On the convergence of the fictitious domain method for wave equation problems

Transcription

1 INSTITUT NATIONAL DE RECHERCHE EN INFORMATIQUE ET EN AUTOMATIQUE On te convergence of te fictitious omain meto for wave equation problems E. Bécace J. Roríguez C. Tsogka N 580 Janvier 006 Tème NUM ISSN ISRN INRIA/RR FR+ENG apport e recerce

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3 On te convergence of te fictitious omain meto for wave equation problems E. Bécace, J. Roríguez, C. Tsogka Tème NUM Systèmes numériques Projet POEMS Rapport e recerce n 580 Janvier pages Abstract: Tis paper eals wit te convergence analysis of te fictitious omain meto use for taking into account te Neumann bounary conition on te surface of a crack or more generally an object in te context of acoustic an elastic wave propagation. For bot types of waves we consier te first orer in time formulation of te problem known as mixe velocity-pressure formulation for acoustics an velocity-stress formulation for elastoynamics. Te convergence analysis for te iscrete problem epens on te mixe finite elements use. We consier ere two families of mixe finite elements tat are compatible wit mass lumping. Wen using te first one wic is less expensive an correspons to te coice mae in a previous paper, it is sown tat te fictitious omain meto oes not always converge. For te secon one a teoretical convergence analysis is presente in te acoustic case an numerical convergence is sown bot for acoustic an elastic waves. Key-wors: mixe finite elements, fictitious omain meto, acoustic waves, elastic waves, convergence POEMS, INRIA-Rocquencourt, BP 05, F-7853 Le Cesnay Céex eliane.becace@inria.fr POEMS, ENSTA, 3 boulevar Victor, Paris ceex 5, France jeronimo.roriguez@inria.fr University of Cicago, Dept. Matematics, 5734 University Avenue Cicago, IL 60637, USA tsogka@mat.ucicago.eu Unité e recerce INRIA Rocquencourt Domaine e Voluceau, Rocquencourt, BP 05, 7853 Le Cesnay Ceex France Télépone : Télécopie :

4 Sur la convergence es omaines fictifs pour es problèmes ones Résumé : Cet article concerne l analyse e convergence e la métoe es omaines fictifs utilisée pour prenre en compte une conition aux limites e Neumann sur la surface une fissure ou plus généralement un obstacle ans le context e la propagation ones acoustiques et élastiques. Pour les eux types ones, on consieère la formulation u premier orre en temps u problème, appelée formulation mixte vitesse-pression pour l acoustique et formulation mixte vitesse-contraintes pour l élastoynamique. L analyse e convergence u problème iscret épen es éléments finis mixtes utilisés. Nous consiérons ici eux familles éléments finis mixtes compatibles avec la conensation e masse. Quan on utilise le premier coix qui est moins couteux et correspon au coix fait ans un papier antérieur, il est montré que la métoe es omaines fictifs ne converge pas toujours. Pour le secon coix, une analyse téorique e convergence e la métoe est présentée ans le cas acoustique et la convergence numérique est montrée pour les eux cas acoustique et élastoynamique. Mots-clés : éléments finis mixtes, métoe es omaines fictifs, ones acoustiques, ones élastiques, convergence

