Finite element methods for Maxwells equations: A local a priori - PowerPoint PPT Presentation

5. Weihnachtskolloquium Dec 20, 2016 Finite element methods for Maxwell’s equations: A local a priori estimate Claudio Rojik Vienna University of Technology Institute for Analysis and Scientific Computing

Finite element methods for Maxwell’s equations: A local a priori estimate Motivation Claudio Rojik (TU Wien)

Finite element methods for Maxwell’s equations: A local a priori estimate Motivation A local a priori estimate for Poisson’s equation Assumptions Ω ⊆ R n . . . bounded domain Claudio Rojik (TU Wien) – 1 –

Finite element methods for Maxwell’s equations: A local a priori estimate Motivation A local a priori estimate for Poisson’s equation Assumptions Ω ⊆ R n . . . bounded domain Poisson’s equation − ∆ u = f in Ω � Ω ∇ u · ∇ v dx for u, v ∈ H 1 (Ω) corresponding bilinear form a ( u, v ) := Claudio Rojik (TU Wien) – 1 –

Finite element methods for Maxwell’s equations: A local a priori estimate Motivation A local a priori estimate for Poisson’s equation Assumptions Ω ⊆ R n . . . bounded domain Poisson’s equation − ∆ u = f in Ω � Ω ∇ u · ∇ v dx for u, v ∈ H 1 (Ω) corresponding bilinear form a ( u, v ) := T h . . . triangulation of Ω with element diameter h W h (Ω) ⊆ H 1 (Ω) . . . H 1 -conforming finite element space Claudio Rojik (TU Wien) – 1 –

Finite element methods for Maxwell’s equations: A local a priori estimate Motivation A local a priori estimate for Poisson’s equation Theorem 1 (Nitsche and Schatz 1974) Let B 0 be a ball with radius r and B d be the concentric ball with radius r + d , where d ≫ h . (Of course B d ⊆ Ω must hold.) If u h ∈ W h (Ω) is a FEM-approximation to u ∈ H 1 (Ω) satisfying a ( u − u h , χ ) = 0 for χ ∈ W comp ( B d ) , then h � � � u − χ � H 1 ( B d ) + d − 1 � u − χ � L 2 ( B d ) � u − u h � H 1 ( B 0 ) ≤ C min χ ∈ W h ( B d ) + Cd − 1 � u − u h � L 2 ( B d ) . Claudio Rojik (TU Wien) – 2 –

Finite element methods for Maxwell’s equations: A local a priori estimate Motivation Can we establish a similar a priori estimate if we consider Maxwell’s equations instead of Poisson’s equation? Claudio Rojik (TU Wien) – 3 –

Finite element methods for Maxwell’s equations: A local a priori estimate FEM setting for Maxwell’s equations Claudio Rojik (TU Wien)

Finite element methods for Maxwell’s equations: A local a priori estimate FEM setting for Maxwell’s equations Maxwell’s equations E, H . . . electric field intensity, magnetic field intensity B, D . . . magnetic flux density, displacement current density j, ρ . . . electric current density, charge density Claudio Rojik (TU Wien) – 4 –

Finite element methods for Maxwell’s equations: A local a priori estimate FEM setting for Maxwell’s equations Maxwell’s equations E, H . . . electric field intensity, magnetic field intensity B, D . . . magnetic flux density, displacement current density j, ρ . . . electric current density, charge density Maxwell’s equations curl E = − ∂B ∂t curl H = ∂D ∂t + j div D = ρ div B = 0 Claudio Rojik (TU Wien) – 4 –

Finite element methods for Maxwell’s equations: A local a priori estimate FEM setting for Maxwell’s equations Maxwell’s equations Introduce vector potential A : E = − ∂A ∂t Maxwell’s equations + material laws B = µH, j = σE, D = ǫE ⇒ curl( µ − 1 curl A ) + κA = j µ is the permeability, κ ∈ C is a constant depending on the setting! We will set µ ≡ 1 for simplicity. Claudio Rojik (TU Wien) – 5 –

Finite element methods for Maxwell’s equations: A local a priori estimate FEM setting for Maxwell’s equations Variational framework Assumptions Ω ⊆ R 3 bounded domain with polyhedral boundary div j = 0 κ ∈ C \{ 0 } Claudio Rojik (TU Wien) – 6 –

Finite element methods for Maxwell’s equations: A local a priori estimate FEM setting for Maxwell’s equations Variational framework Assumptions Ω ⊆ R 3 bounded domain with polyhedral boundary div j = 0 κ ∈ C \{ 0 } Weak formulation Find u ∈ H 0 (curl , Ω) such that b ( u, v ) = ( j, v ) L 2 (Ω) ∀ v ∈ H 0 (curl , Ω) where b ( u, v ) := (curl u, curl v ) L 2 (Ω) + κ ( u, v ) L 2 (Ω) . Claudio Rojik (TU Wien) – 6 –

Finite element methods for Maxwell’s equations: A local a priori estimate FEM setting for Maxwell’s equations Finite element spaces: H 1 -conforming T h . . . triangulation of Ω with element diameter h Claudio Rojik (TU Wien) – 7 –

Finite element methods for Maxwell’s equations: A local a priori estimate FEM setting for Maxwell’s equations Finite element spaces: H 1 -conforming T h . . . triangulation of Ω with element diameter h P 1 ( K ) denotes the space of polynomials of degree 1 in K Claudio Rojik (TU Wien) – 7 –

