J Sc Comput (2017) 70:60 84 DOI 10.1007/s10915-016-0242-5 Optmal Quadrlateral Fnte Elements on Polygonal Domans Hengguang L 1 Qnghu Zhang 2 Receved: 30 January 2015 / Revsed: 21 January 2016 / Accepted: 24 June 2016 / Publshed onlne: 4 July 2016 Sprnger Scence+Busness Meda New York 2016 Abstract We propose three quadrlateral mesh refnement algorthms to mprove the convergence of the fnte element method approxmatng the sngular solutons of ellptc equatons, whch are due to the non-smoothness of the doman. These algorthms result n graded meshes consstng of convex and shape-regular quadrlaterals. Wth analyss n weghted spaces, we provde the selecton crtera for the gradng parameter, such that the optmal convergence rate can be recovered for the assocated fnte element approxmaton. Varous numercal tests verfy the theory. In addton to the b-k elements, we also nvestgate the serendpty elements on the graded quadrlateral meshes n the numercal experments. Keywords Corner sngularty Fnte element Graded quadrlateral mesh Error analyss Mathematcs Subject Classfcaton 65N30 65N50 65N15 1 Introducton Let R 2 be a bounded polygonal doman and V ={v },1 l, be ts vertex set. Let T be a conformng mesh of, consstng of convex quadrlaterals. For any quadrlateral K T, H. L was partally supported by the NSF Grants DMS-1158839 and DMS-1418853, and by the Wayne State Unversty Grants Plus Program. Q. Zhang was partally supported by the Natural Scence Foundaton of Chna Grants 11001282 and 11471343, and by Guangdong Provncal Natural Scence Foundaton of Chna under Grant 2015A030306016. B Qnghu Zhang zhangqh6@mal.sysu.edu.cn Hengguang L hl@math.wayne.edu 1 Department of Mathematcs, Wayne State Unversty, Detrot, MI 48202, USA 2 Guangdong Provncal Key Laboratory of Computatonal Scence and School of Data and Computer Scence, Sun Yat-Sen Unversty, Guangzhou 510275, People s Republc of Chna
J Sc Comput (2017) 70:60 84 61 there s a unque blnear mappng F K, such that K = F K ( ˆK ),where ˆK = ( 1, 1) ( 1, 1) s the reference element. For any v L 1 (K ), wedefne ˆv K := v F K L 1 ( ˆK ). (1) Let ˆQ k := span{p s ( ˆx)q t (ŷ)} be the Lagrange fnte element space of degree k 1on ˆK, where p s and q t are polynomals of degree k. Thus, the quadrlateral fnte element space of degree k assocated wth T s S ={v C( ), v K s such that ˆv K ˆQ k for all K T }. (2) In ths paper, we study the quadrlateral fnte element approxmaton to sngular solutons of ellptc equatons due to the non-smoothness of the doman boundary. Although sngular n the usual Sobolev spaces, these functons (solutons) belong to a famly of weghted Sobolev spaces K m μ, where the weght s a functon of the dstance to the vertex set V. In partcular, we propose new algorthms for quadrlateral mesh refnements, such that the resultng quadrlaterals are convex and shape regular. The meshes are approprately graded toward the sngular ponts (vertces V) n a systematc fashon n order for the assocated fnte element method to approxmate the functon u K m μ n the optmal convergence rate. The space K m μ and ts generalzed versons (where the set V, f necessary, s augmented wth addton sngular ponts n ) play an mportant role n analyzng sngular solutons of secondorder ellptc equatons [8,12,13,16 18], such as corner sngulartes, sngulartes from nonsmooth nterfaces n transmsson problems, and sngulartes from sngular potentals n the Schrödnger operator. Assumng the equaton s well defned n (a subspace of) H 1,twas shown that the sngular components of these solutons are n K m μ (or ts generalzed verson), where m depends on the smoothness of the gven data; and μ depends on the dfferental operator and on the geometry of the doman. To smplfy the presentaton, ths paper only concerns corner sngulartes. However, smlar results apply to other problems for whch the sngular soluton are n K m μ (or ts generalzed verson). The development of effectve fnte element methods for sngular solutons of ellptc equatons has been of great nterest n the computatonal communty. The conventonal fnte element method on quas-unform grds s only sub-optmal, due to the uneven dstrbuton of the approxmaton error. Based on dfferent prncples, varous algorthms have been desgned for trangular mesh refnements to capture the local behavor of the sngular soluton and to acheve the optmal convergence rate. See [1,3,6,15,16,19 21] and references theren. As another category of grd, the quadrlateral mesh s also popular, for example, n computatonal flud dynamcs and elastcty [5,24], n hgh-order fnte element post-processng schemes [25], and n the constructon of fnte volume methods [5,26]. Compared wth trangular meshes, quadrlateral meshes are more restrctve on shape condtons for effectve approxmatons [2]. Dealng wth the complex geometry, graded quadrlateral refnement schemes for sngular solutons that mantan good mesh propertes are less obvous. We propose three quadrlateral mesh algorthms (Algorthms 3.1, 3.4 and 3.8) for sngular solutons n K m μ. Each of them s derved from a dfferent dea and gves an explct constructon of a sequence of fnte element spaces. The frst two algorthms are ndrect, snce they are based on addtonal processes of the graded trangular mesh. The thrd algorthm s drect, n the sense that t refnes quadrlateral meshes recursvely wthout usng auxlary trangular meshes. Usng nterpolaton error analyss n weghted spaces, we shall gve a sharp range for the mesh parameter, such that these three constructons all lead to optmal rates of convergence for the numercal soluton n the H 1 norm (Remark 4.8). In addton, we dscuss n detal the dfferences between these three graded quadrlateral meshes, n terms of the com-
62 J Sc Comput (2017) 70:60 84 plexty of the algorthm, mesh shape regularty, and ther effcacy when only a proper subset of ˆQ k s used for the fnte element space, e.g., serendpty elements. All three algorthms are mplemented solvng model problems to llustrate the constructon of the approxmaton space and the effectveness of our approach. The rest of the paper s organzed as follows. In Sect. 2, we ntroduce notaton and revew exstng results that wll be used throughout the paper. In partcular, we defne the weghted space K m μ and show examples for common sngular solutons that belong to Km μ.wealso recall the graded trangular mesh refnements for sngular solutons. In Sec 3, we gve the detaled descrpton of three algorthms for graded quadrlateral mesh refnements and prove mportant geometrc propertes of the resultng mesh. In Sect. 4, we present nterpolaton error analyss on the proposed quadrlateral meshes for sngular functons n the weghted space. In turn, t gves rse to a range for the mesh gradng parameter, for whch the nterpolaton error s asymptotcally mnmzed, and hence the fnte element soluton acheves optmal convergence rate for sngular solutons. In Sect. 5, we conduct a seres of numercal tests for model problems on dfferent domans. The fnte element approxmatons assocated wth the proposed quadrlateral algorthms are compared for the blnear element (k = 1), the b-quadratc element (k = 2), and the quadratc serendpty element. The test results valdate the algorthms by verfyng the theoretcal predcton. In Sect. 6, we summarze the results and gve concludng remarks. Throughout the paper, by A B, we mean that there are constants C 1 > 0andC 2 > 0, such that C 1 A B C 2 A. The generc constant C > 0 n our analyss below may be dfferent at dfferent occurrences. It wll depend on the computatonal doman, but not on the functons nvolved n the estmates or the mesh level n the fnte element algorthms. To fx the notaton, we shall use T and T to denote a trangular mesh and a quadrlateral mesh, respectvely. 2 Prelmnares We frst ntroduce necessary notaton and defntons n ths secton. Some prelmnary estmates are also provded. 2.1 Functon Spaces For any ω, we use the standard notaton H m (ω) for the Sobolev spaces. Namely, for m 0, v H m (ω) := 1/2 α v 2 dx, v H m (ω) := 1/2 v 2 H α =m ω j. (ω) j m where α = (α 1,α 2 ) Z 2 0 s the mult-ndex, α =α 1 + α 2,andL 2 ( ) = H 0 ( ). Let V H 1 ( ) be a closed subspace equpped wth the H 1 norm. Denote by a(, ) the blnear form assocated wth a second-order ellptc operator wth suffcently smooth coeffcents. Suppose a(, ) s both contnuous and coercve on V. We consder the soluton of the ellptc problem u V that s defned by the followng equaton a(u,v)= F(v), v V, (3) where F V represents a functonal n the dual space V of V and s problem dependent.
