Natural hp-bem for the electric field integral equation with singular solutions

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1 Natural hp-bem for the electrc feld ntegral equaton wth sngular solutons Alexe Bespalov Norbert Heuer Abstract We apply the hp-verson of the boundary element method (BEM) for the numercal soluton of the electrc feld ntegral equaton (EFIE) on a Lpschtz polyhedral surface Γ. The underlyng meshes are supposed to be quas-unform trangulatons of Γ, and the approxmatons are based on ether Ravart-Thomas or Brezz-Douglas-Marn famles of surface elements. Non-smoothness of Γ leads to sngulartes n the soluton of the EFIE, severely affectng convergence rates of the BEM. However, the sngular behavour of the soluton can be explctly specfed usng a fnte set of power functons (vertex-, edge-, and vertex-edge sngulartes). In ths paper we use ths fact to perform an a pror error analyss of the hp- BEM on quas-unform meshes. We prove precse error estmates n terms of the polynomal degree p, the mesh sze h, and the sngularty exponents. Key words: hp-verson wth quas-unform meshes, boundary element method, electrc feld ntegral equaton, sngulartes, a pror error estmate AMS Subect Classfcaton: 65N38, 65N15, 78M15, 41A10 1 Introducton In ths paper we study numercal approxmatons of the electrc feld ntegral equaton (EFIE) on a surface Γ = Ω, where Ω R 3 s a Lpschtz polyhedron. The EFIE s a boundary ntegral formulaton of a boundary value problem for the tme-harmonc Maxwell equatons n the doman exteror to Ω. It models the scatterng of electromagnetc waves at a perfectly conductng body. For the numercal soluton of the EFIE we use the Galerkn boundary element method (BEM). It employs H(dv Γ )-conformng famles of surface elements, namely Ravart-Thomas (RT) and Brezz-Douglas-Marn (BDM) elements, to dscretse the varatonal formulaton of the EFIE (called Rumsey s prncple). Ths approach s referred to as the natural BEM for the EFIE. As n fnte element methods, the convergence may be acheved ether by keepng Supported by EPSRC under grant no. EP/E058094/1. School of Mathematcs, Unversty of Manchester, Manchester, M13 9PL, UK. Emal: albespalov@yahoo.com Facultad de Matemátcas, Pontfca Unversdad Católca de Chle, Avenda Vcuña Mackenna 4860, Santago, Chle. Emal: nheuer@mat.puc.cl 1

2 polynomal degrees fxed and refnng the mesh (h-verson), or by fxng the mesh and ncreasng polynomal degrees (p-verson), or by smultaneous h-refnement and p-enrchment (hp-verson). The Galerkn BEM s wdely used n the engneerng practce for the smulaton of electromagnetc scatterng. Moreover, t had been used long before a rgorous theoretcal analyss of the method became avalable. Error analyses of dfferent boundary element schemes for the eddy current problem on smooth obstacles are gven by MacCamy and Stephan n [24, 25]. Despte these early results the BEM-analyss for non-smooth obstacles started much later. In fact, the convergence and a pror error analyss of the h-bem for the EFIE on pecewse smooth (open or closed) surfaces has been developed wthn the last decade (see [13, 22, 16, 10, 15]), and the correspondng results for hgh-order methods (p- and hp-bem) are very recent (see [6, 4, 7]). In partcular, [6] and [7] establsh quas-optmal convergence of the natural hp-bem on meshes of shape-regular elements. An essental ngredent for the proofs n these papers are the proectonbased nterpolaton operators developed by Demkowcz and co-authors [19, 20, 18]. These operators also facltate a pror error analyss, where one needs hp-approxmaton theory n specfc trace spaces. In [4] we prove an a pror error estmate of the hp-bem on quas-unform meshes of affne elements under the assumpton that the regularty of the soluton to the EFIE s gven n Sobolev spaces on Γ. The latter result states that the method converges wth the same rate r n both h and p 1 (here, r denotes the Sobolev regularty order, and p s assumed to be large enough). However, t s evdent from the numercal results reported n [23] that the p-bem converges faster than the h-bem for the EFIE. Ths s smlar to what was observed and proved for the p-bem and the h-bem appled to ellptc problems n three-dmensons (cf. [27, 21, 3, 5]). Wth ths paper we fll a gap n the theory of the BEM for the EFIE on polyhedral surfaces by provng a precse error estmate for the hp-bem on quas-unform meshes. Smlarly to the ellptc case we make use of explct expressons for sngulartes n electromagnetcs felds (these expressons are avalable from [17, 6]). The establshed error estmate shows that the convergence rates n h and p depend on the strongest sngularty exponent, and that the p-bem converges twce as fast as the h-bem on quas-unform meshes. Ths extends the results of [6], where the analyss was restrcted to the p-bem on a plane open surface. It s now well known that approprate decompostons of vector felds (on both the contnuous and dscrete level) are crtcal for the analyss of electromagnetc problems and ther approxmatons. In the case of the EFIE the man dea s to solate the kernel of the dv Γ - operator such that the complementary feld possesses an enhanced smoothness (cf. [10, 9, 7]). The correspondng decompostons of sngular vector felds greatly facltate our analyss n ths paper as well. In partcular, the complementary vector felds are sngular vector functons belongng to H 1/2 (Γ). Then, n the case of the BDM-based BEM, these vector functons can be approxmated component-wse by contnuous pecewse polynomals, and the desred hp-error estmates are derved by usng the correspondng results n [3] for scalar sngulartes (belongng to H 1/2 (Γ)). However, ths smple approach does not work for the RT-based BEM because the dmenson of the underlyng RT-space on the reference element s smaller than the dmenson of the BDM-counterpart. That s why, the results of [3] cannot be appled drectly, and addtonal techncal arguments are needed (see the proof of Lemma 4.2 n Secton 5.2). 2

