POLYNOMIAL APPROXIMATIONS OF REGULAR AND SINGULAR VECTOR FIELDS WITH APPLICATIONS TO PROBLEMS OF ELECTROMAGNETICS

Transcription

1 POLYNOMIAL APPROXIMATIONS OF REGULAR AND SINGULAR VECTOR FIELDS WITH APPLICATIONS TO PROBLEMS OF ELECTROMAGNETICS Alex Besalov School of Mathematics The University of Manchester Collaborators: Norbert Heuer Ralf Hitmair PUC, Santiago, Chile ETH, Zurich, Switzerland Numerical Analysis and Scientific Comuting Seminar University of Manchester 29 October, 2010 A. Besalov Polynomial aroximations of regular and singular vector fields 1/27

2 Outline of the talk 1. Problem formulation. 2. H(div)-conforming -interolation in two dimensions: classical Raviart-Thomas interolation oerators; rojection based interolation oerators; error estimation. 3. Alication: h-bem for 3D roblem of electromagnetic scattering: convergence and error analysis; aroximation of singularities. 4. Conclusions. 5. References. A. Besalov Polynomial aroximations of regular and singular vector fields 2/27

3 Polynomial aroximations of vector fields: roblem formulation Notation Ω R 2 a olygonal domain; Ω = i Kh i ; h > 0 mesh arameter; 1 olynomial degree; u(x) = (u 1 (x), u 2 (x)), x = (x 1, x 2 ) Ω; H r (div, Ω) := {u H r (Ω); div u H r (Ω)}, r 0. A. Besalov Polynomial aroximations of regular and singular vector fields 3/27

4 Polynomial aroximations of vector fields: roblem formulation Notation Ω R 2 a olygonal domain; Ω = i Kh i ; h > 0 mesh arameter; 1 olynomial degree; u(x) = (u 1 (x), u 2 (x)), x = (x 1, x 2 ) Ω; H r (div, Ω) := {u H r (Ω); div u H r (Ω)}, r 0. The roblem Given u H r (div, Ω) with r > 0, find v(x) = (v 1 (x), v 2 (x)) such that v 1 (x), v 2 (x) are iecewise olynomials of degree, v H(div, Ω) = {v L 2 (Ω); div v L 2 (Ω)}, u(x) v(x), i.e., u v H(div, Ω) 0 as h 0 and/or. A. Besalov Polynomial aroximations of regular and singular vector fields 3/27

5 Polynomial aroximations of vector fields: roblem formulation Notation Ω R 2 a olygonal domain; Ω = i Kh i ; h > 0 mesh arameter; 1 olynomial degree; u(x) = (u 1 (x), u 2 (x)), x = (x 1, x 2 ) Ω; H r (div, Ω) := {u H r (Ω); div u H r (Ω)}, r 0. The roblem Given u H r (div, Ω) with r > 0, find v(x) = (v 1 (x), v 2 (x)) such that v 1 (x), v 2 (x) are iecewise olynomials of degree, v H(div, Ω) = {v L 2 (Ω); div v L 2 (Ω)}, u(x) v(x), i.e., u v H(div, Ω) 0 as h 0 and/or. Alications Mixed finite element methods for ellitic roblems FEM for Maxwell s equations in 2D (due to isomorhism of div and curl) H(div)-conforming BEM for Maxwell s equations in 3D A. Besalov Polynomial aroximations of regular and singular vector fields 3/27

6 Polynomial aroximations of vector fields: roblem formulation [Raviart, Thomas 77], [Brezzi, Fortin 91] Reference element K: equilateral triangle T or unit square Q. Polynomial sace on K: the Raviart-Thomas sace of order 1, { (P 1 (T )) 2 x P 1 (T ) if K = T, P RT (K) = P, 1 (Q) P 1, (Q) if K = Q. The roblem Given u H r (div, K) with r > 0, find u P RT (K) and δ (r) such that i) u is well-defined and stable (with resect to ) for any r > 0; ii) u allows to construct H(div)-conforming aroximations on a atch of elements (e.g., u interolates normal comonents of u along K); iii) u u H(div, K) δ (r) u H r (div, K) and δ (r) 0 as. A. Besalov Polynomial aroximations of regular and singular vector fields 4/27

7 Classical H(div)-conforming interolation oerator Π RT [Raviart, Thomas 77], [Brezzi, Fortin 91] u H r (K) H(div, K) the interolant Π RT u is defined by the conditions { (P 2 (T )) 2 if K = T, u Π RT u, v 0,K = 0 v P 2, 1 (Q) P 1, 2 (Q) if K = Q; (u Π RT u) n, w 0,l = 0 w P 1 (l) and l K. Commuting diagram roerty: H r (K) H(div, K) Π RT P RT (K) div div L 2 (K) Π 0 1 P 1 (K), where Π 0 : L 2 (K) P (K) denotes the standard L 2 -rojection onto P (K). A. Besalov Polynomial aroximations of regular and singular vector fields 5/27