J Sc Comput (2017) 70:60 84 63 Recall the fnte element space S from (2). Let S n S be the subspace on the quadrlateral mesh, such that S n V. Then, the fnte element soluton u n S n to Eq. (3)s a(u n,v n ) = F(v n ), v n S n. Thus, by the Céa Theorem, the fnte element soluton s comparable wth the best approxmaton n the fnte element space u u n H 1 ( ) C nf v S n u v H 1 ( ). (4) Remark 2.1 Equaton (3) can be regarded as the varatonal formulaton for many ellptc equatons, n whch the smoothness of the soluton u depends on the smoothness of the boundary, on the boundary condtons, and on the smoothness of F. The regularty of the soluton n turn affects the best approxmaton error n (4). For an ellptc problem wth smooth coeffcents and good boundary condtons, even when F s smooth, the soluton may be sngular n H m near the vertces on a polygonal doman. We next defne a class of weghted spaces to handle these corner sngulartes. Defnton 2.2 (Kondrat ev-type Spaces) Recall the vertex set V ={v } of, 1 l. Letr (x) be the dstance functon from x to the th sngular pont v.let μ := (μ 1,μ 2,...,μ l ) be an l-dmensonal vector. For a constant c, we denote c ± μ := (c ± μ 1, c ± μ 2,...,c ± μ l ). Then, we defne the functon ρ(x) := r (x), and ts exponents ρ c± μ (x) := 1 l Then, for ω, the weghted space s α =m 1 l r (x) c±μ = ρ c 1 l r (x) ±μ. K m μ (ω) := {ρ α μ α v L 2 (ω) for all α m}, v K m μ (ω) := 1/2 ρ m μ α v 2 L 2, v (ω) K m μ (ω) := v 2 K α α m Remark 2.3 The space K m μ has the followng notable local property. Let ω be a regon wthn the neghborhood of the vertex v, whch does not nclude other vertces. Then, v K m μ (ω ) 1/2 r m μ α v 2 L 2. (5) (ω ) α =m However, on a regon ω away from the vertex set V, the weght functon ρ s bounded both above and below from zero. Therefore, the weghted space K m μ and the Sobolev space H m are equvalent. Namely, μ (ω) v K m μ (ω) v H m (ω). (6) Usng ths space, one can obtan the full-regularty estmates for sngular solutons [8,9,11, 13,16,18], whch does not hold n usual Sobolev spaces. Namely, the weghted norm of the sngular soluton contnuously depends on the weghted norm of gven data. 1/2.
64 J Sc Comput (2017) 70:60 84 Example (The Posson Problem) Consder the equaton Defne the blnear form a(w, v) = u = f n, u = 0on. (7) w vdx, w, v H 1 0 ( ). Then, the assocated varatonal soluton u H0 1( ) := {v H 1 ( ), v = 0} satsfes a(u,v)= ( f,v):= f vdx, v H0 1 ( ). (8) Usng the Poncaré nequalty, t can be shown that for f H 1 ( ),(8) unquely defnes the soluton u H0 1 ( ). When has a smooth boundary, we further have the full-regularty estmate u H m+2 ( ) C f H m ( ), m 0. (9) Recall that, however, s a polygon wth the vertex set V := {v }. Then, gven f H m ( ), near each vertex v, the soluton has a partcular sngular expanson [9,11] u = u S + u R = s,t c s,t ψr s ln t r + u R, (10) where u R H m+2 s the regular component, ψ s a smooth functon, and r s the dstance to v.notethatc s,t R,ands, t R + {0} are real numbers dependng on the local geometry of the vertex, such that the sngular component u S = s,t c s,tψr s ln t r / H m+2. Thus, we lose the full regularty estmate n (9) even for a smooth rght hand sde f. In contrast, wth approprate ndces, the followng full-regularty estmates can be obtaned n weghted spaces [16]. Proposton 2.4 For 1 l, let η := π/φ,whereφ s the nteror angle at the vertex v. Then, for any 0 a <η and a := (a 1, a 2,...,a l ), the soluton of Eq. (7) satsfes u K m+2 a+ 1 ( ) C f K m a 1 ( ), where 1 = (1, 1,...,1) s an l-dmensonal constant vector. In partcular, f 0 a 1,we have u K m+2 a+ 1 ( ) C f K m a 1 ( ) C f H m ( ). Remark 2.5 For general ellptc problems, the constant η > 0 may be dfferent from the value gven n Proposton 2.4, but can be determned numercally. In fact, η s the frst egenvalue assocated wth the operator pencl that s determned by the ellptc operator and the local geometry of the doman near the vertex v [14,18]. As the upper bound of the ndex, the parameter η := (η 1,η 2,...,η l ) leads to a sharp regularty descrpton of the soluton n weghted spaces. Remark 2.6 When the soluton has nonzero Drchlet boundary condtons or Neumann boundary condtons on both adjacent sdes of a vertex, smlar to (10), the soluton can be decomposed nto a sngular component u S and a known (or smoother) component u R [16], such that u S K m a+ 1 ( ) for some a 0. Although n ths case u R may not
J Sc Comput (2017) 70:60 84 65 belong to K m a+ 1 ( ) for any a 0, t s ether known or smoother, and therefore can be approxmated accurately. Thus, the convergence rate of the fnte element method s decded by the convergence rate approxmatng the sngular component u S K m ( ). For ths a+ 1 reason, we wll concentrate on the fnte element approxmaton of (3) when the soluton u K m a+ 1 ( ). Assumpton 2.7 From now on, wthout loss of generalty, we assume that there are constants η > 0, such that for any 0 a <η and a := (a 1, a 2,...,a l ), the soluton of Eq. (3) u K k+1 ( ),wherek 1 s the degree of the quadrlateral fnte element space (2). Ths s a+ 1 the case provded the functonal F s suffcently smooth. See Proposton 2.4 for example. 2.2 Graded Trangular Meshes Based on a-pror regularty estmates, varous graded trangular mesh algorthms have been proposed to mprove the convergence rate of the fnte element soluton approxmatng solutons wth corner sngulartes [1,3,4,16]. We now descrbe a smple and explct constructon of trangular meshes that share smlar propertes to other graded trangular meshes. Defnton 2.8 (Graded Trangular Meshes) LetT be a trangulaton of consstng of trangles, whose vertces nclude V, such that no trangle n T has more than one of ts vertces n V. Defne the vector κ = (κ 1,κ 2,...,κ l ),forκ (0, 1/2], a κ-refnement of T, denoted by κ(t), s obtaned by dvdng each edge AB of T n two parts as follows: If nether A nor B s n V, thenwedvdeab nto two equal parts. Otherwse, f A s v V, wedvdeab nto AM and MB such that AM =κ AB. Ths wll dvde each trangle of T nto four trangles (Fg. 1). Gven an ntal trangulaton T 0, the assocated famly of graded trangulatons {T j : j 0} s defned recursvely, T j+1 = κ(t j ). Remark 2.9 It can be shown [16] that wth a careful selecton of the gradng parameter κ, the graded trangulaton T n leads to the fnte element soluton approxmatng the sngular soluton u K k+1 ( ) n the optmal rate, where k 1 s the degree of the trangular Lagrange a+ 1 fnte element space. In addton to ts smplcty, the graded refnements n Defnton 2.8 lead to a number of attrbutes good for practcal computatons, such as shape-regular trangles as well as nested dscrete subspaces. Fg. 1 Graded trangular refnements, left rght: an ntal trangle; a unform refnement; a graded refnement to A, κ A = AM 1 AB = AM 3 AC = M 1 M 3 BC