Finite element methods for Maxwell’s equations: A local a priori estimate FEM setting for Maxwell’s equations Finite element spaces: H 1 -conforming T h . . . triangulation of Ω with element diameter h P 1 ( K ) denotes the space of polynomials of degree 1 in K Definition We define W h (Ω) := { u h ∈ H 1 (Ω) : ( u h ) | K ∈ P 1 ( K ) ∀ K ∈ T h } . The space W h is a H 1 -conforming finite element space. Claudio Rojik (TU Wien) – 7 –

Finite element methods for Maxwell’s equations: A local a priori estimate FEM setting for Maxwell’s equations Finite element spaces: H (curl) -conforming Definition Using R k := ( P k − 1 ) 3 ⊕ { p ∈ ( ˜ P k ) 3 : x · p = 0 } , we define V h (Ω) := { v h ∈ H (curl , Ω) : v h | K ∈ R 1 ∀ K ∈ T h } . The space V h is now H (curl) -conforming. Claudio Rojik (TU Wien) – 8 –

Finite element methods for Maxwell’s equations: A local a priori estimate FEM setting for Maxwell’s equations Finite element spaces: H (div) -conforming Definition Using D k := ( P k − 1 ) 3 ⊕ ˜ P k − 1 x we define Q h (Ω) := { q h ∈ H (div , Ω) : q h | K ∈ D 1 ∀ K ∈ T h } The space Q h is a H (div) -conforming finite element space. Claudio Rojik (TU Wien) – 9 –

Finite element methods for Maxwell’s equations: A local a priori estimate FEM setting for Maxwell’s equations The DeRham-diagram curl div ∇ H 1 (Ω) L 2 (Ω) − → H (curl , Ω) − → H (div , Ω) − → ↓ π h ↓ r h ↓ w h ↓ p 0 ,h ∇ curl div W h (Ω) − → V h (Ω) − → Q h (Ω) − → S h (Ω) Claudio Rojik (TU Wien) – 10 –

Finite element methods for Maxwell’s equations: A local a priori estimate FEM setting for Maxwell’s equations An idea of the a priori estimate We are currently working on establishing an estimate similar to the following: Theorem 2 (Idea) Let B 0 be a ball with radius r and B d be the concentric ball with radius r + d , where h ≪ d ≤ 1 . (Of course B d ⊆ Ω must hold.) If u h ∈ V h (Ω) is a FEM-approximation to u ∈ H (curl , Ω) satisfying b ( u − u h , χ ) = 0 for χ ∈ V comp ( B d ) , then h � d − 1 / 2 � u − χ � H (curl ,B d ) � u − u h � H (curl ,B 0 ) ≤ C min χ ∈ V h ( B d ) � + d − 1 � u − χ � L 2 ( B d ) + Cd − 1 � u − u h � L 2 ( B d ) . Claudio Rojik (TU Wien) – 11 –

Finite element methods for Maxwell’s equations: A local a priori estimate FEM setting for Maxwell’s equations An idea of the a priori estimate Questions and problems: Claudio Rojik (TU Wien) – 12 –

Finite element methods for Maxwell’s equations: A local a priori estimate FEM setting for Maxwell’s equations An idea of the a priori estimate Questions and problems: The bilinear form b contains an L 2 -term with a constant κ . Change assumptions in the theorem suitably? Claudio Rojik (TU Wien) – 12 –

Finite element methods for Maxwell’s equations: A local a priori estimate FEM setting for Maxwell’s equations An idea of the a priori estimate Questions and problems: The bilinear form b contains an L 2 -term with a constant κ . Change assumptions in the theorem suitably? u h ∈ W comp Proof of Theorem 1 → � ( B d ) and a (˜ u − � u h , � u h ) = 0 h = ⇒ | � u h | H 1 ( B d ) ≤ | ˜ u | H 1 ( B d ) Claudio Rojik (TU Wien) – 12 –

Finite element methods for Maxwell’s equations: A local a priori estimate FEM setting for Maxwell’s equations An idea of the a priori estimate Questions and problems: The bilinear form b contains an L 2 -term with a constant κ . Change assumptions in the theorem suitably? u h ∈ W comp Proof of Theorem 1 → � ( B d ) and a (˜ u − � u h , � u h ) = 0 h = ⇒ | � u h | H 1 ( B d ) ≤ | ˜ u | H 1 ( B d ) u − � u h ) = 0 ⇒ ? b (˜ u h , � Claudio Rojik (TU Wien) – 12 –

Finite element methods for Maxwell’s equations: A local a priori estimate FEM setting for Maxwell’s equations An idea of the a priori estimate Questions and problems: The bilinear form b contains an L 2 -term with a constant κ . Change assumptions in the theorem suitably? u h ∈ W comp Proof of Theorem 1 → � ( B d ) and a (˜ u − � u h , � u h ) = 0 h = ⇒ | � u h | H 1 ( B d ) ≤ | ˜ u | H 1 ( B d ) u − � u h ) = 0 ⇒ ? b (˜ u h , � d − 1 / 2 � u − χ � H (curl ,B d ) → only quasi-optimality dependent on d . Improvement possible? Claudio Rojik (TU Wien) – 12 –