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A SEAMLESS REDUCED BASIS ELEMENT METHOD FOR 2D MAXWELL’S PROBLEM: AN INTRODUCTION YANLAI CHEN

†,

JAN S. HESTHAVEN

† , AND

YVON MADAY

‡ †

Abstract. We present a reduced basis element method (RBEM) for the time-harmonic Maxwell’s equation. The RBEM is a Reduced Basis Method (RBM) with parameters describing the geometry of the computational domain, coupled with a domain decomposition method. The basic idea is the following. First, we decompose the computational domain into a series of subdomains, each of which is deformed from some reference domain. Then, we associate with each reference domain precomputed solutions to the same governing partial differential equation, but with different choices of deformations. Finally, one seeks the approximation on a new domain as a linear combination of the corresponding precomputed solutions on each subdomain. Unlike the work on RBEM for thermal fin and fluid flow problems, we do not need a mortar type method to “glue” the various local functions. This “gluing” is done “automatically” thanks to the use of a discontinuous Galerkin method. We present the rationale for the method together with numerical results showing exponential convergence for the simulation of a metallic pipe with both ends open.

1. Introduction. There is a need to rapidly, perhaps even in real time, and accurately predict some quantities of interest under the variation of a set of parameters in applications such as computational optimization, control and design, the development of efficient ways to quantify uncertainties and their impact. In such cases, an output of interest, here denoted by se , is defined by a functional applied to the solution of a parameterized partial differential equation (PDE). Let us write the problem in weak form as ¯ ¯ For an input ν ∈ D ⊂ Rp the output is defined by ¯ ¯ se (ν) := l(ue (ν); ν) ∈ C, ¯ (1) ¯ e ¯ where u (ν) ∈ X e is the exact solution of ¯ ¯ a(ue (ν), v; ν) = f (v; ν), ∀ v ∈ X e , where a and f are bilinear and linear forms, respectively, associated to the PDE, and X e is the space of the exact solution ue . To approximate its solution uN (ν) ' ue (ν), one could use the following finite element (FE) discretization : Given ν ∈ D ⊂ RP , find uN (ν) ∈ X N satisfying aN (uN (ν), v; ν) = f N (v; ν), ∀v ∈ X N . Here X N is the finite element space approximating X e with dim(X N )≡ N , and aN (·, ·; ·) and f N (·; ·) are computable approximations of a(·, ·; ·) and f (·; ·), respectively. We assume uN provides a reference solution (called the truth approximation) that is accurate enough for all ν ∈ D. For that purpose, one usually must choose a very large N . This makes it time-consuming to solve for the truth approximation, in particular when the solution is needed for many instances of ν. The RBM is particularly well suited for this “many-query” scenario to improve the simulation efficiency. It was introduced in [13, 6]. See [17, 14, 7, 4] for recent developments including rigorous a posteriori error estimators and greedy algorithm to form the global approximation spaces. See also the review paper [16] and the extensive reference therein. The first theoretical a priori convergence result for a one dimensional parametric space problem is presented in [10] where exponential convergence of the reduced basis approximation is confirmed. The key idea of the RBM is to store the solutions of the PDE for a specific set of parameters, and then find the reduced basis approximation for a new parameter as a linear combination of these precomputed solutions. The fundamental observation is that the parameter dependent solution ue (ν) is not simply an arbitrary member of the infinite-dimensional space associated with the PDE, but rather that it evolves † Division

of Applied Mathematics, Brown University, 182 George St, Providence, RI 02912, USA Pierre et Marie Curie-Paris6, UMR 7598, Laboratoire J.-L. Lions, Paris, F-75005 France § E-mail addresses: Yanlai [email protected] (Y.Chen) [email protected] (J.Hesthaven) [email protected] (Y.Maday) ‡ Universit´ e