66 J Sc Comput (2017) 70:60 84 3 Graded Quadrlateral Meshes In ths secton, we propose three algorthms to generate graded quadrlateral meshes for the sngular soluton u K k+1 ( ) of Eq. (3). a+ 1 The frst algorthm s based on a smple decomposton of trangles n the trangular mesh. Algorthm 3.1 (Barycenter Refnement) LetT j,0 j n, be the graded trangular mesh from Defnton 2.8. Then, we dvde each trangle T T j nto three quadrlaterals usng the mdpont of each edge and the barycenter of the trangle (Fg. 2). Thus, the quadrlateral mesh T j of,0 j n, conssts of all the quadrlaterals from the decomposton. See also Fg. 3 for example. We now show that the resultng quadrlaterals from Algorthm 3.1 are of good shape qualty. Proposton 3.2 The barycenter refnement of a graded trangulaton T j, 0 j n, drectly leads to a mesh T j that conssts of convex and shape-regular quadrlaterals. Proof We frst show that the barycenter refnement of a trangle results n convex quadrlaterals. Consder the trangle ABC and the quadrlateral M 1 BM 2 M 4 n Fg. 2. The barycenter M 4 s the ntersecton of two medans AM 2 and CM 1. Therefore, M 1 M 4 M 2 = AM 4 C. Note AM 4 C <π,snce AM 4 C s an nteror angle of the trangle AM 4 C. Therefore, M 1 M 4 M 2 <π. Meanwhle, snce M 4 s always an nteror pont of the trangle ABC,t s straghtforward to see that other nteror angles of M 1 BM 2 M 4 are less than π. Hence, the quadrlateral M 1 BM 2 M 4 s convex. For the same reason, quadrlaterals M 3 M 4 M 2 C and AM 1 M 4 M 3 are both convex. Note that successve graded refnements (Defnton 2.8) for a trangle T T 0 generate chld trangles wthn at most four smlarty classes. Therefore, the trangles n T j s wthn Fg. 2 Quadrlaterals from a trangle, left rght: a trangle n the mesh T j ; the resultng quadrlaterals (M 4 s the barycenter) Fg. 3 Mesh refnements for a polygonal doman based on Algorthm 3.1 (κ A = 0.2,κ B = κ C = κ D = 0.5): the ntal trangular mesh T 0 (left), the resultng quadrlateral mesh T 0 (center), and the quadrlateral mesh T 2 (rght)
J Sc Comput (2017) 70:60 84 67 Fg. 4 Trangle parng I (unform refnement): a par n T j (left), the chld pars n T j+1 (center), and the resultng quadrlaterals n T j+1 (rght). Each par s dentfed by the common edge (the dotted lne) at most 4N 0 smlarty classes, where N 0 s the number of ntal trangles n T 0. Thus, the barycentrc refnement of the trangular mesh T j leads to quadrlaterals wthn at most 12N 0 smlarty classes. Hence, we have shown that the resultng mesh T j conssts convex and shape-regular quadrlaterals. Instead of dvdng a trangle, the second proposed method obtans quadrlaterals by combnng adjacent trangles. Let T j,0 j n, be the graded trangular mesh from Defnton 2.8. We say that t s a complete trangle parng on T j, f we can group every two adjacent trangles n T j as a par, such that each trangle s n one and only one such par. Then, we need the followng mesh condton on T 1 to proceed. Assumpton 3.3 There s a complete trangle parng on T 1, such that the unon of the two trangles n each par s a convex quadrlateral. A complete trangle parng on T j can be dentfed by the set E j of common edges of pared trangles. We obtan the set E j on T j,1 j n, as follows. Algorthm 3.4 (Trangle Parng) E 1 s gven by Assumpton 3.3. Step 1 We defne E 2 to be the set of the edges n T 2, such that: ether (1) they are on edges n E 1 ; or (2) they do not ntersect any edge n E 1.SeeFg.4 for example. Step 2 We obtan a complete trangle parng on T 2 determned by E 2. If each par on T 2 forms a convex quadrlateral, go to Step 3; otherwse, return to Step 1 and start wth a dfferent trangle parng on T 1 satsfyng Assumpton 3.3. Step 3 For j 2, suppose E j s gven. Then, we defne E j+1 to be the set of the edges n T j+1, such that: ether (1) they are on edges n E j ; or (2) they do not ntersect any edge n E j. Thus, the assocated quadrlateral mesh T j,1 j n, conssts of the quadrlaterals determned by the trangle parng on T j. Remark 3.5 Based on how the two trangles n a par wll be refned n the next trangular refnement (Defnton 2.8), there are three possble types of parngs for trangles n T j for j 1: (I) both trangles wll be unformly refned (Fg. 4); (II) one trangle wll be specally refned toward a vertex and the other trangle wll be unformly refned (Fg. 5); (III) both trangles wll be specally refned toward a common vertex (Fg. 6). See Fg. 7 as an example for the trangle parng on a polygonal doman. For Algorthm 3.4 to proceed on trangular meshes T j,1 j n, the ntal trangulaton T 0 should be able to lead to a trangle parng on T 1 that satsfes Assumpton 3.3, and consequently lead to a trangle parng on T 2 that enables Step 3 n the algorthm. Ths s n
68 J Sc Comput (2017) 70:60 84 Fg. 5 Trangle parng II (the graded refnement toward an solated vertex): a par n T j (left), the chld pars n T j+1 (center), and the resultng quadrlaterals n T j+1 (rght). Each par s dentfed by the common edge (the dotted lne) Fg. 6 Trangle parng III (the graded refnement toward a common vertex): a par n T j (left), the chld pars n T j+1 (center), and the resultng quadrlaterals n T j+1 (rght). Each par s dentfed by the common edge (the dotted lne) Fg. 7 Mesh refnements for a polygonal doman based on Algorthm 3.4 (κ A = 0.2,κ B = κ C = κ D = 0.5): the resultng quadrlateral mesh T 0 from a trangular mesh T 0 (left), the quadrlateral mesh T 1 (center), and the quadrlateral mesh T 3 (rght). The dotted lnes are the common edges for the pars fact not a very restrctve condton on T 0. As ndcated n the algorthm, we only need to ensure that the pared trangles n T 2 form convex quadrlaterals. We gve a concrete proof of ths argument below. Proposton 3.6 Algorthm 3.4 gves rse to the sets E j of edges, 1 j n, provded that each par of trangles on T 2 forms a convex quadrlateral. In turn, each edge set E j determnes a complete trangle parng on T j that leads to convex and shape-regular quadrlaterals. Proof It suffces to consder the followng two cases for a trangle par (a quadrlateral ABCD)nT 2. Case I (The par does not touch any vertex n V.) Based on Defnton 2.8, after one refnement, each trangle n ths par s decomposed unformly nto four trangles n T 3. Therefore, by Algorthm 3.4, the quadrlateral ABCD T 2 s decomposed and regrouped nto four quadrlaterals n T 3 (see Fg. 4), two (AM 1 M 5 M 4 and M 5 M 2 CM 3 )ofwhchare parallelograms, two (M 4 M 5 M 3 D and M 1 BM 2 M 5 ) of whch are smlar to the orgnal quadrlateral ABCD. Note that by Algorthm 3.4, subsequent unform refnements of parallelograms lead to smlar parallelograms; and for quadrlaterals M 4 M 5 M 3 D and M 1 BM 2 M 5, snce they are smlar to ABCD, subsequent unform refnements lead to