3 The rest of the paper s organsed as follows. In the next secton we formulate the EFIE n ts varatonal form and recall the typcal structure of the soluton to ths model problem. In Secton 3 we ntroduce the hp-verson of the BEM for the EFIE and formulate the man result of the paper (Theorem 3.1) statng convergence rates of the method. Ths result follows from the general approxmaton theorem (Theorem 4.1) establshed n Secton 4. The proof of Theorem 4.1 reles, n partcular, on two techncal lemmas whch are proved n Secton 5. Throughout the paper, C denotes a generc postve constant ndependent of h and p. 2 Formulaton of the problem Let Γ be a Lpschtz polyhedral surface n R 3. Throughout the paper we wll use exactly the same notaton as n [4] for all nvolved dfferental and boundary ntegral operators as well as for Sobolev spaces of scalar functons and tangental vector felds on Γ (all essental defntons are gven n [4, Secton 3.1]). In partcular, we use boldface symbols for vector felds, and the spaces (or sets) of vector felds are denoted n boldface as well. For a gven wave number κ > 0, we denote by Ψ κ (resp., Ψ κ ) the scalar (resp., the vectoral) sngle layer boundary ntegral operator on Γ for the Helmholtz operator κ 2 (see [15, Secton 5]). The varatonal formulaton for the EFIE wll be posed n the followng Hlbert space of tangental vector felds on Γ: X = H 1/2 (dv Γ, Γ) := {u H 1/2 (Γ); dv Γ u H 1/2 (Γ)}, whch s the trace space of H(curl, Ω), where Ω R 3 s a Lpschtz polyhedron such that Γ = Ω (we refer to [11, 12, 14] for the defnton and propertes of ths and other trace spaces on Γ). Let X be the dual space of X (wth dualty parng extendng the L 2 (Γ)-nner product for tangental vector felds). Then, for a gven source functonal f X the varatonal formulaton of the EFIE reads as: fnd a complex tangental feld u X such that a(u, v) := Ψ κ dv Γ u, dv Γ v κ 2 Ψ κ u, v = f, v v X. (2.1) Here, the brackets, denote dualtes assocated wth H 1/2 (Γ) and H 1/2 (Γ). To ensure the unqueness of the soluton to (2.1) we always assume that κ 2 s not an electrcal egenvalue of the nteror problem. Let us recall the typcal structure of the soluton u to problem (2.1), provded that the source functonal f s suffcently smooth (we note that ths regularty assumpton s satsfed for the electromagnetc scatterng wth plane ncdent wave). We use the results of [6, Appendx A]. These results were derved from the regularty theory n [17] for the boundary value problems for Maxwell s equatons n 3D by makng use of trace arguments and some techncal calculatons n the sprt of [29] and [28]. Let V = {v} and E = {e} denote the sets of vertces and edges of Γ, respectvely. For v V, let E(v) denote the set of edges wth v as an end pont. Then the soluton u of (2.1) can be wrtten as u = u reg + u sng, (2.2) 3

