Abdulle, Assyr

Grote, Marcus J.

Stohrer, Christian M.

Infoscience - École polytechnique fédérale de Lausanne

Finite Element Heterogeneous Multiscale Method for the Wave Equation: Long-Time EffectsAssyr Abdulle1, Marcus J. Grote2, Christian M. Stohrer2,∗1 ANMC, Section of Mathematics, EPFL, Lausanne, Switzerland2 Department of Mathematics and Computer Science, University of Basel, Basel, Switzerland∗Email: christian.stohrer@unibas.chAbstractFor limited time the propagation of waves in ahighly oscillatory medium is well-described by thenon-dispersive homogenized wave equation. With in-creasing time, however, the true solution deviatesfrom the classical homogenization limit, as a largesecondary wave train develops unexpectedly. Here,we propose a new finite element heterogeneous mul-tiscale method (FE-HMM), which captures not onlythe short-time macroscale behavior of the wave fieldbut also those secondary long-time dispersive effects.1 Long-Time Wave PropagationLet Ω ⊂ Rn be a domain and T > 0. We considerthe wave equation∂ttuε −∇ · (aε∇uε) = F in Ω× (0, T ),uε(x, 0) = f(x) in Ω,∂tuε(x, 0) = g(x) in Ω,(1)where aε(x) ∈ (L∞(Ω))d×d is symmetric, uniformlyelliptic, and bounded. Here ε > 0 represents a smallscale in the problem, which characterizes the multi-scale nature of the tensor aε(x). We set either homo-geneous Dirichlet or periodic boundary conditions touniquely determine the solution for every ε > 0.1.1 Classical homogenizationAccording to classical homogenization theory, uεconverges to the solution u0 of the “homogenized”wave equation as ε→ 0,∂ttu0 −∇ · (a0∇u0) = F,where the homogenized tensor (or squared veloc-ity field) a0 can only rarely be computed explicitly.Thus, u0 approximates uε but only for short times.For longer times T ∼ ε−2, the homogenized solu-tion becomes increasingly inadequate, since it ne-glects microscopic dispersive effects that accumulateover time, as shown in Figure 1. Here we consider (1)in Ω = (−1, 1) with periodic boundary conditions, letu(x, 0) be a Gaussian pulse with zero initial velocity−1 −0.5 0 0.5 100.51T=2−1 −0.5 0 0.5 1T=100ref.hom.eff.Figure 1: Reference (ref.), homogenized (hom.)and effective (eff.) solution: short-time (left) andlong-time (right).and setaε =√2 + sin(2pixε)with ε =150. (2)The reference solution of (1)–(2) corresponds to adirect numerical simulation (DNS), where the micro-scale is fully resolved. After one revolution (T = 2),the homogenized and the DNS solution coincide. Af-ter fifty revolutions (T = 100), however, the DNSdisplays dispersive effects, which the homogenized so-lution fails to capture.1.2 Effective dispersive equationVarious formal asymptotic arguments were de-rived to elucidate that peculiar inherently dispersivelong-time behavior of waves propagating through astrongly heterogeneous periodic medium [1]. An ef-fective equation that captures those dispersive effectswas recently derived in [2] for the one-dimensionalcase when aε is ε-periodic:∂tt(ueff − ε2b∂xxueff)− a0∂xxueff = F. (3)Again, a0 denotes the homogenized effective coeffi-cient from classical homogenization theory and b > 0.As shown in Figure 1, uε and ueff essentially coincideboth at early and later times.2 FE Heterogeneous Multiscale MethodIn [3], the FE-HMM for elliptic [4] was extended tothe time dependent wave equation. It was shown toconverge to u0 at finite times, yet it failed to capturelong-time dispersive effects in the true solution. Toincorporate those dispersive effects, we not only needan effective bilinear form but also an effective innerproduct, akin to the weak formulation of (3). Bothrequire the numerical solutions of micro problems onsampling domains Iδ of size δ proportional to ε. Analternative HMM scheme, based on the finite differ-ence approximation of an effective flux, was proposedin [5].We now give a description of the algorithm: First,we generate a macro triangulation TH and choose anappropriate macro FE space S(Ω, TH). By macro wemean that H  ε is allowed. Within each macroelement K ∈ TH we choose a quadrature formula{xK,j , ωK,j}. The HMM solution uH is given by thefollowing variational problem:Find uH : [0, T ]→ S(Ω, TH) such that(∂ttuH , vH)H +BH(uH , vH) = (F, vH)for all vH ∈ S(Ω, TH) and,uH(0) = fH , ∂tuH(0) = gH in Ω,(4)where the initial data fH and gh are suitable approx-imations of f and g in S(Ω, TH), (·, ·)H and BH aredefined below. Since (·, ·)H is an inner product andthe bilinear form BH is elliptic and bounded, the FE-HMM is well defined.The FE-HMM inner product is defined by(vH , wH)H :=∑K,jωK,j|Iδ| ·∫Iδ(vH(xK,j) + vh(y))(wH(xK,j) + wh(y)) dy,and the FE-HMM bilinear form byBH(vH , wH) :=∑K,jωK,j|Iδ| ·∫Iδaε(y)(∇vH(xK,j)+∇vh)·(∇wH(xK,j)+∇wh) dy,where vh (resp. wh) is the solution of the micro prob-lemFind vh ∈ S(Iδ, Th) such that∫Iδaε(y)(∇vH(xK,j) +∇vh(y)) · ∇zh(y) dy = 0,for all zh ∈ S(Iδ, Th).(5)Here S(Iδ, Th) is a micro FE space on the samplingdomain Iδ with a micro triangulation Th.−1 −0.5 0 0.5 100.51T=100ref.HMM (see [3])HMM (new)Figure 2: Reference solution (ref.), FE-HMM from[3] and new FE-HMM.3 Numerical ExperimentsWe again apply our FE-HMM, defined in (4), to(1)–(2) as in Figure 1. We use cubic FE at the macro-and the micro-scale, with mesh sizes H = 1/75 andh = ε/20 = 1/1000. Note that linear or quadratic finiteelements could also be used. For time-stepping weuse a standard Leap-Frog scheme, with ∆t = H/10.As shown in Figure 2, the new FE-HMM succeeds incapturing, the long-time effects in the true solution.In contrast, the solution of the FE-HMM of [3] isunable to capture those dispersive effects, since thissolution was proven to converge to the homogenizedsolution, u0, as ε→ 0 on finite time intervals.References[1] F. Santosa and W. W. Symes, A Dispersive Ef-fective Medium for Wave Propagation in Peri-odic Composites, SIAM J. Appl. Math., 51, pp.984–1005.[2] A. Lamacz, Dispersive Effective Models forWaves in Heterogeneous Media, Math. ModelsMethods Appl. Sci., 21 (2011), pp. 1871–1899.[3] A. Abdulle and M. J. Grote, Finite ElementHeterogeneous Multiscale Method for the WaveEquation, Multiscale Model. Simul., 9 (2011),pp. 766–7921.[4] A. Abdulle, The Finite Element HeterogeneousMultiscale Method: a computational strategyfor multiscale PDEs, GAKUTO Internat. Seri.Math. Sci. Appl., 31 2009, pp. 133–182.[5] B. Engquist, H. Holst and O. Runborg , Multi-scale methods for wave propagation in hetero-geneous media, Comm. Math. Sci., 9 2011, pp.33–56.

Finite element heterogeneous multiscale method for the wave equation: dispersive effects

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Finite element heterogeneous multiscale method for the wave equation: dispersive effects

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