J Sc Comput (2017) 70:60 84 69 chld quadrlaterals smlar to those of ABCD. Snce the quadrlateral ABCD T 2 s convex, by nducton, all the chld quadrlaterals of ABCD are convexand shape regular. Case II (The par touches a vertex n V.) Thus, the sngular vertex may be an solated vertex of a trangle (Fg. 5) or on the common edge (Fg. 6). We need to show the chld quadrlaterals for both are convex and shape regular. In the case of Fg. 5, the trangle ABD T 2 s specally refned toward the sngular vertex A wth raton κ A n the next step; and the trangle BCD T 2 s unformly refned. Recall that T 2 s the trangular mesh obtaned from the ntal trangulaton after two graded refnements. Let T T 1 be the parent trangle of the trangle ABD T 2. Thus, ABD s smlar to T wth rato κ A,andAM 1 M 5 M 4 ABD s smlar to ABCD T wth the same rato, snce the graded refnement follows the same rule on T 1 and T 2. Wth the unform refnement n the trangle BCD, M 5 M 2 CM 3 s a parallelogram. For the quadrlateral M 4 M 5 M 3 D, snce the quadrlateral ABCDs convex, we have M 3 DM 4 <π; DM 4 M 5 DAB <πfor κ A 1/2 and M 5 M 3 D = BCD <π; and snce M 1 M 5 //BC,wehave M 4 M 5 M 3 = π M 1 M 5 M 4 <π. Therefore, the quadrlateral M 4 M 5 M 3 D s convex. Wth a smlar argument, so s the quadrlateral M 1 BM 2 M 5. Further subsequent refnements of the quadrlaterals M 1 BM 2 M 5, M 5 M 2 CM 3,andM 4 M 5 M 3 D wll fall nto Case I. Subsequent refnements of the par AM 1 M 5 M 4 create chld quadrlaterals smlar to those of ABCD, snce AM 1 M 5 M 4 s smlar to ABCD. Therefore, all the chld quadrlaterals of ABCD are convex and shape regular. In the case of Fg. 6, both trangles are specally refned toward the common sngular vertex D wth rato κ D. Therefore, the quadrlateral M 4 M 5 M 3 D s smlar to ABCD. Other three quadrlaterals AM 1 M 5 M 4, M 1 BM 2 M 5,andM 2 CM 3 M 5 wll be unformly refned as n Case I n subsequent refnements. Therefore, as for the case n Fg. 5, t suffces to show that these three quadrlaterals are convex. Gven that ABCD s convex, followng a smlar argument as above, t s straghtforward to see that all the nteror angles n these three quadrlaterals are less than π, except for M 1 M 5 M 2, for whch addtonal calculatons are needed. Recall κ D 1/2. Thus, M 1 M 5 B ADB and BM 5 M 2 BDC. Hence, M 1 M 5 M 2 ADC <π. Therefore, all the chld quadrlaterals of ABCD are convex and shape regular, whch completes the proof. Remark 3.7 The frst two Algorthms 3.1 and 3.4 are based on addtonal modfcatons on exstng graded trangular meshes, and therefore both of them preserve the local mesh sze from the graded trangular mesh. However, they are also apparently dfferent. The barycenter refnement has a smple formulaton, but the resultng quadrlaterals n general do not converge to parallelograms. The trangle parng s more complcated and requres moderate ntal mesh condtons on T 0 to proceed. Recall the trangles n T j, f away from the sngular vertex, are unformly refned. Therefore, most of quadrlaterals n T j from Algorthm 3.4 are parallelograms, except for those that are ether covered by more than one ntal trangle n T 0 or close to the sngular vertex (see the mesh T 3 n Fg. 7). We also note that wth the same number of refnements, Algorthm 3.1 leads to more quadrlaterals than Algorthm 3.4, and hence a larger fnte element space. The thrd algorthm drectly refnes a quadrlateral mesh, wthout requrng auxlary trangular meshes. Algorthm 3.8 (Graded 2-refnement) LetT be a conformng quadrlateral mesh of wth convex quadrlaterals. Suppose the vertces n V are ncluded n the vertces n T and each quadrlateral contans at most one vertex of the doman. Defne the vector
70 J Sc Comput (2017) 70:60 84 Fg. 8 Drect quadrlateral refnements, left rght: an ntal quadrlateral; a unform 2-refnement; a graded refnement toward A, κ A = AM 1 AB = AM 4 AD = AM 5 AC κ = (κ 1,κ 2,...,κ l ),forκ steps. (0, 1/2]. We frst decde specal ponts n the followng Step 1 (Edge Pont Selecton). Let AB be an edge n the quadrlateral mesh T. If nether A nor B s n V, then we mark the mdpont M of AB. Otherwse, f A = v V, we mark the pont M on AB, such that AM =κ AB. Step 2 (Interor Pont Selecton). Let K T be a quadrlateral. If K does not contan a vertex n V, then we use two straght lnes to connect the ponts on the opposte edges of K, marked n Step 1. Then, mark the ntersecton of these two lnes. If v V s a vertex of K,letAC be the dagonal of K, such that A = v. Then, we mark the pont M on AC, such that AM =κ AC. Then, we dvde each quadrlateral K nto four quadrlaterals by connectng the marked nteror pont to the four marked edge ponts (see Fg. 8). Gven an ntal mesh T 0 consstng of convex quadrlaterals, the assocated famly of graded quadrlateral meshes {T j : j 0} s defned recursvely, T j+1 = κ(t j ). As for the frst two algorthms, we show that the proposed graded 2-refnement also results n meshes of good qualty. Proposton 3.9 For 0 j n, Algorthm 3.8 leads to a sequence of meshes T j consstng of convex and shape-regular quadrlaterals. Proof It suffces to consder the graded 2-refnements of a sngle quadrlateral n T 0.Fora convex quadrlateral that s away from the sngular vertex, t s known [2] that the unform refnements (the second pcture n Fg. 8) lead to convex and shape-regular quadrlaterals. Thus, t suffces to show ths s the case for a quadrlateral touchng a sngular vertex. For a convex quadrlateral ABCD T 0 wth one vertex A V (the thrd pcture n Fg.8), we may dvde the quadrlateral nto two trangles usng ts dagonal AC. Then, for ths quadrlateral, Algorthm 3.8 and a specal case of Algorthm 3.4 (the trangle parng n Fg.6) produce the same chld quadrlaterals after one refnement. Therefore, the four chld quadrlaterals of ABCD are convex and shape regular. For the three chld quadrlaterals M 1 BM 2 M 5, M 5 M 2 CM 3,andM 5 M 3 DM 4, whch are away from the sngular vertex, they wll be unformly refned n subsequent refnements and hence have chld quadrlaterals wth desred qualty as the stuaton dscussed n the frst paragraph of the proof. Note that the quadrlateral AM 1 M 5 M 4 s smlar to ABCD. Further refnements on AM 1 M 5 M 4 wll repeat the pattern as for ABCD. Namely, AM 1 M 5 M 4 wll have chld quadrlaterals smlar to those