4 where u reg X k := {u H k (Γ); dv Γ u H k (Γ)} wth k > 0 (2.3) (the spaces H k (Γ) and H k (Γ) are defned n a pecewse fashon by localsaton to each face of Γ, and the space X k s equpped wth ts graph norm X k, see [4]), u sng = u e + u v + u ev, (2.4) e E v V v V e E(v) and u e, u v, and u ev are the edge, vertex, and edge-vertex sngulartes, respectvely. In order to wrte explct expressons for the above sngulartes, let us fx a vertex v V and an edge e E(v). Then, on each face Γ ev Γ such that e Γ ev we wll use local polar and Cartesan coordnate systems (r v, θ v ) and (x e1, x e2 ), both wth the orgn at v, such that e = {(x e1, x e2 ); x e2 = 0, x e1 > 0} and for a suffcently small neghbourhood B τ of v there holds Γ ev B τ {(r v, θ v ); 0 < θ v < ω v }. Here, ω v denotes the nteror angle (on Γ ev ) between the edges meetng at v. For smplcty of notaton we wrte out here only the leadng sngulartes n u e, u v, and u ev on the face Γ ev, thus omttng the correspondng terms of hgher regularty (see [6, Appendx A] for complete expansons). For the edge sngulartes u e one has u e = curl Γ ev ( x γe 1 e2 log x e2 se 1 b e 1 (x e1 ) χ e 1(x e1 ) χ e 2(x e2 ) ) + x γe 2 e2 log x e2 se 2 b e 2 (x e1 ) χ e 1(x e1 ) χ e 2(x e2 ), (2.5) where curl Γ ev = ( / x e2, / x e1 ) s the tangental vector curl operator curl Γ restrcted to the face Γ ev (cf. [11, 12]), γ1 e, γe 2 > 1 2, and se 1, se 2 0 are ntegers. Here, χe 1, χe 2 are C cut-off functons wth χ e 1 = 1 n a certan dstance to the end ponts of e and χe 1 = 0 n a neghbourhood of these vertces. Moreover, χ e 2 = 1 for 0 x e2 δ e and χ e 2 = 0 for x e2 2δ e wth some δ e (0, 1 2 ). The functons be 1 χe 1 Hm 1 (e) and b e 2 χe 1 Hm 2 (e) for m 1 and m 2 as large as requred. The vertex sngulartes u v have the form ( ) u v = curl Γ ev r λv 1 v log r v qv 1 χ v (r v ) χ v 1(θ v ) + r λv 2 v log r v qv 2 χ v (r v ) χ v 2(θ v ), (2.6) where λ v 1, λv 2 > 1 2 are real numbers, qv 1, qv 2 0 are ntegers, χv s a C cut-off functon wth χ v = 1 for 0 r v τ v and χ v = 0 for r v 2τ v wth some τ v (0, 1 2 ). The functons χv 1, χv 2 are such that χ v 1 Ht 1 (0, ω v ), χ v 2 Ht 2 (0, ω v ) for t 1, t 2 as large as requred. For the combned edge-vertex sngularty u ev one has u ev = u ev 1 + u ev 2, where ( ) u ev 1 = curl Γ ev x λv 1 γe 1 e1 x γe 1 e2 log x e1 β 1 log x e2 β 2 χ v (r v )χ ev (θ v ) 4

5 ( + x λv 2 γe 2 e1 x γe 2 e2 log x e1 β 3 log x e2 β 4 χ v (r v )χ ev 0 (θ v ) 1 ) (2.7) and ( ) u ev 2 = curl Γ ev x γe 1 e2 log x e2 se 1 b e 3 (x e1, x e2 ) χ e 2(x e2 ) + x γe 2 e2 log x e2 se 2 b e 4 (x e1, x e2 ) χ e 2(x e2 ). (2.8) Here, λ v, γe, se ( = 1, 2), χv, and χ e 2 are as above, β k 0 (k = 1..., 4) are ntegers, β 1 + β 2 = s e 1 + qv 1, β 3 + β 4 = s e 2 + qv 2 wth qv 1, qv 2 beng as n (2.6), χev s a C cut-off functon wth χ ev = 1 for 0 θ v β v and χ ev = 0 for 3 2 β v θ v ω v for some β v (0, mn{ω v /2, π/8}]. The functons b e 3 and be 4, when extended by zero onto IR2+ := {(x e1, x e2 ); x e2 > 0}, le n H m 1 (IR 2+ ) and H m 2 (IR 2+ ), respectvely, wth m 1, m 2 as large as requred. Fnally, the supports of u ev 1 are subsets of the sector S ev = {(r v, θ v ); 0 r v 2τ v, 0 θ v 3 2 β v}. and uev 2 Remark 2.1 () The exponents γ e ( = 1, 2) of the edge and vertex-edge sngulartes n (2.5), (2.7), (2.8) satsfy γ e > 1 2. However, for our approxmaton analyss below t suffces to requre that γ e > 0 ( = 1, 2). Note that γe > 0 and λv > 1 2 ( = 1, 2) are the mnmum requrements to guarantee u X. () As mentoned above, the terms of hgher regularty (.e., wth greater sngularty exponents) are omtted n (2.5) (2.8). These terms are necessary to obtan the regular part u reg X k of decomposton (2.2) as smooth as requred. Ths can be done by consderng suffcently many (omtted) sngularty terms of each type. Remark 2.2 () By (2.5) (2.8) we conclude that any sngular vector feld u s n (2.4) (s = e, v, or ev) can be wrtten as u s = curl Γ w s + v s = curl Γ w s + (v s 1, v s 2) (2.9) wth correspondng (scalar) sngular functons w s, v1 s, vs 2 beng defned on the whole surface Γ. Note that these scalar functons are H 1/2 -regular on Γ,.e., w s H 1/2 (Γ), v s = (v s 1, v s 2) H 1/2 (Γ), s = e, v, ev. () It s mportant to observe that the functons w s, v1 s, vs 2 (s = e, v, or ev) n (2.9) are typcal scalar sngulartes nherent to solutons of the boundary ntegral equatons wth hypersngular operator for the Laplacan on Γ and wth possbly sngular rght-hand sde. Contnuous pecewse polynomal approxmatons of these scalar sngulartes n fractonal-order Sobolev spaces were analysed n [3, 1] and wll be used to prove the man result of the present paper. 3 The hp-verson of the BEM and the man result For the approxmate soluton of (2.1) we apply the hp-verson of the BEM on quas-unform trangulatons of Γ. Our BEM s based on Galerkn dscretsatons wth an approprate famly of 5