8 Classical H(div)-conforming interolation oerator Π RT Error estimation for -interolation on the square Q [Suri 90], [Milner, Suri 92], [Stenberg, Suri 97], [Ainsworth, Pinchedez 02] u Π RT u H(div, Q) (r 1/2 ε) u H r (div, Q), r > 1/2. A. Besalov Polynomial aroximations of regular and singular vector fields 6/27

9 Classical H(div)-conforming interolation oerator Π RT Error estimation for -interolation on the square Q [Suri 90], [Milner, Suri 92], [Stenberg, Suri 97], [Ainsworth, Pinchedez 02] u Π RT u H(div, Q) (r 1/2 ε) u H r (div, Q), r > 1/2. In 3D: H(curl)-conforming Nédélec s elements on the cube [Monk 94], [Ben Belgacem, Bernardi 99] A. Besalov Polynomial aroximations of regular and singular vector fields 6/27

10 Classical H(div)-conforming interolation oerator Π RT Error estimation for -interolation on the square Q [Suri 90], [Milner, Suri 92], [Stenberg, Suri 97], [Ainsworth, Pinchedez 02] u Π RT u H(div, Q) (r 1/2 ε) u H r (div, Q), r > 1/2. In 3D: H(curl)-conforming Nédélec s elements on the cube [Monk 94], [Ben Belgacem, Bernardi 99] Conclusions: lack of stability (with resect to ) for low-regular fields; otimal -estimates can hardly be achieved; it is not clear how to deal with triangular elements. A. Besalov Polynomial aroximations of regular and singular vector fields 6/27

11 Projection-based H(div)-conforming interolation oerator Π div [Demkowicz, Babuška 03] u H r (K) H(div, K) with r > 0, the interolant Π div u is defined as where u 1 = u 2 l K ( l Π div u = u 1 + u 2 + u 3 P RT (K), ) u n φ l the lowest order interolant (φ l P 1 RT (K)), the sum of edge interolants, u 3 an interior interolant (vector bubble function) satisfying div(u (u 1 + u 2 + u 3 )), div v 0,K = 0 v P RT,0 (K), u (u 1 + u 2 + u 3 ), curl φ 0,K = 0 φ P 0 (K). A. Besalov Polynomial aroximations of regular and singular vector fields 7/27

12 Projection-based H(div)-conforming interolation oerator Π div [Demkowicz, Babuška 03] u H r (K) H(div, K), r > 0: Π div u = u 1 + u 2 + u 3 P RT (K). Proerties of the oerator Π div : Π div is well defined and stable (w.r.t. ) for any r > 0; it reserves olynomial vector fields from P RT (K); it works equally well on both triangles and arallelograms; it can be easily generalised to allow variation of olynomial degrees; it makes de Rham diagram commute H 1+r (K) Π 1 P (K) curl H r (K) H(div, K) Π div curl P RT (K) div div L 2 (K) Π 0 1 P 1 (K), where Π 1 : H 1+r (K) P (K) is the H 1 -conforming interolation oerator. A. Besalov Polynomial aroximations of regular and singular vector fields 8/27

13 Projection-based H(div)-conforming interolation oerator Π div [Demkowicz, Babuška 03] u H r (K) H(div, K), r > 0: Interolation error estimation Π div If u H r (div, K) with 0 < r < 1, then there holds u = u 1 + u 2 + u 3 P RT (K). u Π div u H(div, K) C(ε) (r ε) u H r (div, K), 0 < ε < r. A. Besalov Polynomial aroximations of regular and singular vector fields 9/27

14 Projection-based H(div)-conforming interolation oerator Π div [Demkowicz, Babuška 03] u H r (K) H(div, K), r > 0: Interolation error estimation Π div If u H r (div, K) with 0 < r < 1, then there holds u = u 1 + u 2 + u 3 P RT (K). u Π div u H(div, K) C(ε) (r ε) u H r (div, K), 0 < ε < r. Orthogonal (Helmholtz) decomosition of u H r (div, K): u = u 0 + curl ψ, u 0, curl φ 0,K = 0 φ H 1 (K). Hence, one has limited regularity of u 0 and ψ! A. Besalov Polynomial aroximations of regular and singular vector fields 9/27

15 Projection-based H(div)-conforming interolation oerator Π div [Demkowicz, Babuška 03] u H r (K) H(div, K), r > 0: Interolation error estimation Π div If u H r (div, K) with 0 < r < 1, then there holds u = u 1 + u 2 + u 3 P RT (K). u Π div u H(div, K) C(ε) (r ε) u H r (div, K), 0 < ε < r. Conclusions on the interolation error estimation: Π div satisfies a sub-otimal interolation error estimate; error estimate is available only for fields with limited regularity. A. Besalov Polynomial aroximations of regular and singular vector fields 9/27

16 Regular decomositions via Poincaré-tye integral oerators The Poincaré ma R a : C ( K) (C ( K)) 2 for some a = (a 1, a 2 ) K: R a ψ = (R 1, R 2 ), R i (x) := (x i a i ) 1 tψ(a + t(x a)) dt, i = 1, 2. Proerties: div(r a ψ) = ψ ψ C 1 ( K); R a mas P (K) into P RT +1(K); R a cannot be extended to a continuous maing L 2 (K) H 1 (K). 0 A. Besalov Polynomial aroximations of regular and singular vector fields 10/27