5 CONVERGENCE OF THE FICTITIOUS DOMAIN METHOD 3 Contents Introuction 3 Te fictitious omain formulation of te iffraction problem. Te acoustic case 4. Te continuous problem Te semi-iscrete approximation Te fictitious omain meto using te Q iv Q 0 element 8 3. Position of te problem Numerical illustrations Te moifie element Q iv P isc 4. Presentation of te moifie element Some numerical illustrations of te fictitious omain meto using te moifie element Dispersion analysis of te moifie element Damping of te spurious moes Convergence analysis 7 5. Elliptic projection error Error estimates Numerical error estimates 4 7 Extension to te elastoynamic case 5 7. Te continuous problem Te approximate problem Numerical illustrations 9 9 Numerical error estimates 33 Introuction Tis work falls witin te more general framework of eveloping efficient numerical metos for approximating wave propagation in complex meia suc as anisotropic, eterogeneous meia wit cracks or objects of arbitrary sapes. We consier ere scattering of acoustic an elastic waves by perfect reflectors, i.e., objects or cracks wit a omogeneous Neumann bounary conition. To solve tese wave propagation problems in an efficient way we use a fictitious omain approac. Tis approac, also calle te omain embeing meto, consists in extening artificially te solution insie te object so tat te new omain of computation as a very simple sape typically a rectangle in D. To account for te bounary conition, a new auxiliary unknown, efine only at te bounary of te object, is introuce. Te solution of tis extene problem as now a singularity across te bounary of te object wic can be relate to te new unknown. Te main avantage of te meto is tat te mes for te solution on te enlarge omain can be cosen inepenently of te geometry of te object. In particular, one can use regular gris or structure meses wic allows for simple an efficient computations. Special interest as been given to tis approac as it as been sown to lea to efficient numerical metos for a large number of applications e.g [,, 6, 3, 5, 8, 9] an tese last years for time epenent wave propagation problems [0,, 3, 0, 5, ]. Te meto can be re-interprete in terms of optimization teory in wic case te auxiliary unknown appears as a Lagrange multiplier associate to te bounary conition viewe now as an equality constraint in te functional space. Tus te key point of te approac is tat it RR n 580

6 4 E. Bécace & J. Roríguez & C. Tsogka can be applie to essential type bounary conitions, i.e., conitions tat can be consiere as an equality constraint. To o so wit te free surface conition, te ual unknown velocity in te acoustic case an stress tensor in te elastic as to be one of te unknowns. Tis can be one by consiering eiter te ual formulation te formulation wit only one unknown, te ual one or te mixe ual primal formulation. In bot cases, te ual unknown is introuce an seeke for in te space Hiv in wic te Neumann bounary conition v n or σ n = 0 can be consiere as an equality constraint. In tis case, te Lagrange multiplier is noting but te jump of te primal unknown across te bounary of te object. For te approximation of te mixe formulation in te scalar acoustic case, in [4], te autors ave propose mixe finite elements, te so-calle Q iv k+ Q k elements, inspire by Néélec s secon family []. Tese elements are compatible wit mass lumping, an terefore allow for constructing explicit scemes in time. Te generalization of tose elements in te case of elastic waves was introuce in [3] for te velocity-stress formulation. Te above elements ave been use for solving iffraction problems in complex meia ue to te fictitious omain meto in [5]. A non stanar convergence analysis of te Q iv k+ Q k elements as been carrie out in [4] for teir scalar version an in [6] for teir elastoynamic vectorial version. However tis convergence analysis only eals wit te velocity-pressure resp. velocity-stress mixe problem witout object, tat is, it i not concern te convergence of te fictitious omain meto. In tis paper te convergence of te fictitious omain meto is analyze. Te scalar case is consiere first. Section presents te fictitious omain meto an its approximation. In section 3, we aress te question of its convergence wen using te Q iv Q 0 elements for approximating te volume unknown an it is sown in section 3. troug some numerical experiments tat te meto oes not always converge. Motivate by tis negative result, we introuce in section 4 a moifie finite element, te so-calle Q iv P isc. After illustrating wit numerical results te convergence for tis moifie element in section 4., we present its ispersion analysis in section 4.3. Tis analysis sows in particular te presence of spurious moes ue to te enricment of te approximation space for te primal unknown te pressure for te scalar case an te velocity for te elastic case. To get ri of tis non pysical part of te solution we propose to attenuate te spurious moes in section 4.4 by introucing a amping term in te equations. Section 5 is evote to te convergence analysis of te fictitious omain meto wen using te Q iv P isc element. Te teoretical orer of convergence is compare to te numerical one in section 6 for a particular object. In a secon part, te vectorial elastic case is consiere. Te convergence issues are te same as in te scalar case, but te proof of convergence is not a straigtforwar generalization of tis simplifie case, as explaine in section 7. However, numerical results sown in section 8 an te numerical orer of convergence obtaine in section 9 for a particular case, seem to confirm te conjecture tat te moifie element still converges in te elastic case. Te fictitious omain formulation of te iffraction problem. Te acoustic case. Te continuous problem We consier te iffraction of an acoustic wave by an object wit a Neumann type conition for te pressure fiel on its bounary Γ. Te object can be eiter an obstacle wit a close bounary or a crack wit an open bounary see Fig. but for te sake of clarity we will consier ere only tis secon configuration. Te omain of propagation is enote Ω wit an exterior bounary Σ see Fig. an we assume tat C = Ω Γ is a omain of simple geometry, typically a rectangle. Te propagation meium is assume to be anisotropic an te equation satisfie by te pressure fiel is te scalar wave equation. In orer to apply te fictitious omain meto to tis type of bounary conition it is INRIA