6 H(dv Γ, Γ)-conformng surface elements (here, H(dv Γ, Γ) := {u L 2 (Γ); dv Γ u L 2 (Γ)}). In what follows, h > 0 and p 1 wll always specfy the mesh parameter and a polynomal degree, respectvely. For any Ω R n we wll denote ρ Ω = sup{dam(b); B s a ball n Ω}. Let T = { h } be a famly of meshes h = {Γ ; = 1,..., J}. Each mesh s a partton of Γ nto trangular elements Γ such that Γ = J =1 Γ, and the ntersecton of any two elements Γ, Γ k ( k) s ether a common vertex, an entre sde, or empty. In the followng we always dentfy a face of the polyhedron Γ wth a subdoman of R 2. We denote h = dam(γ ) for any Γ h. Furthermore, any element Γ s the mage of the reference trangle K = {(ξ 1, ξ 2 ); 0 < ξ 1 < 1, 0 < ξ 2 < 1 ξ 1 } under an affne mappng T, more precsely Γ = T ( K), x = T (ξ) = B ξ + b, where B R 2 2, b R 2, x = (x 1, x 2 ) Γ, and ξ = (ξ 1, ξ 2 ) K. Then, the Jacoban matrx of T s B R 2 2, and ts determnant J := det(b ) satsfes the relaton J h 2. We consder a famly T of quas-unform shape-regular meshes h on Γ n the sense that there exst postve constants σ 1, σ 2 ndependent of h = max h such that for any Γ h and arbtrary h T there holds h σ 1 ρ Γ, h σ 2 h. (3.1) Whereas the mappng T ntroduced above s used to assocate scalar functons defned on the real element Γ and on the reference trangle K, the Pola transformaton s used to transform vector-valued functons between K and Γ : v = M (ˆv) = 1 J B ˆv T 1, ˆv = M 1 (v) = J B 1 v T. (3.2) Let us ntroduce the needed polynomal sets. By P p (K) we denote the set of polynomals of total degree p on the reference trangle K. We wll use two famles of H(dv Γ, Γ)-conformng surface elements: the Ravart-Thomas (RT) and Brezz-Douglas-Marn (BDM) elements. The correspondng spaces of degree p 1 on the reference trangle K wll be denoted as follows (see, e.g., [8, 26]): P RT p (K) = (P p 1 (K)) 2 ξp p 1 (K); P BDM p (K) = (P p (K)) 2. We wll use the unfed notaton P p (K) whch refers to ether the RT- or BDM-space on K for p 1. Accordngly, all results n ths paper are formulated n a unfed way and are vald for both the RT- and BDM-based boundary element spaces defned on the trangulaton of Γ. Note, however, that n some cases we wll need to provde arguments separately for each type of these boundary elements. Usng transformatons (3.2), we set X hp := {v H(dv Γ, Γ); M 1 (v Γ ) P p (K), = 1,..., J}. (3.3) 6

7 Note that only one type of surface elements (.e., ether the RT- or BDM-elements) s used n (3.3) for all trangles Γ. We wll denote by N = N(h, p) the dmenson of the dscrete space X hp. One has N h 2 for fxed p and N p 2 for fxed h. The hp-verson of the Galerkn BEM for the EFIE reads as: Fnd u hp X hp such that a(u hp, v) = f, v v X hp. (3.4) Due to the nfnte-dmensonal kernel of the dv Γ -operator, the blnear form a(, ) n (2.1) s not X-coercve, and, hence, the unque solvablty of (3.4) cannot be proved by standard arguments. However, the refned analyss n [7] shows that the unque BEM-soluton u hp X hp does exst, and t converges quas-optmally to the exact soluton u X of the EFIE as N(h, p). Ths result s formulated n the followng proposton. Proposton 3.1 [7, Theorem 1.2] There exsts N 0 1 such that for any f X and for arbtrary mesh-degree combnaton satsfyng N(h, p) N 0 the dscrete problem (3.4) s unquely solvable and the hp-verson of the Galerkn BEM converges quas-optmally,.e., u u hp X C nf{ u v X ; v X hp }. (3.5) Here, u X s the soluton of (2.1), u hp X hp s the soluton of (3.4), X denotes the norm n X, and C > 0 s a constant ndependent of h and p. The next theorem s the man result of ths paper. It states convergence rates of the hp-bem wth quas-unform meshes for the EFIE. These convergence rates (n both the mesh parameter h and the polynomal degree p) are gven explctly n terms of the sngularty exponents of the vector felds n (2.5) (2.8). Theorem 3.1 Let u X and u hp X hp be the solutons of (2.1) and (3.4), respectvely. We assume that the source functonal f n (2.1) s suffcently smooth such that representaton (2.2) (2.8) holds for the soluton u of (2.1). Let v 0 V, e 0 E(v 0 ) be a vertex-edge par such that mn{λ v /2, λv /2, γe 0 1, γe 0 2 } = mn v V,e E(v) mn {λv 1 + 1/2, λ v 2 + 1/2, γ e 1, γ e 2} wth λ v and γ e ( = 1, 2) beng as n (2.5) (2.8). Then for any h > 0 and for every p mn {λ v 0 1, λv 0 2, γe , γe } there holds ( ) v h mn{λ /2,λv /2,γe 0 1,γe 0 }( 2 u u hp X C p log p ) β+ν, (3.6) h where β = max {q v se , qv se } f λv 0 = γ e for = 1, 2, max {q v se , qv se 0 2 } f λv 0 1 = γe , λv 0 2 γe , max {q v se 0 1, qv se } f λv 0 1 γe , λv 0 2 = γe , max {q v se 0 1, qv se 0 2 } otherwse 7 (3.7)