17 Regular decomositions via Poincaré-tye integral oerators The Poincaré ma R a : C ( K) (C ( K)) 2 for some a = (a 1, a 2 ) K: R a ψ = (R 1, R 2 ), R i (x) := (x i a i ) tψ(a + t(x a)) dt, i = 1, 2. [Costabel, McIntosh 10]: the regularised Poincaré oerator R : C ( K) (C ( K)) 2, Rψ := θ(a) R a ψ da, 1 0 B where θ C (R 2 ), su θ B K, B θ(a) da = 1, a = (a 1, a 2 ). Proerties: div(rψ) = ψ ψ H r (K), r 0; R mas P (K) into P RT +1(K); R defines a bounded oerator H r 1 (K) H r (K) for any r 0. A. Besalov Polynomial aroximations of regular and singular vector fields 11/27

18 Regular decomositions via Poincaré-tye integral oerators [Costabel, McIntosh 10]: regularised Poincaré integral oerators R : H r 1 (K) H r (K), r 0, A : H r (K) H r+1 (K), r 0, div(rψ) = ψ ψ H r (K); curl(au) = u u H r (div0, K). A. Besalov Polynomial aroximations of regular and singular vector fields 12/27

19 Regular decomositions via Poincaré-tye integral oerators [Costabel, McIntosh 10]: regularised Poincaré integral oerators R : H r 1 (K) H r (K), r 0, A : H r (K) H r+1 (K), r 0, div(rψ) = ψ ψ H r (K); curl(au) = u u H r (div0, K). Lemma 1. Let u H r (div, K), r > 0. Then there exist ψ H r+1 (K) and v H r+1 (K) such that u = curl ψ + v. Moreover, v H r+1 (K) div u H r (K) and ψ H r+1 (K) u H r (K). (1) A. Besalov Polynomial aroximations of regular and singular vector fields 12/27

20 Regular decomositions via Poincaré-tye integral oerators [Costabel, McIntosh 10]: regularised Poincaré integral oerators R : H r 1 (K) H r (K), r 0, A : H r (K) H r+1 (K), r 0, div(rψ) = ψ ψ H r (K); curl(au) = u u H r (div0, K). Lemma 1. Let u H r (div, K), r > 0. Then there exist ψ H r+1 (K) and v H r+1 (K) such that u = curl ψ + v. Moreover, v H r+1 (K) div u H r (K) and ψ H r+1 (K) u H r (K). (1) Proof. 1) div u H r (K) v := R(div u) H r+1 (K) and u = (u R(div u)) + R(div u) = (u R(div u)) + v. A. Besalov Polynomial aroximations of regular and singular vector fields 12/27

21 Regular decomositions via Poincaré-tye integral oerators [Costabel, McIntosh 10]: regularised Poincaré integral oerators R : H r 1 (K) H r (K), r 0, A : H r (K) H r+1 (K), r 0, div(rψ) = ψ ψ H r (K); curl(au) = u u H r (div0, K). Lemma 1. Let u H r (div, K), r > 0. Then there exist ψ H r+1 (K) and v H r+1 (K) such that u = curl ψ + v. Moreover, v H r+1 (K) div u H r (K) and ψ H r+1 (K) u H r (K). (1) Proof. 1) div u H r (K) v := R(div u) H r+1 (K) and u = (u R(div u)) + R(div u) = (u R(div u)) + v. 2) u R(div u) H r (K), div(u R(div u)) = div u div(r(div u)) = 0. A. Besalov Polynomial aroximations of regular and singular vector fields 12/27

22 Regular decomositions via Poincaré-tye integral oerators [Costabel, McIntosh 10]: regularised Poincaré integral oerators R : H r 1 (K) H r (K), r 0, A : H r (K) H r+1 (K), r 0, div(rψ) = ψ ψ H r (K); curl(au) = u u H r (div0, K). Lemma 1. Let u H r (div, K), r > 0. Then there exist ψ H r+1 (K) and v H r+1 (K) such that u = curl ψ + v. Moreover, v H r+1 (K) div u H r (K) and ψ H r+1 (K) u H r (K). (1) Proof. 1) div u H r (K) v := R(div u) H r+1 (K) and u = (u R(div u)) + R(div u) = (u R(div u)) + v. 2) u R(div u) H r (K), div(u R(div u)) = div u div(r(div u)) = 0. 3) ψ := A(u R(div u)) H r+1 (K) and curl ψ = u R(div u). A. Besalov Polynomial aroximations of regular and singular vector fields 12/27

23 Otimal error estimation for H(div)-conforming -interolation Theorem 1. Let u H r (div, K), r > 0. Then there holds u Π div u H(div, K) r u H r (div, K). A. Besalov Polynomial aroximations of regular and singular vector fields 13/27

24 Otimal error estimation for H(div)-conforming -interolation Theorem 1. Let u H r (div, K), r > 0. Then there holds u Π div u H(div, K) r u H r (div, K). Proof. 1) Lemma 1: u = curl ψ + v, ψ H r+1 (K), v H r+1 (K). A. Besalov Polynomial aroximations of regular and singular vector fields 13/27

