2 Ch 1/2 (h 1 e(x i ) 1/2 w H 1 (T i ) + w H 2 (T i ) ). Hence, in both cases, cobining (20), the estiates yield Πw H 1 (T ) C ( w H 1 (T i ) + h w H 2 (T i ) ) + (h 1 w L 2 (T l ) + w H 1 (T l ) + h w H 2 (T l ). ) i,x i Z l,x l A Cobining the estiates above and (19), we obtain fo any p P 1 (U ), v Πv H 1 (T ) v p H 1 (T ) +Π(v p) H 1 (T ) C(h 1 v p L 2 (U ) + v p H 1 (U ) + h v p H 2 (U ) ) Ch v H 2 (U ). Rea 4.7. It is clea that suing up the estiates in Lea 4.6 fo all tiangles nea the z-axis and cobining with Lea 4.5, one has v Πv H 1 Ch v H 2, if the tiangulation T contains quasi-unifo tiangles of size h. Let u h be the finite eleent solution of Eq. (4). Thus, based on Céa s Lea, we have u u h H 1 Cu Πu H 1 Ch u H 2, povided that u H+ 2 H2 on these eshes. Rea 4.8. Note that in ode to obtain an estiate as in Lea 4.6, it is not necessay to include evey tiangle in the neighbohood of xi Z {ē(x i )} T (patch U ). Based on the analysis above, the eo estiate still holds on U, as long as U satisfies the following citeia. 1. Besides T, U should contain evey tiangle T i associated to e(x i ), x i Z, and evey tiangle T l associated to x l, x l A. 2. U is the union of finite ovelapped doains D i, each of which is sta-shaped with espect to a ball of adius Ch, such that the estiate (19) holds by [48]. This allows us to siplify ou pesentation in the subsections below, by odifying the definition of U while eeping the citeia above Appoxiation in the space K a, The appoxiation esults in Rea 4.7 povide the analogy of the best polynoial appoxiation in the usual Sobolev space, when the solution is egula enough (in H+ 2 ). Howeve, it is vey possible that the solution of Eq. (4) possesses singulaities in H+ 2 nea the vetices of the doain, which will destoy the optial convegence ate. Fo now on, we shall extend these appoxiation esults to the space K a, fo possible singulaities and descibe a siple and explicit constuction of a sequence of finite eleent spaces, such that the quasi-optial convegence ate can be achieved fo singula solutions. Recall the opeato Π : H 2 S(T, 1). Fo a function v K 2 {v a, Γ 0 = 0}, a 1, we change its definition on the vetex set of the doain. We define (Πv)(Q i ) = 0 Q i S, and let Πv eain the sae on the othe nodes as in Definition 4.4, since v H 2 (G) fo any G away fo the vetices. Recall the open set Γ 1. Then, fo any edge e(x i ) associated to a node x i Γ 1 (Definition 4.4), we equie that the assigned tiangle Ti does not contain any point fo the vetex set S. A possible selection of T i is given in Rea We fist study the local behavio with espect to dilations of a function v K a,, in the neighbohood V i of Q i S. We conside a new coodinate syste that is a siple tanslation of the old z-coodinate syste, now with Q i at the oigin of the new coodinate syste. Let G λ V i be a subset, such that ϑ ξ l on Gλ. Fo 0 <λ<1, We let G := λg λ. Then, we (21)