8 wth the numbers s e 0 and q v 0 ( = 1, 2) gven n (2.5) and (2.6), respectvely, and ν = { 1 2 f p = mn {λ v 0 1, λv 0 2, γe 0 1 1/2, γe 0 2 1/2}, 0 otherwse. (3.8) If 1 p < mn {λ v 0 1, λv 0 2, γe , γe }, then for any h > 0 there holds u u hp X C h p+1/2. (3.9) Proof. Consderng enough sngularty terms n representaton (2.4) the functon u reg n (2.3) s as regular as needed. Then, due to the quas-optmal convergence (3.5) of the hp-bem wth quas-unform meshes, the asserton follows mmedately from the general approxmaton result gven n Theorem 4.1 below. Remark 3.1 We have only consdered meshes of trangular elements on Γ. If the meshes contan also shape-regular parallelogram elements (.e., affne mages of the reference square), then a pror error estmates of Theorem 3.1 reman vald only n the case of the RT-based BEM. Ths s because all the auxlary results needed for the proof are vald n ths case. Ths, however, s not true for the BDM-based BEM. In partcular, the arguments n the proofs of Proposton 3.1 above and Proposton 4.1 below rely essentally on the fact that the nvolved polynomal spaces form the exact curl dv sequence (on the reference element), a property the BDM-spaces fal to satsfy on the reference square (see, e.g., [18]). Remark 3.2 We have assumed that Γ s a polyhedral (closed) surface. However, all arguments n our proofs carry over only wth mnor modfcatons to the case of a pecewse plane orentable open surface Γ. Note that n ths case there are no restrctons needed on the wave number κ to ensure the unqueness of the soluton to the EFIE; the strongest edge sngulartes n (2.5) have the exponents γ e = 1 1/2 2 ( = 1, 2); the energy space for the EFIE s H 0 (dv Γ, Γ) and the boundary element space X hp conssts of H(dv Γ, Γ)-conformng polynomal vector felds wth normal components vanshng on Γ (see [6, 4] for detals). 4 General approxmaton result In ths secton we prove the followng general hp-approxmaton result for the vector feld u gven by formulas (2.2) (2.8) wth the sngularty exponents γ e and λ v ( = 1, 2) satsfyng the mnmum requrements to guarantee u X. Theorem 4.1 Let the vector feld u be gven by (2.2) (2.8) on Γ wth γ e 1, γe 2 > 0 and λv 1, λv 2 > 1 2 for each edge e and every vertex v. Also, let v 0 V, e 0 E(v 0 ) be such that mn{λ v /2, λv /2, γe 0 1, γe 0 2 } = mn v V,e E(v) mn {λv 1 + 1/2, λ v 2 + 1/2, γ e 1, γ e 2} 8

9 wth λ v and γ e ( = 1, 2) beng as n (2.5) (2.8). Then for any h > 0 and for every p mn {λ v 0 1, λv 0 2, γe , γe }, there exsts uhp X hp such that { u u hp X C max h mn{k,p}+1/2 p (k+1/2), ( ) v h mn{λ /2,λv /2,γe 0 1,γe 0 } 2 p log p ) β+ν, (4.1) h where β and ν are defned by (3.7) and (3.8), respectvely. If 1 p < mn {λ v 0 1, λv 0 2, γe , γe }, then for any h > 0 there exsts uhp X hp such that u u hp X C h mn{k,p}+1/2. (4.2) In order to prove ths theorem one needs to fnd dscrete vector felds belongng to X hp and approxmatng the smooth and sngular parts of u such that the approxmaton errors satsfy the upper bounds n (4.1) and (4.2). We start wth formulatng the followng hp-approxmaton result for regular vector felds on Γ. Ths result wll be used, n partcular, to approxmate the vector feld u reg X k n (2.2). Proposton 4.1 Let P hp : X X hp be the orthogonal proecton wth respect to the norm n X. If u X k wth k > 0, then u P hp u X C h mn{k,p}+1/2 p (k+1/2) u X k (4.3) wth a postve constant C ndependent of h, p, and u. The proof s gven n [4, Theorem 4.1] for the case of RT-spaces, and t carres over wthout essental modfcatons to the case of BDM-spaces on trangular elements (cf. Remark 3.1). Now, we wll study approxmatons of the sngular part u sng n representaton (2.2). By (2.4) (2.8) and due to the arguments n Remark 2.2 (), we conclude that u sng can be wrtten as u sng = curl Γ w + v = curl Γ w + (v 1, v 2 ), (4.4) where w H 1/2 (Γ) and v H 1/2 (Γ). Let us defne the followng dscrete space (of contnuous pecewse polynomals) over the mesh h : S hp (Γ) := {v C 0 (Γ); v Γ T P p (K), = 1,..., J}. We wll also need the followng functons of h and p: f (h, p) := h α p 2α (1 + log(p/h)) β +ν, (4.5) where = 1, 2, α := mn {λ v 0 + 1/2, γ e 0 }, (4.6) 9