7 CONVERGENCE OF THE FICTITIOUS DOMAIN METHOD 5 Ω Γ n Σ Figure : Geometry of te problem. classical e.g. [4] to formulate te problem as a first-orer velocity-pressure system, Fin v, p : x, t Ω [0, T ] vx, t, px, t R R satisfying, wit te initial conitions, ρ p t iv v = f, in Ω, a A v t p = 0, in Ω, b v n = 0, on Γ, c p = 0, on Σ, { pt = 0 = p0, vt = 0 = v 0, were te unknowns p an v enote te pressure an te velocity fiel. Te scalar function ρ an te tensor A caracterize te propagation meium an f represents te external forces. Moreover, we assume tat ρ satisfies 0 < ρ ρx ρ + < +, an A is a secon orer symmetric positive tensor suc tat 0 < κ w Axw w ν w w 0. We also assume tat te support of te initial ata v 0, p 0 an te support of te source f o not intersect Γ, wic means tat 3 suppv 0 suppp 0 C \ Γ, suppft C \ Γ. Te natural variational formulation of tis problem woul be set in some functional spaces tat epen on te sape of te obstacle i.e., epen on Ω. More precisely, te classical variational formulation is, Fin vt, pt X 0 M satisfying, Av w x + ivwp x = 0, w X t 0, Ω Ω 4 ρpq x ivvq x = f, q, q M, t Ω Ω v, p /t=0 = v 0, p 0, t T RR n 580

8 6 E. Bécace & J. Roríguez & C. Tsogka were te functional spaces are efine as, X 0 Ω = {w Hiv; Ω, w n = 0, on Γ}, M = L Ω. Te well poseness of problem 4 results from classical teory on yperbolic PDEs, Teorem. Let f C 0 [0, T ], M, v 0 X 0 Ω, p 0 M satisfying 3. Ten, problem 4 as a unique solution v, p C [0, T ], L Ω C 0 [0, T ], X 0 Ω C [0, T ], M. Te fictitious omain formulation of tis problem consists in taking into account te bounary conition on Γ in a weak way, by introucing a Lagrange multiplier λ efine on Γ. Tis allows for working in functional spaces for te volume unknowns wic o not epen any more on te sape of te obstacle. Te fictitious omain formulation is ten te following, to simplify te notations, we still enote by vt, pt te new unknowns efine now in C 5 Fin vt, pt, λt X M G satisfying, t av, w + bw, p < w n, λ > Γ = 0, w X, t p, q ρ bv, q = f, q, q M, < v n, µ > Γ = 0, µ G, v, p /t=0 = v 0, p 0, were te functional spaces are now efine as, X= XC = Hiv; C, M = L C, G = H / 00 Γ, te bilinear forms as, 6 av, w = p, q η = bw, q = C C C Av w x, v, w X X, η p q x, p, q M M, ivwq x, w, q X M, an te bracket < w n, µ > Γ is te uality prouct between G an G. Note tat, ue to 3, uner assumptions of teorem., te ata also belong to, 7 f C 0 [0, T ], M, v 0 XC, p 0 M. In te following we will enote by, :=, te usual L C scalar prouct. Te well poseness of problem 5 follows from te tree following lemmas. Lemma. Existence We assume te ata v 0, p 0, f satisfy 7. Let v, p C [0, T ], L Ω C 0 [0, T ], X 0 Ω C [0, T ], M be te solution of problem 4. Ten: i p C 0 [0, T ], H Ω an one can efine λ = [p] Γ C 0 [0, T ], G, were [p] Γ enotes te jump of p across Γ. ii v C 0 [0, T ], XC an v, p, λ is a solution of 5. INRIA