25 Otimal error estimation for H(div)-conforming -interolation Theorem 1. Let u H r (div, K), r > 0. Then there holds u Π div u H(div, K) r u H r (div, K). Proof. 1) Lemma 1: u = curl ψ + v, ψ H r+1 (K), v H r+1 (K). 2) Π div u = Π div (curl ψ) + Π div v = curl(π 1 ψ) + Π div v. A. Besalov Polynomial aroximations of regular and singular vector fields 13/27

26 Otimal error estimation for H(div)-conforming -interolation Theorem 1. Let u H r (div, K), r > 0. Then there holds u Π div u H(div, K) r u H r (div, K). Proof. 1) Lemma 1: u = curl ψ + v, ψ H r+1 (K), v H r+1 (K). 2) Π div 3) u Π div u = Π div (curl ψ) + Π div u = curl(ψ Π 1 ψ) + (v Π div v = curl(π 1 ψ) + Π div v. v). A. Besalov Polynomial aroximations of regular and singular vector fields 13/27

27 Otimal error estimation for H(div)-conforming -interolation Theorem 1. Let u H r (div, K), r > 0. Then there holds u Π div u H(div, K) r u H r (div, K). Proof. 1) Lemma 1: u = curl ψ + v, ψ H r+1 (K), v H r+1 (K). 2) Π div 3) u Π div u = Π div (curl ψ) + Π div u = curl(ψ Π 1 ψ) + (v Π div v = curl(π 1 ψ) + Π div v. 4) curl(ψ Π 1 ψ) H(div, K) = curl(ψ Π 1 ψ) L 2 (K) = ψ Π1 ψ H 1 (K) v). r ψ H 1+r (K) r u H r (K). A. Besalov Polynomial aroximations of regular and singular vector fields 13/27

28 Otimal error estimation for H(div)-conforming -interolation Theorem 1. Let u H r (div, K), r > 0. Then there holds u Π div u H(div, K) r u H r (div, K). Proof. 1) Lemma 1: u = curl ψ + v, ψ H r+1 (K), v H r+1 (K). 2) Π div 3) u Π div u = Π div (curl ψ) + Π div u = curl(ψ Π 1 ψ) + (v Π div v = curl(π 1 ψ) + Π div v. 4) curl(ψ Π 1 ψ) H(div, K) = curl(ψ Π 1 ψ) L 2 (K) = ψ Π1 ψ H 1 (K) 5) v Π div v H(div, K) v). r ψ H 1+r (K) r u H r (K). inf v ( v v H(div, K) + Π div (v v ) H(div, K) ) ε (0,1) ( ) inf v v v H ε (K) + div(v v ) L 2 (K) inf v v v H 1 (K). A. Besalov Polynomial aroximations of regular and singular vector fields 13/27

29 Otimal error estimation for H(div)-conforming -interolation Theorem 1. Let u H r (div, K), r > 0. Then there holds u Π div u H(div, K) r u H r (div, K). Proof. 1) Lemma 1: u = curl ψ + v, ψ H r+1 (K), v H r+1 (K). 2) Π div 3) u Π div u = Π div (curl ψ) + Π div u = curl(ψ Π 1 ψ) + (v Π div v = curl(π 1 ψ) + Π div v. 4) curl(ψ Π 1 ψ) H(div, K) = curl(ψ Π 1 ψ) L 2 (K) = ψ Π1 ψ H 1 (K) v). 5) v Π div v H(div, K) inf v r ψ H 1+r (K) r u H r (K). v v H 1 (K) r v H 1+r (K) r div u H r (K). 6) Combine 4) and 5), then use 3). A. Besalov Polynomial aroximations of regular and singular vector fields 13/27

30 Otimal error estimation for H(div)-conforming -interolation Theorem 1. Let u H r (div, K), r > 0. Then there holds u Π div u H(div, K) r u H r (div, K). Immediate (and imortant) extensions and alications Brezzi-Douglas-Marini sace on the reference triangle Otimal h-estimates (by the Bramble-Hilbert argument and scaling) u Π div h u H(div, Ω) h min{r,} r u H r (div, Ω) H(curl)-conforming -interolation oerator in 2D (due to isomorhism of div and curl). Alication: - and h-fem for Maxwell s equations in 2D. A. Besalov Polynomial aroximations of regular and singular vector fields 14/27

31 Alication: h-bem for 3D roblem of electromagnetic scattering Boundary value roblem for PDE (3D roblem of e/m scattering) Assumtions: Ω is a erfectly conducting body in R 3, Γ = Ω is a Lischitz olyhedral surface. Ω e := R 3 \ Ω. A. Besalov Polynomial aroximations of regular and singular vector fields 15/27

