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Efficient high order and domain decomposition methods for the time-harmonic Maxwell’s equations

Marcella Bonazzoli1, Victorita Dolean1,4, Ivan G. Graham3, Frédéric Hecht2, Frédéric Nataf2, Francesca Rapetti1, Euan A. Spence3, Pierre-Henri Tournier2

1University of Nice Sophia Antipolis, Nice, France 2Pierre and Marie Curie University, Paris, France 3University of Bath, UK 4University of Strathclyde, Glasgow, UK

August 31, 2017

Marcella Bonazzoli (UNS, France) High order FEs and DD for Maxwell 31/08/2017 1 / 44

Outline

1

Motivation

2

The boundary value problem

3

High order edge finite elements

4

One-level domain decomposition preconditioners

5

Numerical results for the microwave imaging system

6

Two-level domain decomposition preconditioners

Marcella Bonazzoli (UNS, France) High order FEs and DD for Maxwell 31/08/2017 2 / 44

Outline

1

Motivation

2

The boundary value problem

3

High order edge finite elements

4

One-level domain decomposition preconditioners

5

Numerical results for the microwave imaging system

6

Two-level domain decomposition preconditioners

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: ∇ × (∇ × E) − γ2E = 0, γ = ω√µεσ, εσ := ε − iσ ω ω angular frequency, µ magnetic permeability, ε(x) electric permittivity, σ(x) electrical conductivity of the medium, if σ = 0, γ = ˜ ω = ω√µε the wavenumber. Inverse problem: we know “E” (S-parameters), compute εσ(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

1

Motivation

2

The boundary value problem

3

High order edge finite elements

4

One-level domain decomposition preconditioners

5

Numerical results for the microwave imaging system

6

Two-level domain decomposition preconditioners

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)eiωt), with ω the angular frequency and E the complex amplitude. The computational domain Ω ⊂ R3: 5 rings of 32 ceramic-loaded waveguides around the cylindrical chamber

Γw Γi

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 Γw, impedance b.c. with gj on the transmitting waveguide port Γj, gj = (∇ × E0

j ) × n + iβn × (E0 j × n), TE10 fundamental mode E0 j ,

homogeneous impedance b.c. on the other waveguides ports Γi, i = j.          ∇ × (∇ × E) − γ2E = 0, in Ω, E × n = 0, on Γw, (∇ × E) × n + iβ n × (E × n) = gj, on Γj, (∇ × E) × n + iβ n × (E × n) = 0, on Γi, 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

• Ω
• (∇ × E) · (∇ × v) − γ2E · v
• +
• 160

i=1 Γi

iβ(E × n) · (v × n) =

• Γj

gj · v, ∀v ∈ V , V = {v ∈ H(curl, Ω), v × n = 0 on Γw}, where H(curl, Ω) = {v ∈ L2(Ω)

3, ∇ × v ∈ L2(Ω) 3}

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

1

Motivation

2

The boundary value problem

3

High order edge finite elements

4

One-level domain decomposition preconditioners

5

Numerical results for the microwave imaging system

6

Two-level domain decomposition preconditioners

Marcella Bonazzoli (UNS, France) High order FEs and DD for Maxwell 31/08/2017 12 / 44

Edge finite elements

Finite element discretization: tetrahedral mesh Th of Ω, Vh ⊂ H(curl, Ω)

Low order edge finite elements (degree r = 1, Nédélec)

Given a tetrahedron T ∈ Th, the local basis functions are associated with the oriented edges e = {ni, nj} of T: we = λni∇λnj − λnj∇λni, (the λnℓ are the barycentric coordinates).