J Sc Comput (2017) 70:60 84 71 Fg. 9 Mesh refnements for a polygonal doman based on Algorthm 3.8 (κ A = 0.2,κ B = κ C = κ D = 0.5): the ntal quadrlateral mesh T 0 (left), T 1 (center), and T 3 (rght) of ABCD. Thus, all the chld quadrlaterals of ABCD from subsequent refnements are convex and shape regular. Ths completes the proof. Remark 3.10 We have presented three graded quadrlateral mesh algorthms for the fnte element approxmaton of sngular solutons. The graded 2-refnement drectly refnes a quadrlateral mesh and needs no trangular mesh nformaton. It generalzes the trangular mesh refnement n Defnton 2.8 to quadrlateral meshes. In addton, wth successve refnements, all the quadrlaterals that are away from the sngular vertces wll converge to parallelograms due to the unform refnements (Fg. 9) [2,23]. Ths results n better mesh qualty than the frst two Algorthms 3.1 and 3.4. The dfference n quadrlateral shapes n these three algorthms may lead to dfferent approxmaton propertes n the case that the mapped fnte element space (1) on the reference element ˆK s only a proper subspace of the polynomal space ˆQ k, such as the serendpty elements. Ths wll be llustrated n our numercal tests. 4 Error Analyss Recall from Assumpton 2.7 that there exsts η > 0, 1 l, such that the soluton of Eq. (3) u K k+1 a+ 1 ( ) for any 0 a <η,wherek 1 s the degree of the fnte element space (2). In ths secton, we study the approxmaton property of functons n the fnte element space. 4.1 Mesh Layers We frst study geometrc propertes of the graded mesh. Let T j and T j,0 j n, be the trangular mesh from Defnton 2.8 and the quadrlateral mesh from Algorthm 3.8, respectvely. Both are obtaned after j successve graded refnements of T 0 and T 0 wth the parameter κ. LetT, j T j (resp. K, j T j ), 1 l, be the unon of closed trangles n T j (resp. closed quadrlaterals n T j ) touchng the vertex v V. Namely, T, j (resp. K, j )s the mmedate neghborhood of v n T j (resp. n T j ). We defne the followng regons G, j assocated wth T n close to v G, j = T, j \ T, j+1, for 0 j < n, and G,n = T,n. (11) For the quadrlateral mesh T n,wedefnetheregonsl, j assocated wth T n close to v L, j = K, j \ K, j+1, for 0 j < n, and L,n = K,n. (12)
72 J Sc Comput (2017) 70:60 84 Fg. 10 Mesh layers after two graded refnements: a trangle by Defnton 2.8 (left); a quadrlateral by Algorthm 3.8 (rght) See Fg. 10 for an llustraton of the regons G, j and L, j on a trangle and on a quadrlateral. Then, we defne the mesh layers for the quadrlateral meshes as follows. Defnton 4.1 (Mesh Layers) LetT n be the quadrlateral mesh obtaned from one of the three proposed algorthms on. LetT n be the trangular mesh on that generates T n n Algorthms 3.1 and 3.4. Recall G, j from (11). Then, the mesh layers assocated wth T n are sets of quadrlaterals that are classfed by ther dstances to the vertex set V. (I) If T n s obtaned from the barycenter refnement (Algorthm 3.1), we defne the layers L, j,0 j n, as the regon close to v L, j := G, j. (II) If T n s obtaned from the trangle parng (Algorthm 3.4), we defne the layers L, j := {K T n such that (I) K G, j or (II) K G, j = and K G, j 1 = } 1 j n, L,0 := {K T n, K G,0 }. (III) If T n s obtaned from the graded 2-refnement (Algorthm 3.8), we defne the layers L, j as n (12). Note that the mesh layers are sets consstng of quadrlaterals. Let T n be the graded quadrlateral mesh defned n any of the Algorthms (3.1, 3.4 and 3.8). Based on Defnton 2.8 and Algorthm 3.8,onT n, the dameter of the quadrlaterals n L, j satsfes h, j κ j 2 j n. (13) On 0 := \, j L, j, the mesh sze h 2 n. In addton, snce L, j s n the neghborhood of v, by Defnton 2.2 and by the proposed algorthms, the weght functon ρ and the dstance functon r satsfy ρ L, j r L, j κ j, for 0 j < n; and ρ L,n r L,n Cκ n. (14) 4.2 Interpolaton Error Estmates In ths subsecton, as n Defnton 4.1, wedenotebyt n the quadrlateral mesh obtaned from one of the proposed algorthms wth the parameter κ.lets n be the quadrlateral fnte element space (2) ofdegreek assocated wth T n. Thus, by the constructon, the dmenson of the fnte element space dm(s n ) 4 n.invewof(4), we shall obtan the fnte element approxmaton error by analyzng the nterpolaton error. Recall the reference element ˆK and the assocated fnte element space ˆQ k. Recall the blnear mappng F K : ˆK K T n. For a soluton u K k+1 ( ), k 1, by the Sobolev a+ 1
J Sc Comput (2017) 70:60 84 73 embeddng theorem, u s contnuous n except for the vertex set V. Itwasshownn[17], however, that gven a 0, the dscontnutes of u at V are removable and u 0when approachng the vertces. Namely, u can be regarded as a contnuous functon on and u(v ) = 0foranyv V. Thus, n vew of (1), we defne the nodal nterpolaton I K u on a convex quadrlateral K, such that (I K u) K ˆQ k and Then, by (1), we have I K u(x) = u(x) for any node x K. (I K u) K = (I K u) F K = I ˆK (u F K ) = I ˆK (û K ). Defne the global nterpolaton operator I : C( ) S n, such that Iu K = I K u. (15) We start our estmates by recallng the followng standard argument from [7,10]. Lemma 4.2 Let T be a quadrlateral mesh of, consstng of convex and shape-regular quadrlaterals. Let v H k+1 ( ) be a functon, where k 1 s the degree of the fnte element space. Then, for any quadrlateral K T, there exsts C > 0 ndependent of v and the geometry of K, such that where h K s the dameter of K. v I K v H 1 (K ) Ch k K v H k+1 (K ), Recall the mesh layers n Defnton 4.1. Then, we estmate the nterpolaton error n three regons for u K k+1 a+ 1 ( ), 0 a <η (Assumpton 2.7): (I) 0 := \, j L, j ; (II) L, j for 0 j < n; and (III) L,n. In partcular, on 0, we have the followng result. Lemma 4.3 Let T n be the quadrlateral mesh as n Defnton 4.1. On 0, recall the mesh sze h 2 n. Then, there exsts C > 0 ndependent of h and u, such that u Iu H 1 ( 0 ) Ch k u K k+1 a+ 1 ( 0), 0 < a <η. Proof Snce 0 s away from the vertex set V,by(6), K k+1 a+ 1 and H k+1 are equvalent on 0. Then, the desred estmate follows from (15) and Lemma 4.2. To carry out the error analyss on the layer L, j, we need the followng result regardng the dlaton property n the weghted space. Recall the gradng parameter 0 <κ 1/2 on the layer L, j from the proposed algorthms. We consder a new coordnate system that s a smple translaton of the old xy-coordnate system wth the vertex v now at the orgn of the new coordnate system. For 0 j n, we defne the regon wth dlaton L, j j := κ L, j ; (16) and the dlaton of a functon v on L, j n the new coordnate system v (x, y) := v(κ j x,κj y), (x, y) L, j. (17) Ths defnton makes sense, snce v s the orgn n the new coordnate system. Then, we have the followng lemma.