10 β := { q v 0 + s e f λ v 0 = γ e 0 1 2, q v 0 + s e 0 otherwse, { 1 ν := 2 f p = α 1 2, 0 otherwse, and the numbers γ e 0, λv 0, se 0, qv 0 ( = 1, 2) are gven n (2.5) (2.8) for the vertex-edge par (v 0, e 0 ) ntroduced n the formulaton of Theorem 4.1. In the next two lemmas we formulate approxmaton results for the vector felds curl Γ w and v on the rght-hand sde of (4.4). The proofs are gven n Secton 5 below. Lemma 4.1 Let w H 1/2 (Γ) be the scalar sngular functon n representaton (4.4). Then for any h > 0 and p 1, there exsts w hp S hp (Γ) such that curl Γ w hp X hp and there holds { C f1 curl Γ w curl Γ w hp (h, p) f p α X, C h p+1/2 f 1 p < α 1 1 2, (4.9) where f 1 (h, p) and α 1 are defned by (4.5) and (4.6), respectvely. Lemma 4.2 Let v H 1/2 (Γ) be the sngular vector feld n representaton (4.4). Then for any h > 0 and p 1, there exsts v hp X hp satsfyng { C f2 v v hp (h, p) f p α X, C h p+1/2 f 1 p < α 2 1 2, (4.10) where f 2 (h, p) and α 2 are defned by (4.5) and (4.6), respectvely. Now we are able to prove Theorem 4.1. Proof of Theorem 4.1. For the regular vector feld u reg X k n (2.2) we use the orthogonal proecton P hp : X X hp wth respect to the norm n X to defne u hp reg := P hp u reg X hp. Then we have by Proposton 4.1 (4.7) (4.8) u reg u hp reg X C h mn{k,p}+1/2 p (k+1/2). (4.11) Snce the sngular part of decomposton (2.2) can be wrtten as n (4.4), we use approxmatons w hp S hp (Γ) and v hp X hp from the above two lemmas to defne u hp sng := curl Γ w hp + v hp X hp. Then, applyng the trangle nequalty, we obtan by (4.9) and (4.10) { C max u sng u hp {f1 sng (h, p), f 2 (h, p)} f p mn {α 1, α 2 } 1 2 X, C h p+1/2 f 1 p < mn {α 1, α 2 } 1 2. (4.12) Now, we set u hp := u hp reg + u hp sng X hp. Combnng estmates (4.11) and (4.12), applyng the trangle nequalty, and usng expressons (4.5) and (4.6) for the functons f (h, p) and the parameters α, respectvely, we prove the desred estmates n (4.1) and (4.2). 10

11 5 Proofs of techncal lemmas In ths secton we prove Lemmas 4.1 and Proof of Lemma 4.1 Recallng Remark 2.2 (), we use the results of [3, 1] to fnd the desred pecewse polynomal w hp S hp (Γ) such that the norm w w hp H 1/2 (Γ) s bounded as n (4.9) (see Theorems 5.1, 5.2 n [3] and Theorem 4.1 n [1]). Then, recallng the fact that the the operator curl Γ : H 1/2 (Γ) H 1/2 (Γ) s contnuous (see [12]), we derve the estmate n (4.9): curl Γ w curl Γ w hp X = curl Γ (w w hp ) 1/2 H (Γ) C w w hp H 1/2 (Γ) { C f1 (h, p) f p α 1 1 2, C h p+1/2 f 1 p < α It remans to prove that curl Γ w hp X hp. In fact, t s easy to check (see [6, p. 615]) that ( M 1 (curl Γ w hp Γ ) = curl (w hp Γ T ), where curl = ξ 2, ξ 1 ). Hence, recallng that w hp Γ T P p (K), we conclude that M 1 (curl Γ w hp Γ ) (P p 1 (K)) 2 P p (K) (ths s because (P p 1 (K)) 2 s a subset of both P BDM p (K) and P RT p (K)). Moreover, curl Γ w hp H(dv Γ, Γ), because dv Γ (curl Γ w hp ) 0 on Γ. Therefore, curl Γ w hp X hp, and the proof s fnshed. 5.2 Proof of Lemma 4.2 Let v = (v 1, v 2 ) be the second term n decomposton (4.4) of the sngular vector feld u sng. Agan, usng the results n [3, 1], we fnd contnuous pecewse polynomal approxmatons to the scalar components v 1, v 2 of v: for any h > 0 and every p 1, there exst v hp 1, vhp 2 S hp (Γ) such that for = 1, 2 there holds v v hp H 1/2 (Γ) { C f2 (h, p) f p α 2 1 2, C h p+1/2 f 1 p < α (5.1) wth postve constants C > 0 ndependent of h and p. Let v hp = (v hp 1, vhp 2 ). Then vhp H(dv Γ, Γ). Furthermore, n the case of BDM-elements we observe that for any Γ there holds M 1 (v hp Γ ) = J B 1 (v hp Γ ) T (P p (K)) 2 = P BDM (K). p 11