• riented edges,

they are vector functions, they ensure the continuity of the tangential component across inter-element interfaces, degrees of freedom: ξe : w →

1 |e|

• e w · te,

duality: ξe(we′) = δee′

4

1

e 2 e 3 e 4 e 6 e 5 1 2 3 e

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 ∈ Th, for all oriented edges e of T and for all multi-indices k = (k1, k2, k3, k4) of weight k = k1 + k2 + k3 + k4, define: w{k,e} = λkwe, where λk = (λn1)k1(λn2)k2(λn3)k3(λn4)k4. Only barycentric coordinates! Still Vh ⊂ H(curl, Ω)

4

1

e 2 e 3 e 4 e 6 e 5 1 2 3 e

E.g. degree r = 2 → k = 1 λ1we1, λ2we1, λ3we1, λ4we1, λ1we2, λ2we2, λ3we2, λ4we2, λ1we3, λ2we3, λ3we3, λ4we3, λ1we4, λ2we4, λ3we4, λ4we4, λ1we5, λ2we5, λ3we5, λ4we5, λ1we6, λ2we6, λ3we6, λ4we6. Select linearly independent basis functions! (dim = 20)

[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

1

e 2 e 3 e 4 e 6 e 5 1 2 3 e

e1 = {1, 2}, e2 = {1, 3}, e3 = {1, 4}, e4 = {2, 3}, e5 = {2, 4}, e6 = {3, 4}, f1 = {2, 3, 4}, f2 = {1, 3, 4}, f3 = {1, 2, 4}, f4 = {1, 2, 3}. degree r = 2 : Edge-type basis functions: w1 = λ1we1, w2 = λ2we1, w3 = λ1we2, w4 = λ3we2, w5 = λ1we3, w6 = λ4we3, w7 = λ2we4, w8 = λ3we4, w9 = λ2we5, w10 = λ4we5, w11 = λ3we6, w12 = λ4we6, Face-type basis functions: w13 = λ4we4, w14 = λ3we5, w15 = λ4we2, w16 = λ3we3, w17 = λ4we1, w18 = λ2we3, w19 = λ3we1, w20 = λ2we2. 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 ∈ Th for degree r ≥ 1 as the functionals: ξe : w → 1 |e|

• e

(w · te) q, ∀ q ∈ Pr−1(e), ∀ e ∈ E(T), ξf : w → 1 |f |

• f

(w · tf ,i) q, ∀ q ∈ Pr−2(f ), ∀ f ∈ F(T), tf ,i two sides of f , i = 1, 2, ξT : w → 1 |T|

• T

(w · tT,i) q, ∀ q ∈ Pr−3(T), tT,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

High order edge finite elements

The degrees of freedom (dofs)

4

1

e 2 e 3 e 4 e 6 e 5 1 2 3 e

e1 = {1, 2}, e2 = {1, 3}, e3 = {1, 4}, e4 = {2, 3}, e5 = {2, 4}, e6 = {3, 4}, f1 = {2, 3, 4}, f2 = {1, 3, 4}, f3 = {1, 2, 4}, f4 = {1, 2, 3}. degree r = 2: for e = {ni, nj}, P1(e) = span(λni, λnj); P0(f ) = span(1); no volume dofs Edge-type dofs: ξ1 : w →

1 |e1|

• e1(w · te1) λ1,

ξ2 : w →

1 |e1|

• e1(w · te1) λ2,

. . . ξ11 : w →

1 |e6|

• e6(w · te6) λ3,

ξ12 : w →

1 |e6|

• e6(w · te6) λ4,

Face-type dofs: ξ13 : w →

1 |f1|

• f1(w · te4),

ξ14 : w →

1 |f1|

• f1(w · te5),

. . . ξ19 : w →

1 |f4|

• f4(w · te1),

ξ20 : w →

1 |f4|

• f4(w · te2).

Same choice as for generators

Marcella Bonazzoli (UNS, France) High order FEs and DD for Maxwell 31/08/2017 17 / 44

High order edge finite elements

Adding a new finite element to FreeFem++

We added the edge elements of degree 2, 3 in 3d as new finite elements to the open source domain specific language FreeFem++ (Edge13d, Edge23d, available with load "Element_Mixte3d").