32 Alication: h-bem for 3D roblem of electromagnetic scattering Boundary value roblem for PDE (3D roblem of e/m scattering) Assumtions: Ω is a erfectly conducting body in R 3, Γ = Ω is a Lischitz olyhedral surface. Ω e := R 3 \ Ω. Time-harmonic Maxwell s equations in Ω e : { { curl E iω µh = 0 curl curl E k 2 curl H + iω εe = 0 or E = 0, k := ω µε, H = (iωµ) 1 curl E. Boundary condition on Γ : γ τ (E) = γ τ (E in ), where E in is some incident lane wave with wave number k, γ τ : u u ν Γ. Silver-Müller radiation condition at. A. Besalov Polynomial aroximations of regular and singular vector fields 15/27

33 Alication: h-bem for 3D roblem of electromagnetic scattering Boundary value roblem for PDE (3D roblem of e/m scattering) Assumtions: Ω is a erfectly conducting body in R 3, Γ = Ω is a Lischitz olyhedral surface. Ω e := R 3 \ Ω. Time-harmonic Maxwell s equations in Ω e : { { curl E iω µh = 0 curl curl E k 2 curl H + iω εe = 0 or E = 0, k := ω µε, H = (iωµ) 1 curl E. Boundary condition on Γ : γ τ (E) = γ τ (E in ), where E in is some incident lane wave with wave number k, γ τ : u u ν Γ. Silver-Müller radiation condition at. E, H H loc (curl, Ω e ) := {u L 2 loc (Ω e); curl u L 2 loc (Ω e)}. A. Besalov Polynomial aroximations of regular and singular vector fields 15/27

34 Alication: h-bem for 3D roblem of electromagnetic scattering Boundary integral equation: weak formulation, unique solvability, regularity u jum of the magnetic field H at the interface Γ; u = γ τ (H tot ) u X := γ τ (H loc (curl, Ω e )); X = H 1/2 (div Γ, Γ) := {u H 1/2 (Γ); div Γ u H 1/2 (Γ)}. A. Besalov Polynomial aroximations of regular and singular vector fields 16/27

35 Alication: h-bem for 3D roblem of electromagnetic scattering Boundary integral equation: weak formulation, unique solvability, regularity u jum of the magnetic field H at the interface Γ; u = γ τ (H tot ) u X := γ τ (H loc (curl, Ω e )); X = H 1/2 (div Γ, Γ) := {u H 1/2 (Γ); div Γ u H 1/2 (Γ)}. The BIE: the electric field integral equation (EFIE) for u. A. Besalov Polynomial aroximations of regular and singular vector fields 16/27

36 Alication: h-bem for 3D roblem of electromagnetic scattering Boundary integral equation: weak formulation, unique solvability, regularity u jum of the magnetic field H at the interface Γ; u = γ τ (H tot ) u X := γ τ (H loc (curl, Ω e )); X = H 1/2 (div Γ, Γ) := {u H 1/2 (Γ); div Γ u H 1/2 (Γ)}. The BIE: the electric field integral equation (EFIE) for u. Variational form of the EFIE (Rumsey s rincile): find u X such that a(u, v) := γ tr (V k div Γ u), div Γ v k 2 π τ (V k u), v = f, v v X, where π τ : u ν (u ν) Γ, f := π τ (E in ), and V k, V k are single layer oerators for given k. A. Besalov Polynomial aroximations of regular and singular vector fields 16/27

37 Alication: h-bem for 3D roblem of electromagnetic scattering Boundary integral equation: weak formulation, unique solvability, regularity u jum of the magnetic field H at the interface Γ; u = γ τ (H tot ) u X := γ τ (H loc (curl, Ω e )); X = H 1/2 (div Γ, Γ) := {u H 1/2 (Γ); div Γ u H 1/2 (Γ)}. The BIE: the electric field integral equation (EFIE) for u. Variational form of the EFIE (Rumsey s rincile): find u X such that a(u, v) := γ tr (V k div Γ u), div Γ v k 2 π τ (V k u), v = f, v v X, where π τ : u ν (u ν) Γ, f := π τ (E in ), and V k, V k are single layer oerators for given k. Remark. The form a(u, v) is not X-coercive due to infinite-dimensional kernel of div Γ. A. Besalov Polynomial aroximations of regular and singular vector fields 16/27

38 Alication: h-bem for 3D roblem of electromagnetic scattering Boundary integral equation: weak formulation, unique solvability, regularity u jum of the magnetic field H at the interface Γ; u = γ τ (H tot ) u X := γ τ (H loc (curl, Ω e )); X = H 1/2 (div Γ, Γ) := {u H 1/2 (Γ); div Γ u H 1/2 (Γ)}. The BIE: the electric field integral equation (EFIE) for u. Variational form of the EFIE (Rumsey s rincile): find u X such that a(u, v) := γ tr (V k div Γ u), div Γ v k 2 π τ (V k u), v = f, v v X. [Buffa, Costabel, Schwab 02], [Hitmair, Schwab 02]: if k 2 is not an electrical eigenvalue of the interior roblem, then there exists a unique u X. A. Besalov Polynomial aroximations of regular and singular vector fields 16/27