15 H. Li / Jounal of Coputational and Applied Matheatics 235 (2011) Fig. 3. One κ-efineent fo a tiangle with a vetex Q i S. define the dilation of a function v on G λ in the new coodinate syste as follows v λ (, z) := v(λ,λz), fo all (, z) G λ V i. (This definition aes sense, since Q i is the oigin in the new coodinate syste.) We shall need the following dilation lea. Lea 4.9. Fo 0 <λ< 1, let G λ V i be an open subset, and G := λg λ V i. Then, if Q i { = 0}, v λ K a, (Gλ) = λ a 3/2 v K a, (G); if Q i { = 0}, C 1 λ a 1 v K a, (G) v λ K a, (Gλ) C 2 λ a 1 v K a, (G), fo constants C 1,C 2 > 0 depending on, v K a, (V i), = 0, 1, 2. Poof. The poof is based on the change of vaiables s = λ, t = λz. Note that on both G λ V i and G V i, ϑ(, z) is equal to the distance fo (, z) to Q i, theefoe ϑ(, z) = λ 1 ϑ(s, t). Then, if Q i { = 0}, v λ (, z) 2 Ka, (Gλ) = ϑ j+ a (, z) j v z λ(, z) 2 ddz j+ = j+ Gλ G = λ 2a 3 j+ λ a j ϑ j+ a (s, t)λ j+ j s t v(s, t) 2 λ 3 sdsdt G ϑ j+ a (s, t) j s t v(s, t) 2 sdsdt = λ 2a 3 v 2 K a, (G). On the othe hand, if Q i { = 0}, we notice A 1 B on V i, fo constants A and B depending on the doain. Theefoe, we have, Av(, z) 2 Ka, (D) ϑ j+ a (, z) j v(, z z) 2 ddz Bv(, z) 2 Ka, (D), j+ D whee D V i is any subset of V i. Applying the new coodinate syste with Q i at the oigin as above, we thus have v λ (, z) 2 Ka, (Gλ) A 1 ϑ j+ a (, z) j v z λ(, z) 2 ddz j+ = A 1 j+ Gλ G = A 1 λ 2a 2 j+ λ a j ϑ j+ a (s, t)λ j+ j s t v(s, t) 2 λ 2 dsdt G ϑ j+ a (s, t) j s t v(s, t) 2 dsdt A 1 Bλ 2a 2 v 2 K a, (G). We note that the inequality in the opposite diection can be justified with the sae pocess, which copletes the poof. Fo V i := B(Q i, l), let Tξ V i be a tiangle with the biggest edge of length = ξ and Q i is a vetex of T ξ. Denote by T κξ T ξ the sub-tiangle of T ξ that has Q i as a vetex and has all sides paallel to the sides of T ξ. Theefoe, T κξ is siila with T ξ with the atio of siilaity κ,0<κ<1. Then, T ξ is divided into the sall tiangle T κξ that has the coon vetex Q i with T ξ and the tapezoid between the two paallel edges (Fig. 3). Let G := T ξ T κξ V i be the tapezoid. Recall that the tiangulation T of contains shape-egula tiangles and satisfies (17). Suppose all the tiangles T i T, satisfying T i G =, fo a quasi-unifo tiangulation T G of G. Rea In the case Ḡ { = 0} =, let Z G ={T } be the union of tiangles T T G, such that T { = 0} =. To siplify ou pesentation, fo evey T Z G that contains an endpoint of the segent Ḡ { = 0}, we define a new patch U N with the sae citeia as the patch U in Rea 4.8 as follows. Fo each endpoint x i of the segent Ḡ { = 0}, we assign the associated edge e(x i ) to be on one of the paallel edges of the tapezoid G accodingly. Thus, fo each z-node x i of T G and its associated edge e(x i ), we ae able to assign a tiangle T i T G to e(x i ), such that Ti contains the edge e(x i ) as in Definition 4.4. Fo the desciption, it is clea that T i Z G and is away fo the vetex Q i. In addition, we assue that fo