12 Therefore, v hp X hp n ths case. Moreover, snce v H 1/2 (Γ) and v hp S hp (Γ), we use estmate (5.1) and the contnuty of the operator dv Γ : H 1/2 (Γ) H 1/2 (Γ) to obtan v v hp X = v v hp 1/2 H (Γ) + dv Γ(v v hp ) H 1/2 (Γ) C v v hp H 1/2 (Γ) C 2 =1 v v hp H 1/2 (Γ) { C f2 (h, p) f p α 2 1 2, C h p+1/2 f 1 p < α (5.2) Unfortunately, n the case of RT-elements ths component-wse approxmaton of v does not work snce the dmenson of the RT-space on the reference trangle s smaller than the dmenson of the BDM-space, and, n general, M 1 (v hp Γ ) / P RT p (K), so that v hp = (v hp 1, vhp 2 ) / X hp. In ths case we follow the procedure descrbed n [3] (for the scalar case). More precsely, we use approprate h-scaled cut-off functons and represent the vector feld v as the sum of a sngular vector feld ϕ wth small support n the vcnty of the edges and a suffcently smooth vector feld ψ. In the rest of ths subsecton we wll demonstrate how ths procedure works n the vector case. We wll gve a concse step-by-step outlne of the procedure referrng frequently to [3] for partcular error estmates and other techncal detals. Step 1: decomposton of v. Let h 0 = (σ 1 σ 2 ) 1 h wth σ 1, σ 2 from (3.1). Usng the scaled cut-off functons χ e 2 (x e2/h 0 ) and χ v (r v /h 0 ) wth χ e 2 and χv from (2.5) and (2.6), respectvely, one can decompose v as follows (cf. [3, eqs. (5.4), (5.21), (6.4)]) v = ϕ + ψ, (5.3) where supp ϕ v V e E(v) (Āe Āv), ϕ H 1/2 (Γ), ψ X m (see (2.3) for the notaton), and ψ vanshes n small (h-dependent) neghbourhoods of each vertex and each edge of Γ. Here, A e s the unon of elements at one edge e,.e., Ā e := { Γ ; Γ e ø} (note that the endponts of e are not ncluded n e), A v s the unon of elements at a vertex v,.e., Ā v := { Γ ; v Γ }, m s suffcently large and depends on the parameters t 2 and m 2 specfed for the sngulartes u v and u ev 2, respectvely. Step 2: approxmaton of ϕ for p = 1. If p = 1, then one can approxmate ϕ by zero. One has ϕ hp 0 X hp, and, recallng the contnuty of the operator dv Γ, we derve ϕ ϕ hp X = ϕ H 1/2 (Γ) + dv Γ ϕ H 1/2 (Γ) C ϕ H 1/2 (Γ). (5.4) We wll obtan h-estmates for the norms of ϕ n H s (Γ) wth s = 0 and s = 1 2 +ε (for suffcently small ε > 0) by usng the fact that ϕ has a small support. Frst, we apply Lemma 3.1 of [2] and Lemma 3.5 of [3] to localse the norm ϕ H 1/2+ε(Γ) to the faces Γ f Γ and to the elements Γ Γ f, respectvely. Then, we use scalng on each element Γ supp ϕ (cf. [3, Lemma 3.1, eqs. (5.6), (5.11), (5.23), (6.7)]). As a result we have ϕ 2 H s (Γ) C ϕ 2 H s (Γ f ) f: Γ f Γ 12