[Bonazzoli, Dolean, Hecht, Rapetti, Explicit implementation strategy of high order edge finite elements and Schwarz preconditioning for the time-harmonic Maxwell’s equations. Preprint HAL <hal-01298938>]

To add a new finite element, write a C++ plugin that defines: the basis functions (and their derivatives) locally in a tetrahedron, an interpolation operator, it requires degrees of freedom ξi in duality with the basis functions ˜ wj: Πh : Y ⊂ H(curl, T) → Vh(T), u → uh =

m

• j=1

cj ˜ wj, with cj = ξj(u) Indeed if ξi(˜ wj) = δij, then ξi(uh) = ci = ξi(u)

Marcella Bonazzoli (UNS, France) High order FEs and DD for Maxwell 31/08/2017 18 / 44

High order edge finite elements

Restoring duality (for r > 1)

To restore duality between basis functions ~ wj and dofs ξi: ˜ Vij = ξi(~ wj) = δij take linear combinations of the old basis functions wj with coefficients given by the entries of V −1. Properties of the matrix Vij = ξi(wj): V does not depend on the metrics of the tetrahedron T (but pay attention to orientation!), V is blockwise lower triangular, V −1 entries are integer numbers.

[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 19 / 44

High order edge finite elements

Restoring duality

degree r = 2, basis functions and dofs listed as before (ordering and choice!):

V −1 =                                   4 −2 −2 4 4 −2 −2 4 4 −2 −2 4 4 −2 −2 4 4 −2 −2 4 4 −2 −2 4 −4 −2 2 −2 2 4 8 −4 2 −2 −4 −2 −4 −2 −4 8 −4 −2 2 −2 2 4 8 −4 2 −2 −4 −2 −4 −2 −4 8 −4 −2 2 −2 2 4 8 −4 2 −2 −4 −2 −4 −2 −4 8 −4 −2 2 −2 2 4 8 −4 2 −2 −4 −2 −4 −2 −4 8                                   Marcella Bonazzoli (UNS, France) High order FEs and DD for Maxwell 31/08/2017 20 / 44

Outline

1

Motivation

2

The boundary value problem

3

High order edge finite elements

4

One-level domain decomposition preconditioners

5

Numerical results for the microwave imaging system

6

Two-level domain decomposition preconditioners

Marcella Bonazzoli (UNS, France) High order FEs and DD for Maxwell 31/08/2017 21 / 44

Domain decomposition preconditioning

The matrix A of the discretized linear system is ill conditioned ⇒ a preconditioner is mandatory for iterative solvers. Linear systems solvers: direct solvers (e.g. LU): robust, but high memory cost, difficult to parallelize, iterative solvers (e.g. CG, GMRES): low memory cost, easy to parallelize, but not robust. ⇒ Domain Decomposition (DD) preconditioner for the iterative solver (GMRES) DD methods: subproblems that can be solved concurrently and using direct solvers Domain Ω decomposed into Nsub

• verlapping subdomains Ωi

(mesh partitioner SCOTCH and add layers of overlap)

Marcella Bonazzoli (UNS, France) High order FEs and DD for Maxwell 31/08/2017 22 / 44

One-level domain decomposition preconditioners

Decomposition of Ω into Nsub overlapping subdomains Ωi ⇒ decomposition of the set of unknowns N into Nsub subsets Ni

Optimized Restricted Additive Schwarz preconditioner (ORAS)

M−1 = M−1

1,ORAS = Nsub

• i=1

RT

i DiA−1 i

Ri, Ri restriction matrix from N to Ni, Ai matrix of the local subproblem on Ωi with impedance conditions as transmission conditions between subdomains: (∇ × E) × n + i˜ ω n × (E × n), Di partition of unity matrix for Ni (Nsub

i=1 RT i DiRi = I),

RT

i

extension matrix from Ni to N.

Marcella Bonazzoli (UNS, France) High order FEs and DD for Maxwell 31/08/2017 23 / 44

One-level domain decomposition preconditioners

Construction of the partition of unity matrices {Di}1iNsub for edge finite elements: build piecewise linear partition of unity functions {χi}1iNsub for the linear nodal finite elements, such that Nsub

i=1 χi = 1,

χi and its derivative should be equal to zero on the border of Ωi, values of the diagonal of Di for edge elements: for each edge-type dof (resp. face-type, volume-type dof) interpolate χi at the midpoints of its edge (resp. face, tetrahedron).

