Approximating the divergence of electromagnetic fields by edge elements Patrick Ciarlet online access to recent Refs: http:/www.ensta.fr/˜ciarlet POEMS, ENSTA ParisTech, France RICAM, October 2016 – p. 1/24
Outline Maxwell equations and a priori regularity of the fields Discretization and error estimates on the divergence of the fields Variational formulations Numerical illustrations Conclusion and perspectives RICAM, October 2016 – p. 2/24
Outline Maxwell equations and a priori regularity of the fields Discretization and error estimates on the divergence of the fields Variational formulations Numerical illustrations Conclusion and perspectives RICAM, October 2016 – p. 2/24
Stationary/static problem Let Ω be a Lipschitz, polyhedral domain with connected boundary ∂ Ω . Given source terms f ∈ L 2 (Ω) ( div f = 0 ) and ̺ ∈ H − 1 (Ω) , solve: Find E ∈ L 2 (Ω) with curl E ∈ L 2 (Ω) s.t. µ − 1 curl E � � = f in Ω ; curl div ε E = ̺ in Ω ; E × n = 0 on ∂ Ω . NB. With coefficients ε, µ > 0 a.e. ; ε, ε − 1 , µ, µ − 1 ∈ L ∞ (Ω) . RICAM, October 2016 – p. 3/24
Stationary/static problem Let Ω be a Lipschitz, polyhedral domain with connected boundary ∂ Ω . Given source terms f ∈ L 2 (Ω) ( div f = 0 ) and ̺ ∈ H − 1 (Ω) , solve: Find E ∈ L 2 (Ω) with curl E ∈ L 2 (Ω) s.t. µ − 1 curl E � � = f in Ω ; curl div ε E = ̺ in Ω ; E × n = 0 on ∂ Ω . NB. With coefficients ε, µ > 0 a.e. ; ε, ε − 1 , µ, µ − 1 ∈ L ∞ (Ω) . The problem is well-posed in H 0 ( curl ; Ω) : � E � H ( curl ;Ω) � � f � L 2 (Ω) + � ̺ � H − 1 (Ω) . RICAM, October 2016 – p. 3/24
Helmholtz decomposition Let V N (Ω , ε ) := { v ∈ H 0 ( curl , Ω) | div ε v = 0 } . RICAM, October 2016 – p. 4/24
Helmholtz decomposition Let V N (Ω , ε ) := { v ∈ H 0 ( curl , Ω) | div ε v = 0 } . According to the Helmholtz decomposition of H 0 ( curl , Ω) e.g. [Monk’03]: E = E 0 + ∇ φ, E 0 ∈ V N (Ω , ε ) , φ ∈ H 1 0 (Ω) . NB. The decomposition is orthogonal wrt ( ε · |· ) + ( µ − 1 curl · | curl · ) . RICAM, October 2016 – p. 4/24
Helmholtz decomposition Let V N (Ω , ε ) := { v ∈ H 0 ( curl , Ω) | div ε v = 0 } . According to the Helmholtz decomposition of H 0 ( curl , Ω) e.g. [Monk’03]: E = E 0 + ∇ φ, E 0 ∈ V N (Ω , ε ) , φ ∈ H 1 0 (Ω) . NB. The decomposition is orthogonal wrt ( ε · |· ) + ( µ − 1 curl · | curl · ) . One may characterize E 0 and ∇ φ separately: Find E 0 ∈ V N (Ω , ε ) s.t. µ − 1 curl E 0 � � = f in Ω ; curl Find φ ∈ H 1 0 (Ω) s.t. div ε ∇ φ = ̺ in Ω . RICAM, October 2016 – p. 4/24