13 C f: Γ f Γ : Γ Γ f ( ) h 2s ϕ 2 L 2 (Γ ) + ϕ 2 H s (Γ ) Ch 2(α 2+1/2 s) (1 + log(1/h)) 2 β 2 for s {0, 1/2 + ε}. Here, α 2 and β 2 are defned by (4.6) and (4.7), respectvely. between H 0 (Γ) and H 1/2+ε (Γ), we obtan by (5.4) Hence, usng the nterpolaton ϕ ϕ hp X Ch α 2 (1 + log(1/h)) β 2. (5.5) Step 3: approxmaton of ϕ for p 2. We approxmate each component ϕ of ϕ by a pecewse polynomal ϕ hp S h,p 1 (Γ) ( = 1, 2). Here we can use the results of [3, 1] for each type of sngularty: there exst ϕ hp S h,p 1 (Γ) such that for = 1, 2 there holds (cf. [3, eqs. (5.12), (5.13), (5.24), (6.9)]) ϕ ϕ hp H 1/2 (Γ) C hα 2 (p 1) 2α 2 (1 + log p 1 h ) β 2 C h α 2 p 2α 2 (1 + log p h ) β 2, (5.6) where α 2 and β 2 are the same as n (5.5). Settng ϕ hp = (ϕ hp 1, ϕhp 2 ) we observe that ϕhp H(dv Γ, Γ), and for any element Γ one has M 1 (ϕ hp Γ ) = J B 1 (ϕ hp Γ ) T (P p 1 (K)) 2 P RT (K). Hence, ϕ hp X hp. Moreover, snce ϕ H 1/2 (Γ) and ϕ hp S h,p 1 (Γ) for = 1, 2, we estmate by analogy wth (5.4) and usng (5.6): p ϕ ϕ hp X C ϕ ϕ hp H 1/2 (Γ) 2 C ϕ ϕ hp H 1/2 (Γ) C hα 2 p 2α 2 (1 + log p h ) β 2. (5.7) =1 Comparng (5.5) and (5.7) we observe that one can use estmate (5.7) also n the case p = 1. Step 4: approxmaton of ψ. Recallng that ψ X m we apply Proposton 4.1: there exsts ψ hp X hp such that ψ ψ hp X C h mn{k,p}+1/2 p (k+1/2) ψ X k, 0 k m. (5.8) We also recall that ψ vanshes n the h-neghbourhood of all edges and vertces of Γ. Therefore, havng explct expressons of the sngular vector feld v (and thus, of ψ) on each face Γ f Γ, one can derve upper bounds for the norms ψ H k (Γ f ) and dv Γ f ψ H k (Γ f ) n terms of the mesh sze h, the parameter k, and the sngularty exponents. For the norm of ψ ths can be done component-wse usng the same calculatons as n [3] (see, e.g., nequaltes (5.15) and (6.10) theren): ψ H k (Γ f ) C h α2+1/2 k (log(1/h)) β 2 + ν 2, α k m, (5.9) 13

14 where α 2 and β 2 are defned by (4.6) and (4.7), respectvely, ν 2 = 1 2 f k = α and ν 2 = 0 otherwse. To estmate the norm dv Γ f ψ H k (Γ f ) we observe that the operator dv Γ f reduces all sngularty exponents by one (whle preservng the structure of the correspondng sngularty). Then, usng smlar calculatons as ndcated above, we obtan dv Γ f ψ H k (Γ f ) C h α 2 1/2 k (log(1/h)) β 2 + ν 2, α k m, (5.10) where α 2, β 2 are the same as n (5.9), whereas ν 2 = 1 2 f k = α and ν 2 = 0 otherwse. By (5.8)-(5.10) we conclude that ψ ψ hp X C h mn{k,p}+α 2 k p (k+1/2) (log(1/h)) β 2 + ν 2, α k m (5.11) wth the same α 2, β 2, and ν 2 as n (5.10). Let p > 2α Snce m s large enough, we can select an nteger k satsfyng 2α < k mn {m, p}. Then mn {k, p} = k, and p (k+1/2) p 2α 2. If α < p 2α (.e., p s bounded), we choose an nteger k (α 2 1 2, p], and f p = α 2 1 2, then we take k = p = α In both these cases mn {k, p} = k, and p (k+1/2) C p 2α 2. Thus, for any p α 2 1 2, selectng k as ndcated above we fnd by (5.11) ψ ψ hp X C h α 2 p 2α 2 (log(1/h)) β 2 +ν 2 (5.12) wth α 2, β 2, and ν 2 beng defned by (4.6), (4.7), and (4.8), respectvely. Step 5: approxmaton of v = ϕ + ψ. Let us defne v hp := ϕ hp + ψ hp X hp, where ϕ hp and ψ hp are approxmatons constructed above (see Steps 2 4). Then combnng estmates (5.7) and (5.12) and usng the trangle nequalty we prove (4.10) n the case p α It remans to consder the case 1 p < α In ths case one does not need decomposton (5.3). Observe that for each face Γ f one has dv Γ f v H k (Γ f ) wth 1 k < α 1 2. Therefore, v X k wth 1 k < α 1 2, and applyng Proposton 4.1 we fnd vhp X hp satsfyng v v hp X C h mn {k,p}+1/2 v X k. Hence, selectng k [p, α 1 2 ) we arrve at the desred upper bound n (4.10), and the proof s fnshed. References [1] A. Bespalov, A note on the polynomal approxmaton of vertex sngulartes n the boundary element method n three dmensons, J. Integral Equatons Appl., 21 (2009), pp

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