[Tournier, Aliferis, B., De Buhan, Darbas, Dolean, Hecht, Jolivet, El Kanfoud, Migliaccio, Nataf, Pichot, Semenov, Microwave tomographic imaging of cerebrovascular accidents by using High-Performance Computing. Preprint <hal-01343687>]

Efficient parallel implementation in HPDDM, which has an interface with FreeFem++ (https://github.com/hpddm/hpddm)

Marcella Bonazzoli (UNS, France) High order FEs and DD for Maxwell 31/08/2017 24 / 44

Outline

1

Motivation

2

The boundary value problem

3

High order edge finite elements

4

One-level domain decomposition preconditioners

5

Numerical results for the microwave imaging system

6

Two-level domain decomposition preconditioners

Marcella Bonazzoli (UNS, France) High order FEs and DD for Maxwell 31/08/2017 25 / 44

Numerical results

Plastic-filled cylinder (εcyl

r

= εcyl

σ /ε0 = 3, non dissipative!)

immersed in the matching solution (εgel

r

= 44 − 20i) inside the imaging chamber: (Ceramic-loaded waveguides with εcer

r

= 59)

[Bonazzoli, Dolean, Rapetti, Tournier. Parallel preconditioners for high order discretizations arising from full system modeling for brain microwave imaging. Int.J.Numer.Model. 2017] Marcella Bonazzoli (UNS, France) High order FEs and DD for Maxwell 31/08/2017 26 / 44

Numerical results

Norm of the real part of the solution for one transmitting antenna (at frequency f = 1 GHz) in the second ring from the top: 32 transmitting waveguides → 32 right-hand sides, 160 receiving waveguides

Marcella Bonazzoli (UNS, France) High order FEs and DD for Maxwell 31/08/2017 27 / 44

Numerical results

Reference solution with 114 million (complex-valued) unknowns. 512 subdomains with 1 process and 2 threads per subdomain → 1024 cores,

• n Curie supercomputer.

Edge finite elements of degree 1, 2 for different mesh sizes:

0.01 0.1 1 50 100 150 200 250 300 350 400 450 500 550

2.4M 8.5M 21M 43M 74M 1.5M 5.2M 13M 27M 46M

relative error time to solution (s) Degree 1 Degree 2

Relative error Err =

• j,i |Sij−Sref

ij |2

• j,i |Sref

ij |2

(the Sij are the measurable quantities, calculated from the solution E) For Err ∼ 0.1 degree 1: 21 M unks, 130 s degree 2: 5 M unks, 62 s

Marcella Bonazzoli (UNS, France) High order FEs and DD for Maxwell 31/08/2017 28 / 44

Outline

1

Motivation

2

The boundary value problem

3

High order edge finite elements

4

One-level domain decomposition preconditioners

5

Numerical results for the microwave imaging system

6

Two-level domain decomposition preconditioners

Marcella Bonazzoli (UNS, France) High order FEs and DD for Maxwell 31/08/2017 29 / 44

Two-level domain decomposition preconditioners

Seek independence of the iteration count from the wavenumber ˜ ω = ω√µε ⇒ two-level domain decomposition preconditioners: coarse-grid correction combined with the one-level preconditioner. Two ingredients: algebraic formula to combine the coarse grid correction with the

• ne-level preconditioner (e.g. additive or hybrid),

rectangular full column rank matrix Z, whose columns span the coarse space ⇒ reduced size problem built with Z: the coarse problem

Marcella Bonazzoli (UNS, France) High order FEs and DD for Maxwell 31/08/2017 30 / 44

