Efficient high order and domain decomposition methods for the time-harmonic Maxwell’s equations Marcella Bonazzoli 1 , Victorita Dolean 1 , 4 , Ivan G. Graham 3 , Frédéric Hecht 2 , Frédéric Nataf 2 , Francesca Rapetti 1 , Euan A. Spence 3 , Pierre-Henri Tournier 2 1 University of Nice Sophia Antipolis, Nice, France 2 Pierre and Marie Curie University, Paris, France 3 University of Bath, UK 4 University of Strathclyde, Glasgow, UK August 31, 2017 Marcella Bonazzoli (UNS, France) High order FEs and DD for Maxwell 31/08/2017 1 / 44
Outline Motivation 1 The boundary value problem 2 High order edge finite elements 3 One-level domain decomposition preconditioners 4 Numerical results for the microwave imaging system 5 Two-level domain decomposition preconditioners 6 Marcella Bonazzoli (UNS, France) High order FEs and DD for Maxwell 31/08/2017 2 / 44
Outline Motivation 1 The boundary value problem 2 High order edge finite elements 3 One-level domain decomposition preconditioners 4 Numerical results for the microwave imaging system 5 Two-level domain decomposition preconditioners 6 Marcella Bonazzoli (UNS, France) High order FEs and DD for Maxwell 31/08/2017 3 / 44
Motivation Brain imaging for strokes detection and monitoring (ANR project MEDIMAX) Two types of brain stroke: ischemic and hemorrhagic Establish in the shortest possible time the type of stroke to choose the correct treatment (opposite in the two situations!): they result in different variations of the complex electric permittivity of brain tissues. Marcella Bonazzoli (UNS, France) High order FEs and DD for Maxwell 31/08/2017 4 / 44
Motivation Brain imaging for strokes detection and monitoring Microwave imaging system prototype (EMTensor GmbH): cylindrical chamber with 5 rings of 32 antennas (rectangular waveguides) The measured data are used as input for an inverse problem to determine the complex electric permittivity of the medium. Marcella Bonazzoli (UNS, France) High order FEs and DD for Maxwell 31/08/2017 5 / 44
Motivation The inverse and the direct problems Second order time-harmonic Maxwell’s equation for the electric field: γ = ω √ µε σ , ε σ := ε − i σ ∇ × ( ∇ × E ) − γ 2 E = 0 , ω ω angular frequency, µ magnetic permeability, ε ( x ) electric permittivity, σ ( x ) electrical conductivity of the medium, ω = ω √ µε the wavenumber. if σ = 0, γ = ˜ Inverse problem: we know “ E ” (S-parameters), compute ε σ ( x ) . Direct (or forward) problem: we know the equation coefficients, compute E . Marcella Bonazzoli (UNS, France) High order FEs and DD for Maxwell 31/08/2017 6 / 44
Motivation From the inverse to the direct problem The inversion tool requires repeated solves of direct problem ⇒ accurate and fast direct problem solver Accurate: high order edge finite elements (for a given precision need considerably fewer unknowns), Fast: domain decomposition preconditioner for the iterative solver (can be parallelized). Marcella Bonazzoli (UNS, France) High order FEs and DD for Maxwell 31/08/2017 7 / 44
Outline Motivation 1 The boundary value problem 2 High order edge finite elements 3 One-level domain decomposition preconditioners 4 Numerical results for the microwave imaging system 5 Two-level domain decomposition preconditioners 6 Marcella Bonazzoli (UNS, France) High order FEs and DD for Maxwell 31/08/2017 8 / 44
The boundary value problem Time-harmonic formulation: electric field E ( x , t ) = Re ( E ( x ) e i ω t ), with ω the angular frequency and E the complex amplitude. The computational domain Ω ⊂ R 3 : 5 rings of 32 ceramic-loaded waveguides around the cylindrical chamber Γ i Γ w Alternately each waveguide j transmits a signal Marcella Bonazzoli (UNS, France) High order FEs and DD for Maxwell 31/08/2017 9 / 44
The boundary value problem Boundary conditions on Γ = ∂ Ω : metallic b.c. on the cylinder and waveguides walls Γ w , impedance b.c. with g j on the transmitting waveguide port Γ j , g j = ( ∇ × E 0 j ) × n + i β n × ( E 0 j × n ) , TE 10 fundamental mode E 0 j , homogeneous impedance b.c. on the other waveguides ports Γ i , i � = j . ∇ × ( ∇ × E ) − γ 2 E = 0 , in Ω , E × n = 0 , on Γ w , ( ∇ × E ) × n + i β n × ( E × n ) = g j , on Γ j , ( ∇ × E ) × n + i β n × ( E × n ) = 0 , on Γ i , i � = j , where n is the unit outward normal to ∂ Ω , β propagation wavenumber. One boundary value problem for each transmitting waveguide j Marcella Bonazzoli (UNS, France) High order FEs and DD for Maxwell 31/08/2017 10 / 44