Helmholtz decomposition Let V N (Ω , ε ) := { v ∈ H 0 ( curl , Ω) | div ε v = 0 } . According to the Helmholtz decomposition of H 0 ( curl , Ω) e.g. [Monk’03]: E = E 0 + ∇ φ, E 0 ∈ V N (Ω , ε ) , φ ∈ H 1 0 (Ω) . NB. The decomposition is orthogonal wrt ( ε · |· ) + ( µ − 1 curl · | curl · ) . One may characterize E 0 and ∇ φ separately: Find E 0 ∈ V N (Ω , ε ) s.t. µ − 1 curl E 0 � � = f in Ω ; curl Find φ ∈ H 1 0 (Ω) s.t. div ε ∇ φ = ̺ in Ω . In what follows, we focus on E 0 ; ∇ φ can be handled similarly [Jr-Wu-Zou’14, §§3-4]. RICAM, October 2016 – p. 4/24
Regularity of the fields E 0 ∈ V N (Ω , ε ) ⊂ X N (Ω , ε ) := { v ∈ H 0 ( curl , Ω) | div ε v ∈ L 2 (Ω) } . RICAM, October 2016 – p. 5/24
Regularity of the fields E 0 ∈ V N (Ω , ε ) ⊂ X N (Ω , ε ) := { v ∈ H 0 ( curl , Ω) | div ε v ∈ L 2 (Ω) } . µ − 1 curl E 0 ∈ X T (Ω , µ ) := { v ∈ H ( curl , Ω) | div µ v ∈ L 2 (Ω) , µ v · n | ∂ Ω = 0 } . RICAM, October 2016 – p. 5/24
Regularity of the fields E 0 ∈ V N (Ω , ε ) ⊂ X N (Ω , ε ) := { v ∈ H 0 ( curl , Ω) | div ε v ∈ L 2 (Ω) } . µ − 1 curl E 0 ∈ X T (Ω , µ ) := { v ∈ H ( curl , Ω) | div µ v ∈ L 2 (Ω) , µ v · n | ∂ Ω = 0 } . Theorem [Costabel-Dauge-Nicaise’99]: Consider ε, µ − 1 ∈ W 1 , ∞ (Ω) . If Ω is convex then X N (Ω , ε ) ⊂ H 1 (Ω) and X T (Ω , µ ) ⊂ H 1 (Ω) . RICAM, October 2016 – p. 5/24
Regularity of the fields E 0 ∈ V N (Ω , ε ) ⊂ X N (Ω , ε ) := { v ∈ H 0 ( curl , Ω) | div ε v ∈ L 2 (Ω) } . µ − 1 curl E 0 ∈ X T (Ω , µ ) := { v ∈ H ( curl , Ω) | div µ v ∈ L 2 (Ω) , µ v · n | ∂ Ω = 0 } . Theorem [Costabel-Dauge-Nicaise’99]: Consider ε, µ − 1 ∈ W 1 , ∞ (Ω) . If Ω is convex then X N (Ω , ε ) ⊂ H 1 (Ω) and X T (Ω , µ ) ⊂ H 1 (Ω) . If Ω is non-convex then ∃ δ Dir max , δ Neu max ∈ ]1 / 2 , 1[ s.t. max H δ (Ω) , max H δ (Ω) . X N (Ω , ε ) ⊂ ∩ 0 ≤ δ<δ Dir and X T (Ω , µ ) ⊂ ∩ 0 ≤ δ<δ Neu RICAM, October 2016 – p. 5/24
Regularity of the fields E 0 ∈ V N (Ω , ε ) ⊂ X N (Ω , ε ) := { v ∈ H 0 ( curl , Ω) | div ε v ∈ L 2 (Ω) } . µ − 1 curl E 0 ∈ X T (Ω , µ ) := { v ∈ H ( curl , Ω) | div µ v ∈ L 2 (Ω) , µ v · n | ∂ Ω = 0 } . Theorem [Costabel-Dauge-Nicaise’99]: Consider ε, µ − 1 ∈ W 1 , ∞ (Ω) . If Ω is convex then X N (Ω , ε ) ⊂ H 1 (Ω) and X T (Ω , µ ) ⊂ H 1 (Ω) . If Ω is non-convex then ∃ δ Dir max , δ Neu max ∈ ]1 / 2 , 1[ s.t. max H δ (Ω) , max H δ (Ω) . X N (Ω , ε ) ⊂ ∩ 0 ≤ δ<δ Dir and X T (Ω , µ ) ⊂ ∩ 0 ≤ δ<δ Neu Following [Jr-Wu-Zou’14], let ε, µ − 1 ∈ W 1 , ∞ (Ω) . RICAM, October 2016 – p. 5/24
Regularity of the fields E 0 ∈ V N (Ω , ε ) ⊂ X N (Ω , ε ) := { v ∈ H 0 ( curl , Ω) | div ε v ∈ L 2 (Ω) } . µ − 1 curl E 0 ∈ X T (Ω , µ ) := { v ∈ H ( curl , Ω) | div µ v ∈ L 2 (Ω) , µ v · n | ∂ Ω = 0 } . Theorem [Costabel-Dauge-Nicaise’99]: Consider ε, µ − 1 ∈ W 1 , ∞ (Ω) . If Ω is convex then X N (Ω , ε ) ⊂ H 1 (Ω) and X T (Ω , µ ) ⊂ H 1 (Ω) . If Ω is non-convex then ∃ δ Dir max , δ Neu max ∈ ]1 / 2 , 1[ s.t. max H δ (Ω) , max H δ (Ω) . X N (Ω , ε ) ⊂ ∩ 0 ≤ δ<δ Dir and X T (Ω , µ ) ⊂ ∩ 0 ≤ δ<δ Neu Following [Jr-Wu-Zou’14], let ε, µ − 1 ∈ W 1 , ∞ (Ω) . To fix ideas, suppose that Ω is non-convex and define δ max := min( δ Dir max , δ Neu max ) . Choose a regularity exponent δ ∈ ]1 / 2 , δ max [ . RICAM, October 2016 – p. 5/24