Two-level domain decomposition preconditioners

Two-level preconditioners

M−1

2

= QM−1

1 P + H,

H = ZE −1Z † († the conjugate transpose) M−1

1

the one-level preconditioner, Z a rectangular matrix with full column rank (coarse space), E = Z †AZ “coarse grid” matrix, H = ZE −1Z † “coarse grid correction” matrix, if P = Q = I: additive two-level preconditioner, if P = I − AH, Q = I − HA: hybrid two-level preconditioner, or balancing Neumann Neumann (BNN). Here definition of Z based on a coarser mesh of diameter Hcs: the interpolation matrix from VHcs to Vh

Marcella Bonazzoli (UNS, France) High order FEs and DD for Maxwell 31/08/2017 31 / 44

Two-level domain decomposition preconditioners

The convergence analysis for wave propagation problems is challenging

Sign-indefinite time-harmonic Maxwell problem

• ∇ × (∇ × E) − ˜

ω2E = F in Ω E × n = 0

• n Γ = ∂Ω

We developed a theory for the dissipative case (σ > 0):

Problem with absorption

• ∇ × (∇ × E) − (˜

ω2 + iξ)E = F in Ω E × n = 0

• n Γ = ∂Ω

Work for the Helmholtz equation −∆u − (˜ ω2 + iξ)u = f :

[Graham, Spence, Vainikko, Domain decomposition preconditioning for high-frequency Helmholtz problems with

• absorption. Math.Comp. 2017]

Marcella Bonazzoli (UNS, France) High order FEs and DD for Maxwell 31/08/2017 32 / 44

Two-level domain decomposition preconditioners

Based on the coercivity of the sesquilinear form aξ with absorption ξ ∈ R \ 0 aξ(E, v) =

• Ω

∇ × E · ∇ × v −

• Ω

(˜ ω2 + iξ)E · v

Coercivity of aξ

There exists a constant ρ > 0 independent of ˜ ω and ξ such that |aξ(v, v)| ≥ Im (Θaξ(v, v)) ≥ ρ |ξ| ˜ ω2

• v2

curl,˜ ω

for all ˜ ω > 0 and v ∈ H0(curl, Ω), where Θ = −z/|z|, z :=

• ˜

ω2 + iξ (with branch cut on the positive real axis). ˜ ω-weighted inner product and norm: (v, w)curl,˜

ω = (∇ × v, ∇ × w)L2(Ω) + ˜

ω2(v, w)L2(Ω) vcurl,˜

ω = (v, v)1/2 curl,˜ ω

Marcella Bonazzoli (UNS, France) High order FEs and DD for Maxwell 31/08/2017 33 / 44

Two-level domain decomposition preconditioners

Paper in preparation

[Bonazzoli, Dolean, Graham, Spence, Tournier. Domain decomposition preconditioning for the high-frequency time-harmonic Maxwell equations with absorption]

Convergence rates for GMRES with the two-level Additive preconditioner M−1

ξ,AS = Nsub

• ℓ=0

(Rℓ)T(Aℓ

ξ)−1Rℓ,

Aℓ

ξ = RℓAξ(Rℓ)T,

R0 = Z T. explicit in wavenumber ˜ ω, absorption ξ, coarse-grid diameter Hcs, subdomain diameter Hsub and overlap size δ. Main theorems: upper bound on the norm of M−1

ξ,ASAξ,

lower bound on the field of values of M−1

ξ,ASAξ,

⇒ apply Elman-type estimates for the convergence of GMRES

Marcella Bonazzoli (UNS, France) High order FEs and DD for Maxwell 31/08/2017 34 / 44

Two-level domain decomposition preconditioners

Weighted GMRES method: the residual rm is minimized in the norm induced by V, WD ˜

ω = (vh, wh)curl,˜

ω

(vh, wh ∈ Vh with coefficient vectors V, W)

Theorem (GMRES convergence for left preconditioning)

Ω convex polyhedron. Given ˜ ω0 > 0, there exists C > 0 such that, if (i) ˜ ω ≥ ˜ ω0, (ii) max

• (˜

ωHsub), (˜ ωHcs)

• ˜

ω2 |ξ|

• ≤ C1
• 1 +

Hcs

δ

2−1

|ξ| ˜ ω2

• ,

(iii) m ≥ C

• ˜

ω2 |ξ|

3 1 + Hcs

δ

2 log 12

a

• ,

then rmD ˜

ω

r0D ˜

ω

≤ a for any 0 < a < 1. Particular case: when |ξ| ∼ ˜ ω2 (maximum absorption) and δ ∼ Hcs (generous overlap), Condition (ii) is satisfied with Hsub ∼ Hcs ∼ ˜ ω−1, and then bound (iii) implies convergence independent of ˜ ω.