1

on a lower-dimensional manifold induced by the parametric dependence. Under this assumption we can expect that as ν (∈ D ⊂ Rq ) varies, the set of all solutions ue (ν) can be well approximated by a finite and low dimensional vector space. How to choose the initial set of parameters used to compute the basis functions is crucial to the method. It is guided by the a posteriori error estimators, which are also used to certify the quality of the approximations. In this paper, we are going to concentrate on the case of ν in (1) describing the geometries. This special parameter allows us to combine domain decomposition ideas with RBM to obtain a method called reduced basis element method. It was introduced in [11] and later applied to a thermal fin problem [12] and the Stokes problem [9]. We shall here, for the first time, apply it to Maxwell’s equation. The first ingredient of RBEM is a domain decomposition approach to generalize a discretization method, originally designed over a simple geometry, to a complicated one. The second ingredient is the RBM mentioned above. The extension from RBM to RBEM is like the one from a single element method to a multielement method. We first decompose the computational domain into a series of subdomains that are deformed from several generic, reference building blocks. As a precomputation, we associate with each reference domain solutions to the same problem, but with different choices of the parameters describing the deformations. Finally, we seek the approximation on a new shape as a linear combination of the precomputed solutions mapped from the reference block onto each particular subdomain of interest. We focus on the time-harmonic Maxwell’s problem. The motivation is the design of waveguides, where pipes of different shapes have to be connected together. For the RBEM introduced in [11, 12, 9], a mortar type method is needed to “glue” the local functions on the subdomains due to the non-conformity of the method. This “gluing” is not necessary for our method. Since it is formulated in such a way that the reduced basis element space is a subspace of the finite element space, and the connection between neighboring elements are taken care of by the numerical fluxes. The paper is organized as follows. In section 2, we formulate the RBEM. Some numerical results are given in section 3 to show the superior convergence toward the truth approximation with a seamless “gluing” of the decomposed subdomains. Finally, concluding remarks are presented in section 4. 2. Reduced Basis Element Method. In this section, we discuss the reduced basis element method for electromagnetics. We first formulate the RBM with geometry as a parameter and lay out the numerical scheme for the truth approximation. Then, we study the RBEM. Finally, we discuss the a posteriori error estimate for the RBEM. 2.1. Reduced Basis Method with Geometry As a Parameter. We are seeking the frequencydomain solution of the two-dimensional Maxwell’s equations in normalized differential form,  ³ ´ bξ + 1 ∂ ∂ Ebη − ∂ Ebξ = iωJξ , −²ω 2 E µ ∂η ∂ξ ∂η ³ ´ (1) −²ω 2 E bη − 1 ∂ ∂ Ebη − ∂ Ebξ = iωJη . µ ∂ξ ∂ξ ∂η As shown by Figure 2.1, we want to solve the problem on a domain, Ωa , which consists of three α β α PML

PML

Γi

θ

Fig. 2.1. Actual Decomposed Domain 2

1

Ω1

Ω2

Ω3

0.7 Γi

PML

PML

0.3 0 Fig. 2.2. Decomposed Reference Domain

(xi , 1) Fi θi

Li

ai

(ξi , ηi )

(xi , 0)

Fig. 2.3. A generic mapping on one subdomain.

subdomains, denoted from left to right by Ωa1 , Ωa2 and Ωa3 . A dipole antenna is located in Ωa1 . To the left of the antenna is B´erenger’s perfectly matched layer (PML), see [3, 5], also [1], which is also used in the uller condition is enforced on the exterior boundary of the PML. The symmetric part of Ωa3 . A Silver-M¨ other boundaries are assumed to be perfectly electrically conducting (PEC) metallic wall, i.e., boundary bξ n bη n condition E ˆη − E ˆ ξ = 0 is enforced. Here, (ˆ nξ , n ˆ η ) denotes the unit outward normal. This models a metallic pipe with segments of varying shapes and both ends open. As the parameter ν := (α, β, θ) change, it appears that we need to change the computational domain. In fact, the computation is always done on a reference domain, Ω, as shown by Figure 2.1. The b an element deformation of the i−th subdomain is denoted by Fi , i.e. Ωai = Fi (Ωi ). We denote by K in Ωa corresponding to a reference domain element K. The map Fi , shown in Figure 2.1, is defined as follows ¶¶ µ ai µµ ¶ µ ¶ ¶µ 0 x x − xi ξi Li sin θi Fi = + . ai y y ηi 1 Li cos θi with (ξi , ηi )T defined recursive by µ ¶ µ ¶ ξ1 0 = , η1 0