74 J Sc Comput (2017) 70:60 84 Lemma 4.4 Recall the dstance functon r to the vertex v. Then, for v K m μ (L, j ),m 0, we have α m r α μ α v 2 L 2 (L, j ) = κ2 j (1 μ ) α m r α μ α v 2 L 2 (L, j ), 0 j n. (18) Proof Snce L, j s n the neghborhood of the vertex v, the norm of the space K m μ (L, j ) has an equvalent expresson shown n (5). Thus, for v K m μ (L, j ), the left-hand-sde term of (18) s vald (fnte). Then, the Eq. (18) follows drectly from the scalng argument. We now gve the error estmates on the mesh layers L, j for 0 j < n. Lemma 4.5 Let T n be the quadrlateral mesh as n Defnton 4.1. Then, for 0 j < n, there exsts C > 0 ndependent of j and u, such that u Iu H 1 (L, j ) Cκ ja 2 k( j n) u K k+1 a+ 1 (L, j ), 0 < a <η. Proof Recall the regon L, j from (16). Note that the Km μ norm and the H m norm are equvalent on L, j, snce the weght functon ρ 1onL, j. Recall the mesh sze h, j κ j 2 j n on L, j from (13). Then, by the estmates n (18), Lemma 4.2,(14), (13), and (5), we have u Iu 2 H 1 (L, j ) C r α 1 α (u Iu) 2 L 2 (L, j ) = C r α 1 α (u Iu ) 2 L 2 (L, j ) α 1 α 1 C u Iu 2 H 1 (L, j ) C(h, j κ j ) 2k u 2 H k+1 (L, j ) C(h, j κ j ) 2k u 2 K k+1 1 C(h, j κ j ) 2k u 2 K k+1 (L, j ) C(h, j κ j ) 2k r α 1 α u 2 L 2 (L, j ) 1 α k+1 Cκ 2 ja (h, j κ j ) 2k r α 1 a α u 2 L 2 (L, j ) α k+1 Cκ 2 ja (h, j κ j ) 2k u 2 K k+1 a+ 1 (L, j ) Cκ2 ja 2 2k( j n) u 2 K k+1 a+ 1 (L, j ). Ths completes the proof. Now, we are ready to analyze the nterpolaton error on the last layer L,n. Lemma 4.6 Let T n be the quadrlateral mesh as n Defnton 4.1. There exsts C > 0 ndependent of n and u, such that u Iu H 1 (L,n ) Cκ na u K k+1 a+ 1 (L,n), 0 < a <η. Proof Let u (x, y) = u(κ nx,κn y) be the dlaton (17)ofu wth v as the orgn. Then, u K k+1 a+ 1 (L,n ) by Lemma 4.4,whereL,n sgvenn(16). Note that the dameter dam(l,n ) 1. Let χ : L,n [0, 1] be a smooth functon that s equal to 0 n a neghborhood of v,buts equal to 1 at all the other nodal ponts n L,n. We ntroduce the auxlary functon v = χu on L,n. Consequently, we have for m 0 v 2 K m 1 (L,n ) = χu 2 K m 1 (L,n ) C u 2 K m 1 (L,n ), (19) where C depends on m and the smooth functon χ. Moreover, snce u(v ) = 0, by the defnton of v, wehave Iv = Iu = (Iu) on L,n. Note that the K m 1 norm and the H m norm are equvalent for v on L,n,sncev = 0nthe neghborhood of the vertex v. Then, by (5), Lemma 4.4, (19), and (14), we have (L, j )
J Sc Comput (2017) 70:60 84 75 u Iu 2 H 1 (L,n ) C u Iu 2 K 1 1 (L,n) C = C α 1 α 1 r α 1 α (u Iu) 2 L 2 (L, j ) r α 1 α (u Iu ) 2 L 2 (L, j ) C u v + v Iu 2 K 1 1 (L,n ) C ( u v 2 K 1 1 (L,n ) + v Iu 2 K 1 1 (L,n ) ) = C ( u v 2 K 1 1 (L,n ) + v Iv 2 K 1 1 (L,n ) ) C ( u 2 K 1 1 (L,n ) + v 2 K k+1 1 ( u 2 K 1 1 (L,n) + u 2 K k+1 (L 1,n ) Ths completes the proof. ) ( C u (L,n ) 2 K 1 1 (L,n ) + u 2 K k+1 1 ) Cκ 2na u 2 K k+1 a+ 1 (L,n). (L,n ) ) Based on these estmates, we now gve the range of the gradng parameter κ, for whch we obtan the optmal convergence rate for the fnte element nterpolaton Iu approxmatng u K k+1 a+ 1 ( ). Theorem 4.7 Recall the vector η n Assumpton 2.7, and the soluton u K k+1 ( ) for a+ 1 0 < a <η, 1 l. In the proposed Algorthms (3.1, 3.4 and 3.8), for any 0 < a <η, choose the gradng parameter κ = mn(2 k/a, 1/2). Letdm(S n ) be the dmenson of the fnte element space of degree k assocated wth the resultng graded quadrlateral mesh T n. Then, we have u Iu H 1 ( ) C dm(s n ) k/2 u K k+1 a+ 1 ( ). Proof Recall that the dmenson of the fnte element space dm(s n ) 4 n after n refnements. Wth the choce κ = mn(2 k/a, 1/2), wehaveκ 2 k/a. Replacng κ wth 2 k/a n the estmates n Lemmas 4.5 and 4.6, and addng up the estmates n Lemmas 4.3, 4.5, and 4.6 for dfferent regons of the doman, we have u Iu H 1 ( ) C2 nk u K k+1 a+ 1 ( ) C dm(s n) k/2 u K k+1 a+ 1 ( ). Remark 4.8 When the soluton u of Eq. (3) satsfes Assumpton 2.7, as an mmedate consequence of (4)andTheorem4.7, we have recovered the optmal rate of convergence for the fnte element soluton approxmatng corner sngulartes u u n H 1 ( ) C dm(s n ) k/2, (20) by usng the proposed algorthms for quadrlateral mesh refnements. Regularty s a local property. Some norms equvalent to the weghted norm on dfferent local regons are gven n (5) and(6). In fact, the nterpolaton error analyss n Lemmas 4.3, 4.5, and4.6 can be used as long as the soluton u s locally n K m μ. In the case that the soluton has a smoother part, whch may not be n K m μ near some vertces n V (Remark 2.6), ts nterpolaton Iu conssts of the nterpolaton of the smoother part and the nterpolaton of the sngular part (whch belongs to K m μ ). Ths wll also lead to the desred result as n Theorem 4.7 and n (20), as long as the nterpolaton error for the smoother part s optmal.