16 5170 H. Li / Jounal of Coputational and Applied Matheatics 235 (2011) each a-node x l of T Z G, thee exists at least one tiangle T l T G with x l as a vetex and Tl { = 0} =. Then, we define the new patch fo evey tiangle T Z G, U N := U G, fo U fo (18). Note that evey U N G includes all the tiangles needed in the poof of Lea 4.6 and is the union of finite ovelapped doains. Hence, by Rea 4.8, the estiates in Lea 4.6 still hold if we eplace U by U N. We then have the following estiate nea Q i. Lea Fo 0 <κ<1, let G = T ξ T κξ,q i, V i, T G, and U N be as defined above. Let h be the esh size of T G, Then, v Πv K 1 1, (G) C(κ)ξ a (h/ξ)v K 2 a+1, (G), fo all v K 2 a+1, (V i), a 0, with C(κ) independent of ξ, h, and v. Poof. Recall the new coodinate syste with Q i as the oigin. Let G λ = λ 1 G. Recall the dilation function v λ (, z) = v(λ,λz). Note that by the definition of Πv, (Πv) λ = Π(v λ ) on G λ. Then, we choose λ = ξ/ l, such that Gλ V i. Fo a vetex Q i { = 0}, if Ḡ λ { = 0} =, note that (U N ) λ G λ. On the othe hand, if Ḡ λ { = 0} =, by the popety (17) of T, the distance to the z-axis fo G λ, (G λ ) Ch/λ. We thus apply Leas 4.5 and 4.6 accodingly to the egion G λ, based on its elation with the z-axis, v Πv K 1 1, (G) = λ1/2 v λ (Πv) λ K 1 1, (G λ) = λ 1/2 v λ Π(v λ ) K 1 1, (G λ) M 2λ 1/2 v λ Π(v λ ) H 1 (Gλ) CM 2 λ 1/2 (h/λ)v λ H 2 (Gλ) CM 1M 2 λ 1/2 (h/λ)v λ K 2 1, (G λ) = CM 1 M 2 (h l/ξ)vk 2 1, (G) C(κ)ξ a (h/ξ)v K 2 a+1, (G), whee we used the fact that the spaces H and K 1, ae equivalent on G λ (Lea 2.13), the dilation fo Lea 4.9, and the last inequality is fo Lea Fo a vetex Q i { = 0}, the poof is siila. Note on V i, Πv is actually the nodal intepolate of v. With the coesponding estiate in Leas 4.5 and 4.9, we conclude the poof by 1 v Πv K 1 (G) C v 1 λ (Πv) λ 1, K 1 1, (G λ) = C 1 1 v λ Π(v λ ) K 1 1, (G λ) M 2C 1 1 v λ Π(v λ ) H 1 (Gλ) CM 2 C 1 1 (h/λ)v λ H 2 (Gλ) CM 1M 2 C 1 1 (h/λ)v λ K 2 1, (G λ) CM 1 M 2 C 1 1 C 2(h l/ξ)vk 2 1, (G) C(κ)ξ a (h/ξ)v K 2 a+1, (G) Constuction of the finite eleent spaces In this subsection, we constuct a sequence of eshes T n and the finite eleent spaces S n := S(T n, 1) H 1 () {v Γ 0 = 0} associated to the linea Lagange tiangle, such that the finite eleent appoxiations fo Eq. (4) u n := u Sn S n satisfy u u n H 1 () C di(s n ) 1/2 f L 2 (), even if the solution u H+ 2. We shall achieve this quasi-optial ate of convegence by consideing a suitable gading technique close to the points in S. The poof is based on the eo estiates in weighted spaces H and K a,, established in the pevious subsections. To be oe pecise, we constuct the eshes T n by successive efineents fo an initial tiangulation. Theefoe, they ae nested and will have the sae nube of tiangles as the eshes obtained by the usual idpoint efineents. Fo now on, we let η = in(1, Σ i, λ i,1 + 1/4), which satisfies Theoe 3.5. We assue that in Eq. (4), the ight hand side f L 2. Theefoe, by Theoes 3.3 and 3.5, the unique solution u of Eq. (4) satisfies u K 2 a+1,+ {v Γ 0 = 0}, fo 0 a <η. We now intoduce ou efineent pocedue. Definition Let κ (0, 1/2] and T be a tiangulation of such that no two vetices of belong to the sae tiangle of T. Then the κ-efineent of T, denoted by κ(t ), is obtained by dividing each edge AB of T in two pats as follows. If neithe A no B is in the vetex set S, then we divide AB into two equal pats. Othewise, if A is in S, we divide AB into AC and CB such that AC =κ AB. This will divide each tiangle of T into fou tiangles (Fig. 3).