Marcella Bonazzoli (UNS, France) High order FEs and DD for Maxwell 31/08/2017 35 / 44

Numerical experiments

Ω unit cube, regular or METIS decomposition into subdomains, mesh diameter h ∼ ˜ ω−3/2, or h ∼ 2π/(g ˜ ω) if order 2 FEs are used, F = [f , f , f ], f = − exp(−400((x − 0.5)2 + (y − 0.5)2 + (z − 0.5)2)), GMRES with right preconditioning (tolerance 10−6), random initial guess ⇒ all frequencies are present in the error, (two-level) preconditioners AS, RAS, HRAS, OHRAS.

Marcella Bonazzoli (UNS, France) High order FEs and DD for Maxwell 31/08/2017 36 / 44

Numerical experiments

Computations in FreeFem++ Parallel code run on the French Curie (TGCC-CEA) and OCCIGEN (CINES) supercomputers. We assign each subdomain to one processor. The local problems in each subdomain and the coarse space problem solved with a direct solver (MUMPS). We consider subdomain diameter Hsub ∼ ˜ ω−α, coarse mesh diameter Hcs ∼ ˜ ω−α′, 0 < α, α′ <= 1, precondition Aξprob with the preconditioner M−1

ξprec, 0 ≤ ξprob ≤ ξprec

Marcella Bonazzoli (UNS, France) High order FEs and DD for Maxwell 31/08/2017 37 / 44

Numerical experiments

# : number of iterations 2-level (1-level) Optimal particular case setting (with α = 1): Hsub ∼ Hcs ∼ ˜ ω−α, δ ∼ Hsub, ξprob = ξprec = ˜ ω2, PEC bc on ∂Ω α = 1 = α′ ˜ ω n Nsub ncs #AS #RAS #HRAS 10 4.6 ×105 1000 7.9×103 53(57) 26(37) 12 15 1.5 ×106 3375 2.6×104 59(64) 28(42) 12 20 1.2 ×107 8000 6.0×104 76(105) 29(57) 17 α = 0.8 = α′ ˜ ω n Nsub ncs #AS #RAS #HRAS 10 3.4 ×105 216 1.9 ×103 37(38) 21(21) 13 15 1.9 ×106 512 4.2 ×103 37(38) 22(22) 14 20 7.1 ×106 1000 7.9 ×103 62(63) 31(31) 20

Marcella Bonazzoli (UNS, France) High order FEs and DD for Maxwell 31/08/2017 38 / 44

Numerical experiments

# : number of iterations 2-level Hsub ∼ ˜ ω−α, Hcs ∼ ˜ ω−α′, δ ∼ Hsub, ξprob = ξprec = ˜ ω2, PEC bc on ∂Ω α = 0.8, α′ = 1 ˜ ω n Nsub ncs #AS #RAS #HRAS 10 3.4 ×105 216 7.9 ×103 37 20 11 20 7.1 ×106 1000 6.0 ×104 57 24 11 30 4.1 ×107 3375 2.0 ×105 53 32 16 40 2.0 ×108 6859 4.6 ×105 53 33 16

Marcella Bonazzoli (UNS, France) High order FEs and DD for Maxwell 31/08/2017 39 / 44