Regularity of the fields E 0 ∈ V N (Ω , ε ) ⊂ X N (Ω , ε ) := { v ∈ H 0 ( curl , Ω) | div ε v ∈ L 2 (Ω) } . µ − 1 curl E 0 ∈ X T (Ω , µ ) := { v ∈ H ( curl , Ω) | div µ v ∈ L 2 (Ω) , µ v · n | ∂ Ω = 0 } . Theorem [Costabel-Dauge-Nicaise’99]: Consider ε, µ − 1 ∈ W 1 , ∞ (Ω) . If Ω is convex then X N (Ω , ε ) ⊂ H 1 (Ω) and X T (Ω , µ ) ⊂ H 1 (Ω) . If Ω is non-convex then ∃ δ Dir max , δ Neu max ∈ ]1 / 2 , 1[ s.t. max H δ (Ω) , max H δ (Ω) . X N (Ω , ε ) ⊂ ∩ 0 ≤ δ<δ Dir and X T (Ω , µ ) ⊂ ∩ 0 ≤ δ<δ Neu Following [Jr-Wu-Zou’14], let ε, µ − 1 ∈ W 1 , ∞ (Ω) . To fix ideas, suppose that Ω is non-convex and define δ max := min( δ Dir max , δ Neu max ) . Choose a regularity exponent δ ∈ ]1 / 2 , δ max [ . NB. If Ω is convex, then δ = 1 . RICAM, October 2016 – p. 5/24
Outline Maxwell equations and a priori regularity of the fields Discretization and error estimates on the divergence of the fields Variational formulations Numerical illustrations Conclusion and perspectives RICAM, October 2016 – p. 6/24
Edge element discretization Let ( T h ) h be a shape regular family of tetrahedral meshes of Ω . Define X h := { v h ∈ H 0 ( curl ; Ω) | v h | K = a K + b K × x , ∀ K ∈ T h } . RICAM, October 2016 – p. 7/24
Edge element discretization Let ( T h ) h be a shape regular family of tetrahedral meshes of Ω . Define X h := { v h ∈ H 0 ( curl ; Ω) | v h | K = a K + b K × x , ∀ K ∈ T h } . Assume ( ⋆ ) ∀ h, � E 0 − E 0 ,h � H ( curl ;Ω) � inf v h ∈X h � E 0 − v h � H ( curl ;Ω) . RICAM, October 2016 – p. 7/24
Edge element discretization Let ( T h ) h be a shape regular family of tetrahedral meshes of Ω . Define X h := { v h ∈ H 0 ( curl ; Ω) | v h | K = a K + b K × x , ∀ K ∈ T h } . Assume ( ⋆ ) ∀ h, � E 0 − E 0 ,h � H ( curl ;Ω) � inf v h ∈X h � E 0 − v h � H ( curl ;Ω) . Edge element interpolation ( δ ∈ ]1 / 2 , δ max [ ), cf. [Alonso-Valli’99], [Jr-Zou’99]: � E 0 − E 0 ,h � H ( curl ;Ω) � h δ � f � L 2 (Ω) . RICAM, October 2016 – p. 7/24
Edge element discretization Let ( T h ) h be a shape regular family of tetrahedral meshes of Ω . Define X h := { v h ∈ H 0 ( curl ; Ω) | v h | K = a K + b K × x , ∀ K ∈ T h } . Assume ( ⋆ ) ∀ h, � E 0 − E 0 ,h � H ( curl ;Ω) � inf v h ∈X h � E 0 − v h � H ( curl ;Ω) . Edge element interpolation ( δ ∈ ]1 / 2 , δ max [ ), cf. [Alonso-Valli’99], [Jr-Zou’99]: � E 0 − E 0 ,h � H ( curl ;Ω) � h δ � f � L 2 (Ω) . QUESTION : What of � div ε ( E 0 − E 0 ,h ) � ? RICAM, October 2016 – p. 7/24
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