39 Alication: h-bem for 3D roblem of electromagnetic scattering Boundary integral equation: weak formulation, unique solvability, regularity u jum of the magnetic field H at the interface Γ; u = γ τ (H tot ) u X := γ τ (H loc (curl, Ω e )); X = H 1/2 (div Γ, Γ) := {u H 1/2 (Γ); div Γ u H 1/2 (Γ)}. The BIE: the electric field integral equation (EFIE) for u. Variational form of the EFIE (Rumsey s rincile): find u X such that a(u, v) := γ tr (V k div Γ u), div Γ v k 2 π τ (V k u), v = f, v v X. [Buffa, Costabel, Schwab 02], [Hitmair, Schwab 02]: if k 2 is not an electrical eigenvalue of the interior roblem, then there exists a unique u X. [Costabel, Dauge 00] s (0, 1 2 ) such that u Hs (div Γ, Γ). A. Besalov Polynomial aroximations of regular and singular vector fields 16/27

40 Alication: h-bem for 3D roblem of electromagnetic scattering Galerkin BEM: discrete formulation, unique solvability, quasi-otimality Surface discretisation: a family of quasi-uniform meshes h = {Γ j } on Γ. Discrete subsace: X N H(div Γ, Γ) X is based on Raviart-Thomas saces. P RT (K) = (P 1 (K)) 2 ξp 1 (K), K is the ref. element. N = N(h, ), h > 0 is the mesh arameter, 1 is a olynomial degree. A. Besalov Polynomial aroximations of regular and singular vector fields 17/27

41 Alication: h-bem for 3D roblem of electromagnetic scattering Galerkin BEM: discrete formulation, unique solvability, quasi-otimality Rumsey s rincile: find u X such that a(u, v) = f, v v X. Galerkin BEM: find u N X N such that a(u N, v) = f, v v X N. A. Besalov Polynomial aroximations of regular and singular vector fields 18/27

42 Alication: h-bem for 3D roblem of electromagnetic scattering Galerkin BEM: discrete formulation, unique solvability, quasi-otimality Rumsey s rincile: find u X such that a(u, v) = f, v v X. Galerkin BEM: find u N X N such that a(u N, v) = f, v v X N. Theorem 2 [B., Heuer, Hitmair 10]. For N = N(h, ) large enough, the discrete roblem is uniquely solvable, and the h-bem converges quasi-otimally, i.e., u u N X inf{ u v X ; v X N }. A. Besalov Polynomial aroximations of regular and singular vector fields 18/27

43 Alication: h-bem for 3D roblem of electromagnetic scattering Galerkin BEM: discrete formulation, unique solvability, quasi-otimality Rumsey s rincile: find u X such that a(u, v) = f, v v X. Galerkin BEM: find u N X N such that a(u N, v) = f, v v X N. Theorem 2 [B., Heuer, Hitmair 10]. For N = N(h, ) large enough, the discrete roblem is uniquely solvable, and the h-bem converges quasi-otimally, i.e., u u N X inf{ u v X ; v X N }. Main ingredients in the roof: decomositions X = V W, V H 1/2 (Γ), W = H 1/2 (div Γ 0, Γ) and X N = V N W N, W N W; rojection based interolation oerators [Demkowicz, Babuška 03]; the regularised Poincaré integral oerators [Costabel, McIntosh 10]. A. Besalov Polynomial aroximations of regular and singular vector fields 18/27

44 Alication: h-bem for 3D roblem of electromagnetic scattering A riori error estimation (based on the Sobolev regularity) Theorem 3 [B., Heuer 10]. There exists s > 0 such that u H s (div Γ, Γ) and u u N X h 1/2+min{s,} (s+1/2) u H s (div Γ, Γ). A. Besalov Polynomial aroximations of regular and singular vector fields 19/27

45 Alication: h-bem for 3D roblem of electromagnetic scattering A riori error estimation (based on the Sobolev regularity) Theorem 3 [B., Heuer 10]. There exists s > 0 such that u H s (div Γ, Γ) and u u N X h 1/2+min{s,} (s+1/2) u H s (div Γ, Γ). Proof: use H(div Γ, Γ)-conforming rojection-based interolation oerator : Hs (div Γ, Γ) X N and recall that X = H 1/2 (div Γ, Γ). Π div h A. Besalov Polynomial aroximations of regular and singular vector fields 19/27

46 Alication: h-bem for 3D roblem of electromagnetic scattering A riori error estimation (based on the Sobolev regularity) Theorem 3 [B., Heuer 10]. There exists s > 0 such that u H s (div Γ, Γ) and u u N X h 1/2+min{s,} (s+1/2) u H s (div Γ, Γ). Proof: use H(div Γ, Γ)-conforming rojection-based interolation oerator : Hs (div Γ, Γ) X N and recall that X = H 1/2 (div Γ, Γ). Π div h Ste 1: u u N X u Π div h u X h1/2 1/2 u Π div h u H(div Γ, Γ). A. Besalov Polynomial aroximations of regular and singular vector fields 19/27