µ

ξi+1 ηi+1

¶

µµ = Fi

xi + Li 0

¶¶ .

b to H(curl, K), we apply the Piola transform To maintain tangential components and map H(curl, K) bη , −E bξ )T : (see [2] and [15] for a more complete presentation) to (E à ! µµ ¶¶ µ ¶ bη 1 E Ey Ey = Pi := JFi , bξ −Ex −Ex |JFi | −E where JFi is the Jacobian matrix of the map Fi . After the application of the Piola transform Pi , we 3

obtain, on the reference domain Ωi , the following system of equations  ´ ³ ∂Ey Li ∂Ex  iωµH + −  z ai sin θi ∂x ∂y´ = 0,   ³ Li 1 ∂Hz i²ω ai sin θi³Ex − (cot θi )Ey − µ ∂y ´= Jx ,    Li ∂Hz ∂Hz  = Jy . i²ωEy + µ1 ai sin θi ∂x − (cot θi ) ∂y Next, we account for the PML by modifying the system as follows (see [3, 5] for details).  ³ ´ ∂Ey µσ Li ∂Ex  )H + − = 0, (iωµ +  z ai sin θi ∂x ∂y   ³ ² ´ Li iσ iσ ∂Hz iσ i²ω ai sin θi (1 − ²ω )Ey − µ1 (1 − ²ω ) = (1 − ²ω )Jx , θi Ex − cot  ³ ´ ∂y   L ∂H ∂H iσ 1 i z z i²ω(1 − )Ey + = Jy . ²ω µ ai sin θi ∂x − (cot θi ) ∂y

(2)

(3)

Here, σ is a piecewise quadratic C 1 −function of x (constant along the y−axis). It is identically zero in the non-PML region and monotonically increasing from the PML/non-PML interfaces to both boundaries on the left end and right end. If we let       0 e3 × E Hz iσ )Jx  , S =  (1 − ²ω F (u) =  e1 × H  u =  Ex  Ey e2 × H Jy ´T ³ ∂Ey ∂Hz ∂Hz x and the system (3) can be rewritten as we have ∇ · F (u) = ∂E ∂y − ∂x , − ∂y , ∂x A(Li , ai , θi ) u + B(Li , ai , θi ) ∇ · F (u) = S, where, for any given functions L(x), a(x), and θ(x), the matrices A(L(x), a(x), θ(x)) and B(L(x), a(x), θ(x)) are defined by     − a(x)L(x) 0 0 µiω + µ² σ 0 0 sin θ(x)   i²ωL(x) iσ 1 0 0 0 − cot θ(x)(i²ω + σ)  , B =  A= . µ (1 − ²ω ) a(x) sin θ(x) L(x) 1 0 0 i²ω + σ 0 µ cot θ(x) µa(x) sin θ(x) S S Now, we can state the equation on the domain Ω := Ω1 Ω2 Ω3 as, A(L(x), a(x), θ(x)) u + B(L(x), a(x), θ(x)) ∇ · F (u) = S,

(4)

where,SL(x) is a piecewise function defined to be equal to the width of Ωi on Ωi , a(x) is equal to α S on Ω1 Ω3 and β on Ω2 , θ(x) is equal to π2 on Ω1 Ω3 and θ on Ω2 . Given a mesh Th , we define the following finite element space ¡ ¢3 3 X N = {v ∈ L2 (Th ) : for all elements K ∈ Th , v|K ∈ (Pk (K)) }, and use a discontinuous Galerkin method [8] to solve (4) as in [4] with the same numerical fluxes, but without the elimination of Hz since this can not be done with PML and Silver-M¨ uller boundary condition present. As a result, we obtain the following system aN (uN (ν), v; α, β, θ) = f N (v), ∀ v ∈ X N .