76 J Sc Comput (2017) 70:60 84 Remark 4.9 Followng Theorem 4.7, all three proposed algorthms for graded quadrlateral refnements gve rse to the optmal convergence rate n the numercal soluton when the mapped fnte element space contans ˆQ k (see (2)) on the reference element ˆK.However, when the mapped fnte element space s a proper subset of ˆQ k, these three algorthms may gve dfferent convergence rates, even we choose the same gradng parameter κ. For example, when serendpty elements are used, the quadrlaterals need to converge to parallelograms to acheve the optmal convergence rate [2]. In ths case, the trangle parng (Algorthm 3.4) s better than the barycenter refnement (Algorthm 3.1), and the graded 2-refnement (Algorthm 3.8) s preferred among all three algorthms. See the related dscussons n Remarks 3.7 and 3.10. We also menton [22] and reference theren for error estmates of nonconformng elements on quadrlateral meshes that do not converge to parallelograms. Our algorthms are desgned to approxmate a class of sngular solutons that belong to the weghted space (Defnton 2.2). For other types of sngulartes and n the case that the locaton of the sngularty s unknown, new numercal developments may be necessary and adaptve fnte element schemes based on a-posteror analyss are a feasble opton. 5 Numercal Tests We present three sets of numercal tests to llustrate the theoretcal estmates summarzed n Theorem 4.7, Remarks 4.8, and 4.9. Ths n turn justfes our proposed quadrlateral mesh algorthms approxmatng sngular solutons. In order to have a detaled comparson of the three algorthms, we also report test results for quadratc serendpty elements on graded meshes. For the numercal experments, we consder the followng ellptc equaton u = f n, u = g on (21) on two polygonal domans (Fg. 11): the L-shaped doman ( AOB = 3π/2) and the quadrlateral doman ( AOB = 2π/3).Inthetestsbelow,wealwaysletthevertexO be the frst vertex of the doman, namely, v 1 = O. In(21), f and g are gven functons that we wll specfy later. Fg. 11 Two computatonal domans: the L-shaped doman for the frst test set (left) and the quadrlateral for the second test set (rght)
J Sc Comput (2017) 70:60 84 77 5.1 Set I: Blnear Elements In the frst set of tests, we focus on the fnte element space of degree one (k = 1n(2)) assocated wth the three proposed quadrlateral meshes on the L-shaped doman (Fg. 11). In Eq. (21), we let f and g be functons such that the exact soluton s ( u = r 2 2 ( 3 sn θ π ) ), (22) 3 2 where (r,θ)s the polar coordnate at the orgn O = (0, 0). Ths represents a typcal corner sngularty near the reentrant corner at O: u H 5 3 ɛ ( ) / H 2 ( ) for ɛ>0, and therefore the fnte element method s not able to acheve the optmal convergence rate on a quasunform quadrlateral mesh. Let ω be a neghborhood of O, whch s away from other vertces of the doman. Let χ : [0, 1] be a smooth cut-off functon such that χ = 1 near O and χ = 0 outsde ω. Then, the soluton can be decomposed nto two parts u = u S + u R,wherethe sngular part u S = χu / H 2 ( ). Consequently, u R H 2 ( ). Recall v 1 = O and note that u S = 0on. Then, based on Proposton 2.4 and the fact that u S = 0 near other vertces, u S K 2 a+ 1 ( ) for any 0 < a <η,whereη 1 = π/ AOB = 2/3, and η, > 1, can be arbtrarly large. Therefore, by Theorem 4.7 and Remark 4.8, n order to obtan the optmal convergence rate n the numercal soluton, n the three proposed algorthms, t s suffcent to choose the gradng parameter 0 <κ 1 < 2 3/2 0.354 (23) and κ = 1 for other vertces. Ths, n all the cases, results n specally-graded quadrlateral meshes toward the vertex O and quas-unform meshes n other parts of the doman. We tested the H 1 -convergence rates of the fnte element solutons assocated wth the three proposed meshes for dfferent values of κ 1. Some quadrlateral meshes from each of the three algorthms are llustrated n Fgs. 12, 13 and 14. In each column of Tables 1, 2 and 3, we lst the errors u u j H 1 ( ) for each value of κ 1,whereu j s the fnte element soluton on the mesh after j refnements. Next to the error, dsplayed n the parentheses s the error reducton rate that s calculated by ( ) u u j 1 H log 1 ( ) 2. (24) u u j H 1 ( ) Thus, by dm(s j ) 4 j and by the estmate n (20), the optmal convergence rate corresponds to the error reducton rate 1. Fg. 12 Meshes (Algorthm 3.1) for the L-shaped doman: T 0 (left) andt 2 (rght), κ 1 = 0.2
78 J Sc Comput (2017) 70:60 84 Fg. 13 Meshes (Algorthm 3.4) for the L-shaped doman: T 0 (left) andt 3 (rght), κ 1 = 0.2 Fg. 14 Meshes (Algorthm 3.8) for the L-shaped doman: T 0 (left) andt 3 (rght), κ 1 = 0.2 From Tables 1, 2 and 3, t s clear that all the graded meshes can mprove the convergence of the fnte element soluton. In partcular, we obtan the optmal convergence rate (error reducton rate 1) approxmatng the sngular soluton (22) forκ 1 = 0.1, 0.2, 0.3; and we lose the optmal convergence rate for κ = 0.4 and0.5. Note that there s a large gap on the convergence rates between κ 1 = 0.3 andκ 1 = 0.4; and on meshes wth κ 1 = 0.4, although they are graded, the convergence rate s not optmal. Ths s n strong agreement wth our theoretcal predcton n (23): 0 <κ 1 < 0.354 s the range of the gradng parameter n order to acheve the optmal convergence n the energy norm. Another nterestng observaton s that although smple to formulate, Algorthm 3.1 leads to much more degrees of freedom (DOF) than the other two algorthms for comparable actual errors. 5.2 Set II: B-Quadratc Elements The second set tests the performance of the numercal approxmaton from the fnte element space of degree two [k = 2n(2)] on the proposal quadrlateral meshes. Here, we consder Eq. (21) on a quadrlateral doman wth vertces (0, 0), (1, 1 ), ( 1, 1 ),and(0, 1 +cot( 2 3 3 3 9 π)) (Fg. 11). Wth AOB = 2π/3, we assgn functons f and g, such that the exact soluton s u = r 3 2 sn ( 3 2 ( θ π 6 ) ), (25) where (r,θ) s the polar coordnate at the orgn O = (0, 0). Note that on quas-unform meshes, the fnte element soluton does not lead to a second-order convergence rate, due to the lack of regularty n the soluton (u H 5 2 ɛ ( ) for any ɛ>0butu / H 3 ( )). Smlar to the frst set of tests, usng a cut-off functon near the sngular vertex v 1 = O,we can wrte u = u S + u R,foru R H 3 ( ). Snce the sngular part u S = χu = 0on, based on Proposton 2.4, u S K 2 a+ 1 ( ) for any 0 < a <η,whereη 1 = π/ AOB = 1.5, and η,