17 H. Li / Jounal of Coputational and Applied Matheatics 235 (2011) We now intoduce ou sequence of eshes. Recall that l > 0 was intoduced in Eq. (7), and 4 l is not geate than the distance fo a vetex Q S to an edge of that does not contain it. Definition Suppose the initial esh T 0 is such that each edge in the esh has length l/2 and each point in S is the vetex of a tiangle in T 0. In addition, we chose T 0 such that thee is no tiangle in T 0 that contains oe than one point in S. Then we define by induction T n+1 = κ(t n ) (see Definition 4.12). Rea Note that nea the vetices, ou efineent coincides with the one intoduced in [16,41,31]. In addition, we ay use diffeent κ s at diffeent vetices as in [28] to ipove the shape egulaity of the tiangles (see [49] fo exaple). We now investigate the appoxiation popeties affoded by the tiangulation T n close to a point Q i S. We also fix a tiangle T T 0 that has Q i as a vetex. We denote by T κ j = κ j T T the sall tiangle belonging to T j that is siila to T with atio κ j, with Q i as a vetex. Then T κ j T κ j 1. Moeove, since κ 1/2 and the diaete of T is l/2, we have T κ j V i, j 1, by the definition of V i. Let N be the level of efineents. In all the stateents below, let h 2 N, in the sense that they have copaable agnitudes. We fist have the estiate on the last tiangle that contains the vetex Q i. Lea Let 0 <κ 2 1/a, fo any 0 < a <η. We conside the sall tiangle T κ N = κ N T T with vetex Q i, obtained afte N efineents. Recall the definition of Πu on the tiangulation T N fo u K 2 (V a+1, i) {v Γ0 = 0} in (21). Then, if the vetex Q i { = 0}, we have u Πu K 1 1, (T κ N ) Chu K 2 a+1, (T κ N ) ; if the vetex Q i { = 0}, we have u Πu K 1 1, (T κ N ) Chu K 2 a+1, (U κ N ) ; fo u K 2 a+1, (V i) {v Γ0 = 0}, whee h 1/2 N and U N is the patch associated to T κ N in Rea 4.10, and C depends on the shape egulaity of T 0 and κ. Poof. Define u λ (, z) = u(λ,λz) with Q i as the oigin. If the vetex Q i { = 0}, let λ = κ N and U λ := λ 1 U κ N. Then, T U λ. Let χ : U λ [0, 1] be a sooth function that is equal to 0 in a neighbohood of Q i and is equal to 1 at all the nodal points of U λ diffeent fo the vetex Q i. We intoduce the auxiliay function v = χu λ on U λ. Note that v H 2 (U λ), since v = 0 in the neighbohood of Q i. Consequently, fo = 0, 1, 2, v 2 K 1, (U λ) =χu λ 2 K 1, (U λ) Cu λ 2 K 1, (U λ), whee C depends on and the choice of the nodal points. Moeove, by the definitions of v and the opeato Π, we note Πv = Π(u λ ) = (Πu) λ on U λ. Then, u Πu K 1 1, (T κ N ) = λ1/2 u λ v + v Π(u λ ) K 1 1, (T) λ 1/2 (u λ v K 1 1, (T) +v Π(u λ) K 1 1, (T)) = λ 1/2 (u λ v K 1 1, (T) +v Πv K 1 1, (T)) Cλ1/2 (u λ K 1 1 (T) +v K 2 1, (U λ) ) Cλ 1/2 (u λ K 1 1, (T) +u λ K 2 1, (U λ) ) = C(u K 1 1, (T κ N ) +u K 2 1, (U κ N ) ) Cκ Na u K 2 a+1, (U κ N ) Chu K 2 a+1, (U κ N ). The fist and the sixth elations above ae due to Lea 4.9; the fouth is due to Lea 4.6 and the fact that the K 1, -no and the H -no ae equivalent fo v, since v = 0 in the neighbohood of Q i; the seventh is based on Lea 2.12 and the fact that the size of U κ N is copaable with κ N. The estiate fo a vetex Q i { = 0} is siila, but equies anothe inequality in Lea 4.9 and the estiate in Lea 4.5, since Πu is actually the nodal intepolation. Let χ : T [0, 1] be a sooth function that is equal to 0 in a neighbohood of Q i and is equal to 1 at all the othe nodal points of T. Let λ = κ N and v = χu λ. Then, u Πu K 1 1, (T κ N ) C 1 1 u λ v + v (Πu) λ K 1 1, (T) whee C 1 and C 2 ae fo Lea 4.9. C 1 1 (u λ v K 1 1, (T) +v Πv 1 K 1 (T)) C1 C(u λ 1, K 1 1, (T) +v K 2 (T)) 1, C 1 1 C(u λ K 1 1, (T) +u 1 λ K 2 (T)) C1 CC 2(u 1, K 1 1, (T κ N ) +u K 2 1, (T κ N ) ) Cκ Na u K 2 a+1, (T κ N ) Chu K 2 a+1, (T κ N ),