(5)

Here, aN (uN (ν), v; α, β, θ) = (A uN (ν), v)Th +hBFb(uN (ν))n, vi∂Th −(BF (uN (ν)), ∇v)Th , and f N (v) = b e1 × H, b e2 × H) b T , n the unit outward normal vector, (·, ·)T denotes the (S, v)Th , where Fb = (e3 × E, h broken inner product on the elements and (·, ·)∂Th the broken inner product on the set of faces of all the elements. The standard RBM is then applied to build the reduced basis space. The accuracy of the reduced basis solution is certified by the residual-based a posteriori error estimate. This error estimate is cheap to obtain on-line. It also guides the selection of the parameters and thus the building of the reduced basis space in the greedy algorithm. See e.g. [4, 16] for details. 4

2.2. Reduced Basis Element Method: formulation. In this subsection, we discuss in detail how we apply RBEM to our problem to enable a highly efficient simulation on more complicated domains. We are going to concentrate on domains consisting of pipes such as that shown in Figure 2.2. The

β1

α PML

βD

Γi

α PML

θD θ1

Fig. 2.4. A general domain decomposed into D + 2 subdomains.

standard discontinuous Galerkin FE method for the full problem would result in system: ¡ ¢ aN uN (ν), v; (α; β1 , θ1 ; · · · ; βD , θD ) = f N (v), ∀v ∈ X N .

(6)

We observe, see e.g. Figure 3.3, that the global solution has a certain amount of “repetitiveness” in the middle domains. The two ends have different behavior because of the PML material. Moreover, there is a dipole antenna in the left end. This motivates us to treat the three parts differently, to decompose the domain, and to make assumptions in the following way: We can subdivide the middle part into many blocks. Since the solution has similar patterns on all these blocks, we can mimic RBM, and only seek the solution on each block in a space spanned by a certain set of precomputed functions instead of the (very rich) FE space. Moreover, these spaces on all the blocks can be assumed to be the same. The natural choice of the elementary block is, of course, rectangles in our case. So we denote the subdomains from left to right by S0 , S1 , · · · , SD , SD+1 . Obviously, S0 corresponds to Ωa1 in Figure 2.1, SD+1 to Ωa3 and all the remaining to Ωa2 . Next, we need to identify an appropriate set of functions as our basic patterns. We take the basis functions in the reduced basis space obtained in section 2.1. Note that we have built a reduced basis space that is spanned by N solutions (on the domain as in Figure 2.1) to (5) at N judiciously chosen parameters [17, 14, 4]: W N = span {uN (ν 1 ), · · · , uN (ν N )}. We can cut each of these solutions into three pieces to obtain three sets of basis functions to be used on S0 , {S1 , · · · , SD }, SD+1 . That is, we define a space YN of dimension N (D + 2), © ª YN = {v ∈ X N such that v|S0 ∈ span uN (ν 1 )|Ω1 , · · · , uN (ν N )|Ω1 © ª v|SD+1 ∈ span uN (ν 1 )|Ω3 , · · · , uN (ν N )|Ω3 © ª v|Si ∈ span uN (ν 1 )|Ω2 , · · · , uN (ν N )|Ω2 for i = 1, · · · , D}.