47 Alication: h-bem for 3D roblem of electromagnetic scattering A riori error estimation (based on the Sobolev regularity) Theorem 3 [B., Heuer 10]. There exists s > 0 such that u H s (div Γ, Γ) and u u N X h 1/2+min{s,} (s+1/2) u H s (div Γ, Γ). Proof: use H(div Γ, Γ)-conforming rojection-based interolation oerator : Hs (div Γ, Γ) X N and recall that X = H 1/2 (div Γ, Γ). Π div h Ste 1: u u N X u Π div h u X h1/2 1/2 u Π div h u H(div Γ, Γ). Careful here: X = H 1/2 (div Γ, Γ) is not the dual sace of H 1/2 (div Γ, Γ) with resect to the H(div Γ, Γ)-inner roduct (unless Γ is smooth)! Use duality face by face. Localisation of H 1/2 (div Γ, Γ) to faces is not trivial. A. Besalov Polynomial aroximations of regular and singular vector fields 19/27

48 Alication: h-bem for 3D roblem of electromagnetic scattering A riori error estimation (based on the Sobolev regularity) Theorem 3 [B., Heuer 10]. There exists s > 0 such that u H s (div Γ, Γ) and u u N X h 1/2+min{s,} (s+1/2) u H s (div Γ, Γ). Proof: use H(div Γ, Γ)-conforming rojection-based interolation oerator : Hs (div Γ, Γ) X N and recall that X = H 1/2 (div Γ, Γ). Π div h Ste 1: Ste 2: u u N X u Π div h u X h1/2 1/2 u Π div h u H(div Γ, Γ). u Π div h u H(div Γ, Γ) hmin{s,} s u H s (div Γ, Γ), s > 0. A. Besalov Polynomial aroximations of regular and singular vector fields 19/27

49 Alication: h-bem for 3D roblem of electromagnetic scattering A riori error estimation (based on the Sobolev regularity) Theorem 3 [B., Heuer 10]. There exists s > 0 such that u H s (div Γ, Γ) and Convergence rates u u N X h 1/2+min{s,} (s+1/2) u H s (div Γ, Γ). For the exterior roblem u H s (div Γ, Γ) with s (0, 1 2 ). h-version: 1 is fixed, N h 2, u u N X = O(N 1/2(s+1/2) ). -version: h is fixed, N 2, u u N X = O(N 1/2(s+1/2) ). Available numerical results [Leydecker; PhD Thesis 06]: the -BEM converges faster than the h-bem. A. Besalov Polynomial aroximations of regular and singular vector fields 20/27

50 Alication: h-bem for 3D roblem of electromagnetic scattering Available numerical results Relative Error in the Energy Norm =1 =2 =4 =6 =8 16 elements 9 elements 1 element Number of Unknowns A. Besalov Polynomial aroximations of regular and singular vector fields 21/27

51 Alication: h-bem for 3D roblem of electromagnetic scattering Precise a riori error estimates for the h-bem Reresentation of the solution to the EFIE: u = u reg + u sing, u reg H m (div Γ, Γ) with m > 0, u sing = u e + u v + u ev. A. Besalov Polynomial aroximations of regular and singular vector fields 22/27

52 Alication: h-bem for 3D roblem of electromagnetic scattering Precise a riori error estimates for the h-bem Reresentation of the solution to the EFIE: u = u reg + u sing, u reg H m (div Γ, Γ) with m > 0, u sing = u e + u v + u ev. Structure of singularities [Costabel, Dauge 00], [B., Heuer 10]: u sing = curl Γ w + v, w H 1/2 (Γ), v = (v 1, v 2 ) H 1/2 (Γ). A. Besalov Polynomial aroximations of regular and singular vector fields 22/27

53 Alication: h-bem for 3D roblem of electromagnetic scattering Precise a riori error estimates for the h-bem Reresentation of the solution to the EFIE: u = u reg + u sing, u reg H m (div Γ, Γ) with m > 0, u sing = u e + u v + u ev. Structure of singularities [Costabel, Dauge 00], [B., Heuer 10]: u sing = curl Γ w + v, w H 1/2 (Γ), v = (v 1, v 2 ) H 1/2 (Γ). Tyical singularities: edge singularities ρ γ log ρ β 1, vertex singularities r λ log r β 2, edge-vertex singularities r λ γ ρ γ log r β 3 ; r distance to a vertex v of Γ, ρ distance to one of the edges e Γ such that ē v; γ > 1 2 (γ 1 2 if Γ is an oen surface), λ > 1 2, integers β 1, β 2, β 3 0. A. Besalov Polynomial aroximations of regular and singular vector fields 22/27

54 Alication: h-bem for 3D roblem of electromagnetic scattering Precise a riori error estimates for the h-bem Reresentation of the solution to the EFIE: u = u reg + u sing, u reg H m (div Γ, Γ) with m > 0, u sing = u e + u v + u ev. Structure of singularities [Costabel, Dauge 00], [B., Heuer 10]: u sing = curl Γ w + v, w H 1/2 (Γ), v = (v 1, v 2 ) H 1/2 (Γ). Structure of aroximations: u h sing = curl Γ w h + v h! X N. A. Besalov Polynomial aroximations of regular and singular vector fields 23/27