18 5172 H. Li / Jounal of Coputational and Applied Matheatics 235 (2011) We now cobine the estiates on T κ N fo Lea 4.15 with the estiates on the sets of the fo T κ j T κ j+1 fo Lea 4.11 to obtain the following estiate on a tiangle T T 0 that has a vetex in S. Poposition Let h 1/2 N and 0 <κ 2 1/a fo 0 < a <η. Then, afte N κ-efineents on T, thee exists a constant C > 0, such that u Πu K 1 1, (T) Chu K 2 a+1, (T U κ N ), fo all u K 2 a+1, {v Γ 0 = 0}. Poof. Definition 4.13 shows that the esh on T κ j 1 T κ j satisfies the assuption in Lea 4.11 and has the size κ j 1 2 j 1 N. Using the notation of Lea 4.11, we have ξ = O(κ j 1 ) on T κ j 1 T κ j. Theefoe, u Πu K 1 1, (T κ j 1 T κ j ) Cκ(j 1)a (κ j 1 2 j 1 N /κ j 1 )u K 2 a+1, (T κ j 1 T κ j ) C2 (j 1) 2 N+(j 1) u K 2 a+1, (T κ j 1 T κ j ) = C2 N u K 2 a+1, (T κ j 1 T κ j ) Chu K 2 a+1, (T κ j 1 T κ j ), whee C depends on κ and T 0, but not on the subset T κ j 1 T κ j. We then coplete the poof by adding up the eo estiates on all the subsets T κ j 1 T κ j,1 j N, and on T κ N fo Lea Rea Denote by T, the union of all the initial tiangles that contain vetices of. Then T is a neighbohood of S. Note that the union of the patches U κ N fo the vetices on the z-axis is a subset of T. Theefoe, suing up the estiates in Poposition 4.16 ove all the tiangles in T gives u Πu K 1 1, (T) Chu K 2 (T), as long as κ is chosen appopiately. a+1, We state the ain esult on the convegence of nueical solutions on ou eshes. Theoe Let 0 < a <η and 0 <κ 2 1/a. Let T n be obtained fo the initial tiangulation by n κ-efineents (Definition 4.13). Let u be the solution of Eq. (4). Denote by S n := S n (T n, 1) H 1 {v Γ0 = 0} the finite eleent space associated to the linea Lagange tiangle and by u n S n the finite eleent solution defined by Eq. (16). Then, thee exists C > 0 depending on the doain and the initial tiangulation, such that u u n K 1 Chf 1, L 2, fo h 2 n, f L 2. Poof. Let T be the union of initial tiangles that contain vetices of as in Rea Recall fo Theoe 3.3 that u K 2 u a+1, K 2 Cf a+1,+ L 2. We obtain u u n K 1 1, Cu Πu K 1 C u Πu 1, K 1 1, (T) +u Πu K 1 1, (T) Ch u K 2 1, () +u K 2 (T)) Chu K 2 Chf a+1, a+1, L 2. The fist inequality is based on Lea 4.1 and the thid inequality is based on Leas 4.5 and 4.6, and Poposition Then, as a diect esult of the theoe above, we have the following quasi-optial convegence ate fo the finite eleent solution. Coollay Let 0 < a < η and 0 < κ 2 1/a. Using the notation and assuptions in Theoe 4.18, we have that u n S(T n, 1) satisfies u u n H 1 C di(s n ) 1/2 f L 2, fo a constant C independent of f and n. Poof. Let T n be the tiangulation of afte nκ-efineents. Then, the nube of tiangles is O(4 n ) based on the constuction of tiangles in diffeent levels. Theefoe, the diension of S n, di(s n ) 4 n, fo Lagange tiangles. Thus, fo Theoe 4.18, the following estiates ae obtained, u u n H 1 u u n K 1 Chf 1, L 2 C2 n f L 2 C di(s n ) 1/2 f L 2. Then, the poof is coplete. Rea Note that the optial ange fo κ is (0, 2 1/η ), in which the finite eleent solution will have the quasioptial ate of convegence. We also notice that a sall κ esults in thin tiangles that ay lead to a lage constant C (see [50]) in the estiate. Theefoe, a good choice of κ is a value close to the uppe bound of the optial ange, such that we have both the quasi-optial ate of convegence fo the finite eleent solution and a bette shape egulaity of the tiangulation.

19 H. Li / Jounal of Coputational and Applied Matheatics 235 (2011) Fig. 4. An initial tiangulation of the L-shape doain 1 (left); the esh afte 3 successive κ-efineents (ight), κ = 0.2. Fig. 5. An initial tiangulation of the doain 2, Q = 170 (left); the esh afte 2 successive κ-efineents (ight), κ = Nueical illustations We pesent soe nueical esults that illustate the effectiveness of ou eshing techniques fo solving the axisyetic bounday value poble (4). These tests convincingly show that ou sequence of eshes achieves the quasioptial ates of convegence in the enegy no H 1. We shall see that a choice of κ in the acceptable ange (0, 2 1/η ) yields quasi-optial ates of convegence, wheeas a choice of κ out of this ange will not give the sae convegence ates. Since κ deceases as η deceases, a good deteination of η will cetainly help us to choose κ, such that we obtain the quasi-optial convegence ate, while avoid thin tiangles if possible Nueical tests We hee conside Eq. (4), with the ight hand side f = 1, on two doains, 1 (Fig. 4) and 2 (Fig. 5), to illustate ou teatents fo vetices away fo the z-axis and fo vetices on the z-axis. See Fig. 6 fo nueical solutions on these doains. 1 is an L-shape doain with an edge on the z-axis (Fig. 4). Ou theoetical esults show that on 1, the solution of (4) is not in H+ 2 at the e-entant cone with vetex Q. Theefoe, a special κ-efineent is needed nea this vetex to ensue the convegence ate pedicted in Coollay To be oe pecise, fo the theoy we developed in Section 3 (Eq. (15)), we can tae any value fo a, such that 0 < a <η= π/1.5π 0.667, which gives 2 1/a < 2 1/η Hence, fo any κ < 0.354, we expect that the κ-efineent nea Q will lead to the quasi-optial convegence ate fo the finite eleent solution. In paticula, since the space H is equivalent to the usual Sobolev space H nea Q on 1, a oe accuate a pio estiate [21] gives u H s( 1) fo s < 1 + η 1.667, whee H s is defined by intepolation [1]. 2 is a polygon with one side on the z-axis (Fig. 5), whee the inteio angle of the cone with Q as the vetex is 170. Fo vetices on the z-axis, the values of η fo appopiate eshes follow anothe foula η = λ 1 + 1/4, whee λ 1 is the sallest eal eigenvalue of the Laplace Beltai opeato discussed in Section 3. This gives η 0.7 < 1, fo Q = 170, which eans the solution is not in H+ 2 nea this vetex. Theefoe, we ay choose any κ<2 1/η nea the vetex Q, in ode to get the quasi-optial ate of convegence. It is also inteesting to note that given the sae inteio angle, the singulaities nea the vetices on the z-axis ae stonge than those nea the vetices away fo the z-axis. Based on the calculation of the paaete a fo each vetex, on both 1 and 2, the solutions ae in H+ 2, except in the neighbohoods of the vetex Q. Theefoe, we use the usual idpoint efineents nea vetices diffeent fo Q on both doains.