(7)

The RBEM is nothing but to seek a solution in the space YN , i.e., the RBEM solution is the solution of the following problem: ¡ ¢ Find uN (ν) ∈ YN such that aN uN (ν), v; (α; β1 , θ1 ; · · · ; βD , θD ) = f N (v), ∀v ∈ YN . (8) Note that no “gluing” is necessary since our original space X N consists of discontinuous functions and the new space YN is naturally a subspace of X N . This dramatic difference from the method in [11, 12, 9] motivates the name of our method – seamless reduced basis element method. 5

2.3. Reduced Basis Element Method: error estimate. To end this section, we briefly remark on the error estimate. Since the space YN is a subspace of X N , the error estimate can be done in exactly the same ways as the RBM. The parameter (α; β1 , θ1 ; · · · ; βD , θD ) has a higher dimension than for the RBM used to build YN . However, this is not much of a problem for relatively small D. 3. Numerical Results. In this paper, we set D = 1 and thus only consider problems with three subdomains. Next, we show numerical results for a two-parameter case and then a three-parameter case. 3.1. Two-parameter Case. We choose two parameters: the length of Ωa1 (i.e. Ωa3 ), α, and the length of Ωa2 , β and let θ = π2 . The parameter domain is set to be (α, β) ∈ [0.48, 1.00] × [0.84, 1.75]. We set Jx = 0, Jy = cos(ω(y − 12 ))δΓi . S S First, we perform a reduced basis analysis on Ω := Ω1 Ω2 Ω3 . We use the standard reduced basis method, see e.g. [4, 16], to obtain 26 bases.To test the validity of our basis, we solve for the reduced basis solution for parameters in a set Ξtest containing 250 randomly selected points. The history of convergence of the reduced basis solution toward the truth approximation is shown in Figure 3.1 (a). Clearly, exponential convergence is observed. (a)

(b)

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−6

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25

N

5

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N

Fig. 3.1. History of convergence for (a) RBM and (b) RBEM. Three pieces and two parameters.

Then, we test RBEM on the same set Ξtest . The maximum, median and minimum of the error between the reduced basis element solution and the truth approximation is plotted in Figure 3.1 (b). The exponential convergence with respect to the number of bases used in the RBEM is observed. This clearly shows that the RBEM is working as expected. It is not surprising that RBEM is providing much more accurate solutions than RBM since the RBEM solution are sought in a higher dimensional (3N ) space. Note that for a given RB dimension, the RBM is performing better. However the construction of 3N basis functions for the RBEM requires only N FE solutions, which considerably reduces the offline effort. Moreover, with the same settings for α, β and D > 1 (addressed in a future paper), RBEM can solve problems on a domain having length up to D × 1.75 + 2.0, but RBM can reach at most 3.75. 4π 3.2. Three-parameter Case. Here, we vary θ in [ 3π 7 , 7 ]. A set of 50 bases are generated through the reduced basis analysis. The RBM and RBEM are tested on a set consisting of 500 randomly chosen 4π points in the parameter domain [0.48, 1.00] × [0.84, 1.75] × [ 3π 7 , 7 ]. The convergence results, shown in Figure 3.2, are similar to the two-parameter case. Moreover, we plot in Figure 3.3 the RBEM solutions for the case with the subdomain lengths being 0.6, 1.75 and 0.6 with θ = 3π 7 . We see that the discontinuity (due to the piecewise Piola transform) in Ex bξ ) after the (piece-wise) application is clearly captured by our method, and then recovered nicely (see E

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Fig. 3.2. History of convergence for (a) RBM and (b) RBEM. Three pieces and three parameters.