The boundary value problem Variational formulation Find E ∈ V such that � � � � ( ∇ × E ) · ( ∇ × v ) − γ 2 E · v + i β ( E × n ) · ( v × n ) � 160 Ω i = 1 Γ i � = g j · v , ∀ v ∈ V , V = { v ∈ H ( curl , Ω) , v × n = 0 on Γ w } , Γ j where 3 , ∇ × v ∈ L 2 (Ω) 3 } H ( curl , Ω) = { v ∈ L 2 (Ω) One problem for each transmitting waveguide j ! (only the rhs is different). Marcella Bonazzoli (UNS, France) High order FEs and DD for Maxwell 31/08/2017 11 / 44
Outline Motivation 1 The boundary value problem 2 High order edge finite elements 3 One-level domain decomposition preconditioners 4 Numerical results for the microwave imaging system 5 Two-level domain decomposition preconditioners 6 Marcella Bonazzoli (UNS, France) High order FEs and DD for Maxwell 31/08/2017 12 / 44
Edge finite elements Finite element discretization : tetrahedral mesh T h of Ω , V h ⊂ H ( curl , Ω) Low order edge finite elements (degree r = 1, Nédélec) Given a tetrahedron T ∈ T h , the local basis functions are associated with the oriented edges e = { n i , n j } of T : w e = λ n i ∇ λ n j − λ n j ∇ λ n i , (the λ n ℓ are the barycentric coordinates). oriented edges, 4 they are vector functions, e 3 e 6 they ensure the continuity of the tangential e 5 component across inter-element interfaces, e 2 e 1 1 1 3 � degrees of freedom: ξ e : w �→ e w · t e , | e | e 4 2 duality: ξ e ( w e ′ ) = δ ee ′ Marcella Bonazzoli (UNS, France) High order FEs and DD for Maxwell 31/08/2017 13 / 44
High order edge finite elements The basis functions Generators of degree r = k + 1 Given T ∈ T h , for all oriented edges e of T and for all multi-indices k = ( k 1 , k 2 , k 3 , k 4 ) of weight k = k 1 + k 2 + k 3 + k 4 , define: w { k , e } = λ k w e , where λ k = ( λ n 1 ) k 1 ( λ n 2 ) k 2 ( λ n 3 ) k 3 ( λ n 4 ) k 4 . Only barycentric coordinates! Still V h ⊂ H ( curl , Ω) E.g. degree r = 2 → k = 1 4 λ 1 w e 1 , λ 2 w e 1 , λ 3 w e 1 , λ 4 w e 1 , λ 1 w e 2 , λ 2 w e 2 , λ 3 w e 2 , λ 4 w e 2 , λ 1 w e 3 , λ 2 w e 3 , λ 3 w e 3 , λ 4 w e 3 , e 3 e 6 e 5 λ 1 w e 4 , λ 2 w e 4 , λ 3 w e 4 , λ 4 w e 4 , λ 1 w e 5 , λ 2 w e 5 , λ 3 w e 5 , λ 4 w e 5 , e 2 e 1 1 λ 1 w e 6 , λ 2 w e 6 , λ 3 w e 6 , λ 4 w e 6 . 3 e 4 Select linearly independent basis functions! (dim = 20) 2 [Rapetti, Bossavit, Whitney forms of higher degree, SIAM J. Num. Anal. , 47(3), 2009] Marcella Bonazzoli (UNS, France) High order FEs and DD for Maxwell 31/08/2017 14 / 44
High order edge finite elements The basis functions 4 e 1 = { 1 , 2 } , e 2 = { 1 , 3 } , e 3 = { 1 , 4 } , e 4 = { 2 , 3 } , e 5 = { 2 , 4 } , e 6 = { 3 , 4 } , e 3 e 6 f 1 = { 2 , 3 , 4 } , f 2 = { 1 , 3 , 4 } , f 3 = { 1 , 2 , 4 } , f 4 = { 1 , 2 , 3 } . e 5 e 2 e 1 1 degree r = 2 : 3 e 4 2 Edge-type basis functions: w 1 = λ 1 w e 1 , w 2 = λ 2 w e 1 , w 3 = λ 1 w e 2 , w 4 = λ 3 w e 2 , w 5 = λ 1 w e 3 , w 6 = λ 4 w e 3 , w 7 = λ 2 w e 4 , w 8 = λ 3 w e 4 , w 9 = λ 2 w e 5 , w 10 = λ 4 w e 5 , w 11 = λ 3 w e 6 , w 12 = λ 4 w e 6 , Face-type basis functions: w 13 = λ 4 w e 4 , w 14 = λ 3 w e 5 , w 15 = λ 4 w e 2 , w 16 = λ 3 w e 3 , w 17 = λ 4 w e 1 , w 18 = λ 2 w e 3 , w 19 = λ 3 w e 1 , w 20 = λ 2 w e 2 . Choice using the global numbers of the vertices! Marcella Bonazzoli (UNS, France) High order FEs and DD for Maxwell 31/08/2017 15 / 44
High order edge finite elements The degrees of freedom (dofs) Revisitation of classical dofs Define the dofs on T ∈ T h for degree r ≥ 1 as the functionals: ξ e : w �→ 1 � ( w · t e ) q , ∀ q ∈ P r − 1 ( e ) , ∀ e ∈ E ( T ) , | e | e ξ f : w �→ 1 � ( w · t f , i ) q , ∀ q ∈ P r − 2 ( f ) , ∀ f ∈ F ( T ) , | f | f t f , i two sides of f , i = 1 , 2 , 1 � ξ T : w �→ ( w · t T , i ) q , ∀ q ∈ P r − 3 ( T ) , | T | T t T , i three sides of T , i = 1 , 2 , 3 . As polynomials q , use products of barycentric coordinates [Bonazzoli, Rapetti, High order finite elements in numerical electromagnetism: dofs and generators in duality, NUMA 2017] Marcella Bonazzoli (UNS, France) High order FEs and DD for Maxwell 31/08/2017 16 / 44
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