Numerical experiments

# : number of iterations 2-level (1-level) Hsub ∼ Hcs ∼ ˜ ω−α, δ ∼ 2h, ξprob = ξprec = ˜ ω2, PEC bc on ∂Ω α = 0.8 = α′ ˜ ω n Nsub ncs #AS #RAS 10 3.4 ×105 216 1.9 ×103 52 (58) 34 (39) 15 1.9 ×106 512 4.2 ×103 57 (65) 39 (47) 20 7.1 ×106 1000 7.9 ×103 61 (71) 43 (51)

Marcella Bonazzoli (UNS, France) High order FEs and DD for Maxwell 31/08/2017 40 / 44

Numerical experiments

Number of iterations 2-level (OHRAS), 1-level, Total Time (GMRES Time) Hsub ∼ ˜ ω−α, Hcs ∼ ˜ ω−α′, δ ∼ 2h, ξprob = 0, impedance bc on ∂Ω

ξprec = ˜ ω, α = 0.6, α′ = 0.9 ˜ ω n Nsub 2-level ncs Time 1-level Time 10 2.6 × 105 27 20 2.9 × 103 16.2(1.6) 37 13.7(2.6) 15 1.5 × 106 125 26 1.0 × 104 25.5(4.0) 70 26.1(8.9) 20 5.2 × 106 216 29 2.1 × 104 52.0(9.1) 94 60.6(25.6) 25 1.4 × 107 216 33 4.4 × 104 145.5(29.5) 105 191.2(88.1) 30 3.3 × 107 343 38 6.9 × 104 380.4(128.4) 132 673.5(443.1) ξprec = 0, α = 0.6, α′ = 0.9 ˜ ω n Nsub 2-level ncs Time 1-level Time 10 2.6 × 105 27 20 2.9 × 103 17.1(1.8) 37 13.7(2.6) 15 1.5 × 106 125 26 1.0 × 104 25.4(3.9) 71 26.5(9.1) 20 5.2 × 106 216 29 2.1 × 104 53.0(9.0) 95 60.8(25.9) 25 1.4 × 107 216 33 4.4 × 104 145.0(29.6) 107 189.3(90.5) 30 3.3 × 107 343 39 6.9 × 104 387.9(132.7) 134 669.4(444.7)

Marcella Bonazzoli (UNS, France) High order FEs and DD for Maxwell 31/08/2017 41 / 44

Numerical experiments

Number of iterations 2-level (OHRAS), 1-level, Total Time (GMRES Time) Degree r = 2, h ∼ 2π/(g ˜ ω), Hsub ∼ (g ˜ ω/(2π))−α, Hcs ∼ 2π/(gcs˜ ω), δ ∼ 2h, ξprob = 0, ξprec = ˜ ω, impedance bc on ∂Ω, irregular subdomains built with METIS

g = 20, α = 0.5, gcs = 2 ˜ ω n Nsub 2-level ncs Time 1-level Time 10 1.1 ×106 125 38 1.3 ×103 37.7(7.5) 80 36.6(14.8) 20 8.3 ×106 343 36 9.3×103 85.8(18.9) 123 161.7(72.1) 30 2.8 ×107 729 41 3.0 ×104 155.7(39.8) 162 267.3(174.5) 40 6.6 ×107 1331 51 7.0 ×104 272.3(77.6) > 200 453.8(305.2)

Marcella Bonazzoli (UNS, France) High order FEs and DD for Maxwell 31/08/2017 42 / 44

Conclusion

The high order finite elements make it possible to attain a given accuracy with much fewer unknowns and much less computing time than the degree 1 approximation. High order edge elements: revisitation of the classical dofs, technique to restore duality between dofs and basis functions, available in the open source language FreeFem++. The parallel implementation in HPDDM of the domain decomposition preconditioner is essential to be able to solve the large scale linear systems arising from the microwave imaging application. Two-level preconditioners do improve the dependence of the iteration counts on the wavenumber (experiments also for the problem without absorption).

Marcella Bonazzoli (UNS, France) High order FEs and DD for Maxwell 31/08/2017 43 / 44

Conclusion and outlook

Thank you for your attention!

Marcella Bonazzoli (UNS, France) High order FEs and DD for Maxwell 31/08/2017 44 / 44