55 Alication: h-bem for 3D roblem of electromagnetic scattering Precise a riori error estimates for the h-bem Reresentation of the solution to the EFIE: u = u reg + u sing, u reg H m (div Γ, Γ) with m > 0, u sing = u e + u v + u ev. Structure of singularities [Costabel, Dauge 00], [B., Heuer 10]: u sing = curl Γ w + v, w H 1/2 (Γ), v = (v 1, v 2 ) H 1/2 (Γ). Structure of aroximations: u h sing = curl Γ w h + v h! X N. Theorem 4. Let σ = min min {λ + 1/2, γ} denote the strongest singularity. e,v: v ē u sing u h sing X { h σ 2σ (1 + log(/h)) β+ν if σ 1 2, h +1/2 if 1 < σ 1 2, { β3 + where β 1 := 2 if λ = γ 1 2, β 3 otherwise, ν := { 1 2 if = σ 1 2, 0 otherwise. A. Besalov Polynomial aroximations of regular and singular vector fields 23/27

56 Alication: h-bem for 3D roblem of electromagnetic scattering Precise a riori error estimates for the h-bem Rumsey s rincile: find u X such that a(u, v) = f, v v X. Galerkin BEM: find u N X N such that a(u N, v) = f, v v X N. Theorem 5. Let σ = Then min min {λ + 1/2, γ} denote the strongest singularity. e,v: v ē u u N X { h σ 2σ (1 + log(/h)) β+ν if σ 1 2, h +1/2 if 1 < σ 1 2, where { β3 + β 1 := 2 if λ = γ 1 2, β 3 otherwise, ν := { 1 2 if = σ 1 2, 0 otherwise. A. Besalov Polynomial aroximations of regular and singular vector fields 24/27

57 Conclusions Otimal error estimates for H(div)- and H(curl)-conforming -interolation in two dimensions Convergence of the h-bem for the electric field integral equation on olyhedral surfaces discretised by shae-regular meshes A riori error estimation for the h-bem on quasi-uniform meshes; recise error estimates in terms of h,, and the singularity exonents A. Besalov Polynomial aroximations of regular and singular vector fields 25/27

58 Conclusions Otimal error estimates for H(div)- and H(curl)-conforming -interolation in two dimensions Convergence of the h-bem for the electric field integral equation on olyhedral surfaces discretised by shae-regular meshes A riori error estimation for the h-bem on quasi-uniform meshes; recise error estimates in terms of h,, and the singularity exonents Some areas not (yet) covered Non-regular (e.g., graded) meshes: convergence of the BEM Exonentially convergent h-bem for electromagnetic roblems Comutational asects and imlementation Numerical disersion errors A osteriori error analysis for the BEM; h-adative schemes... A. Besalov Polynomial aroximations of regular and singular vector fields 25/27

59 References H(div)- and H(curl)-conforming -interolation M. Suri, Math. Com., 54 (1990), L. Demkowicz and I. Babuška, SIAM J. Numer. Anal., 41 (2003), A. B. and N. Heuer, SIAM J. Numer. Anal., 47 (2009), A. B. and N. Heuer, ESAIM: M2AN, doi: /m2an/ The regularised Poincaré-tye integral oerators M. Costabel and A. McIntosh, Math. Z., 265 (2010), Singularities and their aroximation T. von Petersdorff and E. P. Stehan, Math. Methods Al. Sci., 12 (1990). M. Costabel and M. Dauge, Arch. Rational Mech. Anal., 151 (2000), A. B. and N. Heuer, Numer. Mat., 100 (2005), A. B. and N. Heuer, Numer. Mat., 106 (2007), A. B. and N. Heuer, ESAIM: M2AN, 42 (2008), A. B. and N. Heuer, IMA J. Numer. Anal., 30 (2010), A. Besalov Polynomial aroximations of regular and singular vector fields 26/27

60 References Natural BEM for the EFIE A. Buffa, M. Costabel, and C. Schwab, Numer. Mat., 92 (2002), R. Hitmair and C. Schwab, SIAM J. Numer. Anal., 40 (2002), A. Buffa and S. H. Christiansen, Numer. Mat., 94 (2003), F. Leydecker, PhD thesis, Universität Hannover, Germany, A. B. and N. Heuer, Al. Numer. Math., 60 (2010), A. B. and N. Heuer, IMA J. Numer. Anal., 30 (2010), A. B., N. Heuer and R. Hitmair, SIAM J. Numer. Anal., 48 (2010), A. Besalov Polynomial aroximations of regular and singular vector fields 27/27

61 References Natural BEM for the EFIE A. Buffa, M. Costabel, and C. Schwab, Numer. Mat., 92 (2002), R. Hitmair and C. Schwab, SIAM J. Numer. Anal., 40 (2002), A. Buffa and S. H. Christiansen, Numer. Mat., 94 (2003), F. Leydecker, PhD thesis, Universität Hannover, Germany, A. B. and N. Heuer, Al. Numer. Math., 60 (2010), A. B. and N. Heuer, IMA J. Numer. Anal., 30 (2010), A. B., N. Heuer and R. Hitmair, SIAM J. Numer. Anal., 48 (2010), Thank you for your attention! A. Besalov Polynomial aroximations of regular and singular vector fields 27/27