J Sc Comput (2017) 70:60 84 79 Table 1 Convergence hstory for blnear elements on the L-shaped doman wth meshes from Algorthm 3.1 j DOF u u j H 1 ( ) κ 1 = 0.1 κ 1 = 0.2 κ 1 = 0.3 κ 1 = 0.4 κ 1 = 0.5 2 1217 4.126E 02 (0.949) 2.781E 02 (0.982) 2.567E 02 (0.984) 3.235E 02 (0.862) 4.559E 02 (0.682) 3 4737 2.101E 02 (0.993) 1.415E 02 (0.994) 1.320E 02 (0.979) 1.808E 02 (0.856) 2.887E 02 (0.672) 4 18,689 1.046E 02 (1.016) 7.118E 03 (1.001) 6.739E 03 (0.979) 1.005E 02 (0.855) 1.825E 02 (0.668) 5 74,241 5.180E 03 (1.019) 3.565E 03 (1.003) 3.425E 03 (0.982) 5.565E 03 (0.858) 1.152E 02 (0.667) 6 295,937 2.571E 03 (1.013) 1.782E 03 (1.002) 1.734E 03 (0.984) 3.069E 03 (0.861) 7.268E 03 (0.666) Table 2 Convergence hstory for blnear elements on the L-shaped doman wth meshes from Algorthm 3.4 j DOF u u j H 1 ( ) κ 1 = 0.1 κ 1 = 0.2 κ 1 = 0.3 κ 1 = 0.4 κ 1 = 0.5 2 225 4.760E 02 (0.984) 3.746E 02 (1.026) 3.703E 02 (1.008) 4.525E 02 (0.892) 6.125E 02 (0.718) 3 833 2.465E 02 (1.006) 1.926E 02 (1.016) 1.946E 02 (0.983) 2.568E 02 (0.865) 3.899E 02 (0.690) 4 3201 1.246E 02 (1.014) 9.755E 03 (1.011) 1.009E 02 (0.975) 1.444E 02 (0.855) 2.473E 02 (0.677) 5 12,545 6.231E 03 (1.014) 4.903E 03 (1.007) 5.187E 03 (0.975) 8.060E 03 (0.854) 1.564E 02 (0.670) 6 49,665 3.109E 03 (1.011) 2.456E 03 (1.005) 2.649E 03 (0.977) 4.476E 03 (0.855) 9.881E 03 (0.668) Table 3 Convergence hstory for blnear elements on the L-shaped doman wth meshes from Algorthm 3.8 j DOF u u j H 1 ( ) κ 1 = 0.1 κ 1 = 0.2 κ 1 = 0.3 κ 1 = 0.4 κ 1 = 0.5 2 225 4.717E 02 (0.973) 3.851E 02 (0.994) 3.802E 02 (0.981) 4.562E 02 (0.883) 6.125E 02 (0.718) 3 833 2.475E 02 (0.986) 2.009E 02 (0.995) 2.022E 02 (0.965) 2.600E 02 (0.859) 3.899E 02 (0.690) 4 3201 1.267E 02 (0.995) 1.027E 02 (0.997) 1.057E 02 (0.963) 1.466E 02 (0.851) 2.473E 02 (0.677) 5 12,545 6.393E 03 (1.001) 5.190E 03 (0.999) 5.469E 03 (0.966) 8.202E 03 (0.850) 1.564E 02 (0.670) 6 49,665 3.207E 03 (1.003) 2.609E 03 (1.000) 2.805E 03 (0.970) 4.562E 03 (0.853) 9.881E 03 (0.668) > 1, can be arbtrarly large. Therefore, by Theorem 4.7 and Remark 4.8, n order to obtan the optmal convergence rate n the numercal soluton, n the three proposed algorthms, t s suffcent to choose the gradng parameter 0 <κ 1 < 2 2/1.5 0.397 (26) and κ = 1 for other vertces. Examples of meshes from each of the three algorthms are llustrated n Fgs. 15, 16 and 17. We report the actual errors and the error reducton rates defned n (24)nTables4, 5 and 6 for dfferent values of κ 1. From these tables, we see that the numercal soluton approxmates the sngular soluton (25) n the optmal convergence rate (error reducton rate 2) on the three graded meshes for κ 1 = 0.1, 0.2, 0.3, and the convergence rate slows down for κ 1 = 0.4and 0.5. Ths agan clearly verfes our theory. Namely, we can obtan the optmal convergence
80 J Sc Comput (2017) 70:60 84 Fg. 15 Meshes (Algorthm 3.1) for the quadrlateral: T 0 (left) andt 2 (rght), κ 1 = 0.2 Fg. 16 Meshes (Algorthm 3.4) for the quadrlateral: T 0 (left) andt 3 (rght), κ 1 = 0.2 Fg. 17 Meshes (Algorthm 3.8) for the quadrlateral: T 0 (left) andt 3 (rght), κ 1 = 0.2 Table 4 Convergence hstory for b-quadratc elements on the quadrlateral doman wth meshes from Algorthm 3.1 j DOF u u j H 1 ( ) κ 1 = 0.1 κ 1 = 0.2 κ 1 = 0.3 κ 1 = 0.4 κ 1 = 0.5 1 417 2.651E 03 (1.868) 1.343E 03 (2.027) 9.477E 04 (2.150) 1.209E 03 (2.000) 2.101E 03 (1.576) 2 1601 7.373E 04 (1.903) 3.454E 04 (2.019) 2.365E 04 (2.063) 3.248E 04 (1.954) 7.467E 04 (1.538) 3 6273 1.960E 04 (1.940) 8.753E 05 (2.010) 5.923E 05 (2.028) 8.660E 05 (1.936) 2.647E 04 (1.519) 4 24,833 5.038E 05 (1.975) 2.202E 05 (2.006) 1.483E 05 (2.013) 2.295E 05 (1.930) 9.370E 05 (1.509) 5 98,817 1.273E 05 (1.993) 5.520E 06 (2.004) 3.713E 06 (2.006) 6.054E 06 (1.930) 3.315E 05 (1.505) rate as long as the gradng parameter κ 1 s wthn the range n (26). In addton, smlar to the frst test set, we observe that Algorthm 3.1 leads to much more degrees of freedom than the other two algorthms for comparable actual errors.
J Sc Comput (2017) 70:60 84 81 Table 5 Convergence hstory for b-quadratc elements on the quadrlateral doman wth meshes from Algorthm 3.4 j DOF u u j H 1 ( ) κ 1 = 0.1 κ 1 = 0.2 κ 1 = 0.3 κ 1 = 0.4 κ 1 = 0.5 1 81 3.751E 03 (1.439) 2.055E 03 (1.692) 1.832E 03 (2.265) 2.777E 03 (2.091) 4.598E 03 (1.682) 2 289 1.277E 03 (1.694) 6.596E 04 (1.787) 5.011E 04 (2.038) 7.867E 04 (1.983) 1.664E 03 (1.598) 3 1089 3.729E 04 (1.856) 1.842E 04 (1.924) 1.337E 04 (1.992) 2.181E 04 (1.934) 5.951E 04 (1.551) 4 4225 9.882E 05 (1.959) 4.801E 05 (1.983) 3.472E 05 (1.989) 5.957E 05 (1.915) 2.116E 04 (1.526) 5 16,641 2.510E 05 (1.999) 1.219E 05 (2.000) 8.864E 06 (1.992) 1.608E 05 (1.910) 7.500E 05 (1.513) Table 6 Convergence hstory for b-quadratc elements on the quadrlateral doman wth meshes from Algorthm 3.8 j DOF u u j H 1 ( ) κ 1 = 0.1 κ 1 = 0.2 κ 1 = 0.3 κ 1 = 0.4 κ 1 = 0.5 1 81 2.398E 03 (2.200) 1.294E 03 (2.479) 1.666E 03 (2.426) 2.760E 03 (2.101) 4.599E 03 (1.682) 2 289 6.107E 04 (2.151) 3.264E 04 (2.166) 4.146E 04 (2.187) 7.773E 04 (1.992) 1.665E 03 (1.598) 3 1089 1.520E 04 (2.097) 8.313E 05 (2.062) 1.040E 04 (2.085) 2.144E 04 (1.941) 5.952E 04 (1.550) 4 4225 3.800E 05 (2.045) 2.099E 05 (2.031) 2.609E 05 (2.040) 5.832E 05 (1.921) 2.116E 04 (1.526) 5 16,641 9.516E 06 (2.020) 5.264E 06 (2.018) 6.533E 06 (2.020) 1.569E 05 (1.915) 7.502E 05 (1.513) 5.3 Set III: Quadratc Serendpty Elements Serendpty elements have less degrees of freedom than the quadrlateral elements (2) but ther convergence rates depend on the shape of the quadrlaterals [2]. In the thrd test set, we use quadratc serendpty elements solvng the same problem as n the b-quadratc element case, namely, Eq. (21) wth the exact soluton gven n (25). In partcular, on the reference element ˆK =[ 1, 1] 2, the quadratc serendpty elements have eght shape functons assocated wth the 4 vertces and 4 mdponts on the edges of ˆK : 1 4 (1 ξ)(1 η)( 1 ξ η), 1 2 (1 ξ)(1 η)(1 + η), 1 (1 ξ)(1 + η)( 1 ξ + η), 4 1 (1 ξ)(1 + ξ)(1 + η), 2 1 4 (1 + ξ)(1 + η)( 1 + ξ + η), 1 2 (1 + ξ)(1 η)(1 + η), 1 (1 + ξ)(1 η)( 1 + ξ η), 4 1 (1 ξ)(1 + ξ)(1 η). 2 Compared wth the b-quadratc element (nne shape functons on ˆK ), the quadratc serendpty element shall result n less degrees of freedom on the same mesh, and consequently a smaller lnear system. The numercal experments are conducted on the same meshes that are used n the second test set (see for example Fgs. 15, 16, 17). We report the convergence results from the three mesh algorthms wth dfferent values of κ 1 n Tables 7, 8 and 9. For quadratc serendp-