of the inverse Piola transform Pi−1 . The solutions on three patches are “glued” together seamlessly. Note that the Piola transform for Ey in our particular case is identity. 4. Concluding Remarks. In this paper, we have formulated a reduced basis element method to simulate electromagnetic wave propagation in a domain consisting of pipes of different lengths. High efficiency and accuracy of the method is confirmed. The second part of this series is going to be devoted to the study of the multi-element case, i.e. D ≥ 2. Acknowledgments. Research supported by AFOSR Grant FA9550-07-1-0425. REFERENCES [1] S. Abarbanel, D. Gottlieb, and J. S. Hesthaven. Well-posed perfectly matched layers for advective acoustics. J. Comput. Phys. 154(1999), 266 – 283. [2] F. Brezzi and M. Fortin Mixed and hybrid Finite Element Methods. Springer Verlag, 1991. ´renger, A perfectly matched layer for the absorption of electromagnetics waves. J. Comput. Phys., 114 [3] J. P. Be (1994) 185 – 200. [4] Y. Chen, J.S. Hesthaven, Y. Maday and J. Rodr´iguez, Certified reduced basis methods and output bounds for the harmonic Mawwell’s equations. Siam J. Sci. Comput., revision submitted. [5] F. Collino, P. B. Monk, Optimizing the perfectly matched layer. Comput. Methods Appl. Mech. Engrg., 164 (1998) 157 – 171. [6] J. P. Fink and W. C. Rheinboldt, On the error behavior of the reduced basis technique for nonlinear finite element approximations. Z. Angew. Math. Mech., 63(1):21–28, 1983. [7] M. A. Grepl, Y. Maday, N. C. Nguyen, and A. T. Patera, Efficient reduced-basis treatment of nonaffine and nonlinear partial differential equations. Mathematical Modelling and Numerical Analysis, 41(3):575–605, 2007. [8] J.S. Hesthaven and T. Warburton, Nodal Discontinuous Galerkin Methods: Algorithms, Analysis, and Applications. Springer Texts in Applied Mathematics 54, Springer Verlag, New York, 2008. [9] A.E. Løvgren, Y. Maday and E.M. Rønquist, A reduced-basis element method for the steady Stokes problem. ESAIM: M2 AN 40(2006) 529–552. [10] Y. Maday, A. T. Patera, and G. Turinici, Global a priori convergence theory for reduced-basis approximations of single-parameter symmetric coercive elliptic partial differential equations. C. R. Acad. Sci. Paris, Ser. I, 335:289–294, 2002. [11] Y. Maday and E.M. Rønquist, A reduced-basis element method. J. Sci. Comput. 17(2002) 447–459. [12] Y. Maday and E.M. Rønquist, The reduced-basis element method: application to a thermal fin problem. SIAM J. Sci. Comput. 26(2004) 240–258. [13] A.K. Noor, J.M. Peters, Reduced basis technique for nonlinear analysis of structures. AIAA Journal, 18(4):455-462, April 1980 7

Fig. 3.3. RBEM solution with (α, β, θ) = (0.6, 1.75, 3π ): the left column is the solution to the parametrized PDE 7 bξ , E bη ); from top to bottom is real part of Ex (E bξ ), (Ex , Ey ), and the right column is the solution on the actual domain (E bη ), imaginary part of Ex (E bξ ) and imaginary part of Ey (E bη ). real part of Ey (E

[14] C. Prud’homme, D. Rovas, K. Veroy, Y. Maday, A. T. Patera, and G. Turinici, Reliable real-time solution of parametrized partial differential equations: Reduced-basis output bound methods. Journal of Fluids Engineering, 124(1):70–80, March 2002. [15] P.A. Raviart and J.M. Thomas, A mixed finite element method for second order elliptic problems. In: I. Galligani and E. Magenes, editors, Mathematical Aspects of Finite Element Methods, Lecture Notes in Mathematics, Vol. 606. Springer Verlag, 1977. [16] G. Rozza, D.B.P. Huynh, and A.T. Patera, Reduced basis approximation and a posteriori error estimation for affinely parametrized elliptic coercive partial differential equations: Application to transport and continuum mechanics. Arch Comput Methods Eng, 15(3):229-275,2008. [17] K. Veroy, C. Prud’homme, D.V. Rovas, and A.T.Patera, A posteriori error bounds for reduced-basis approximation of parametrized noncoercive and nonlinear elliptic partial differential equations. AIAA, 2003. 8