20 5174 H. Li / Jounal of Coputational and Applied Matheatics 235 (2011) Table 1 Copaison of convegence ates fo diffeent values of κ. j \ κ Convegence histoy on 1 Convegence histoy on 2 e : 0.1 e : 0.2 e : 0.3 e : 0.4 e : 0.5 e : 0.1 e : 0.2 e : 0.3 e : 0.4 e : e = log 2 uj u j 1 H 1 u j+1 u j H 1. Fig. 6. The nueical solution on 1 (left); the nueical solution on 2 (ight). Table 1 lists the convegence ates of the finite eleent solutions fo the axisyetic poble on 1 and 2, espectively, fo tiangulations with diffeent values of κ nea the vetex Q. These esults veify ou theoetical pediction: the quasi-optial convegence ates can be obtained fo κ<0.354 on the L-shape doain 1 and obtained fo κ<0.372 on the polygon 2. The left ost colun (values of j) in Table 1 epesents the efineent levels. Let u j be the finite eleent solution on the esh afte j efineents. The quantities pinted out in othe coluns in the table ae the convegence ates defined by e = log 2 uj u j 1 H 1 u j+1 u j H 1, which is a easonable appoxiation of the exact convegence ate. Recall h (1/2) j fo the esh afte j efineents. Then, we see that on 1, fo appopiate gaded eshes (κ < 0.354), the convegence ates ae h 1, while on unifo eshes (κ = 0.5), the convegence ates have slowed down to h 0.7, which is vey close to the theoetical ate fo ou estiates above and sees to get close and close to On 2, fo the discussion above, we have found that the convegence ates of the discete solutions should be quasioptial (h 1 ) as long as κ< 0.372, which atches the nueical esults in Table 1 pefectly. In addition, the convegence ates in the colun fo κ = 0.3 and κ = 0.4 have a lage gap, indicating the citical value of κ fo the good convegence ates lies between 0.3 and 0.4, which, once again, veifies the theoy Suay As a bief suay, we have tested ou ethod fo the odel poble on two doains, 1 and 2, fo singulaities of diffeent types. All the esults in Table 1 convincingly show that the theoetical ate of convegence is consistent with ou nueical coputations. Theefoe, fo the axisyetic poble (4), with the egulaity of the solution deteined in tes of weighted Sobolev spaces K a,, the nueical solutions have the convegence ate di(s n) 1/2, on coectly gaded eshes. Standad quasi-unifo eshes exhibit ates of convegence that ae less than optial when the solution fails to be in H+ 2 (which happens if η<1). The finest esh in ou nueical tests above is obtained afte 10 successive efineents of the coasest esh and has oughly eleents. The peconditioned conjugate gadient (PCG) ethod is used to solve the esulting syste of algebaic equations. Besides the application on the FEM, ou egulaity esults ay be useful fo the analysis