9 CONVERGENCE OF THE FICTITIOUS DOMAIN METHOD 7 Proof. i If v, p C [0, T ], L Ω C 0 [0, T ], X 0 Ω C [0, T ], M is te solution of 4, te re-interpretation of te variational formulation sows tat it satisfies in particular -b in L Ω. Since t v C 0 [0, T ], L Ω we euce tat p C 0 [0, T ], L Ω an terefore p C 0 [0, T ], H Ω. It is ten possible to efine its trace on Γ an efine λ. ii v is in C 0 [0, T ], X 0 Ω an by efinition of X 0 Ω it satisfies v n = 0 on Γ wic implies in particular tat v n is continuous troug Γ tus v C 0 [0, T ], XC. Again using te re-interpretation of te variational formulation, one can see tat -a is satisfie in L Ω, -b in L Ω an - in H / Σ. We ten easily ceck tat it satisfies 5, by multiplying -a wit a function q M, -b wit a function w X, integrating by parts an using te efinition of λ. Lemma. Energy ientity. If v, p, λ is a solution of 5, te energy satisfies te following ientity, 8 E = av, v + p ρ, E t Lemma.3 Te following inf-sup conition is satisfie, = f, p. 9 k > 0, µ G, w X, < w n, µ > Γ k µ G w X. Proof. Tis result as been prove in [0] for a close obstacle Γ, an te corresponing space G = H / Γ. It is straigtforwar to aapt te proof to te present case, extening te open curve Γ to a close curve Γ, since for any function µ H / 00 Γ one can efine its extension by zero µ H / Γ. We ten apply te result for µ an using µ G e = µ G we obtain te result for µ. Teorem. Let f C 0 [0, T ], M, v 0 X, p 0 M satisfying 3. Problem 5 amits a unique solution v, p, λ C [0, T ], L C C 0 [0, T ], X C [0, T ], M C 0 [0, T ], G. Proof. Te existence follows from lemma.. Te energy ientity 8 implies te uniqueness of v, p an te uniqueness of λ is a consequence of te inf-sup conition 9. Remark On te regularity of te solution. For regular ata, one can expect more regularity on te solution. However, in general, te space regularity of te volumic part of te solution is at most, vt H ε iv, C, pt H ε C, ε > 0, an tis is obtaine for regular enoug ata an a regular geometry of te crack. Tis is ue to te fact tat te unknowns are efine on te wole omain C witout consiering te geometry of te obstacle. Te regularity in Ω i.e. outsie te obstacle is in general iger an epens on te geometry of te obstacle. In particular, for ata v 0, p 0, f satisfying 7, we ave for a close bounary: p /Ω t H Ω, λt H 3/ Γ for an open bounary, ue to te singular beavior near te tip of te crack [7] te solution beaves as r, r being ere te istance to te tip, we ave p /Ω t H 3/ ε Ω, λt H ε Γ, ε > 0. RR n 580

10 8 E. Bécace & J. Roríguez & C. Tsogka. Te semi-iscrete approximation For te approximation in space of tis problem, we introuce finite imensional spaces X X, M M an G H G satisfying te approximation properties, 0 lim inf v w 0 w X X = 0, v X, lim inf p q 0 q M M = 0, p M, lim inf λ µ H G = 0, λ G. H 0 µ H G H Te semi-iscrete problem is ten, Fin v t, p t, λ H t X M G H suc tat, t av, w + bw, p < w n, λ H > Γ = 0, w X, t p, q ρ bv, q = f, q, q M, v t = 0 = v,0, p t = 0 = p,0, < v n, µ H > Γ = 0, µ H G H, were v,0, p,0 X M is an approximation of te exact initial conition. Te question is : ow to coose te approximate spaces in orer to insure te convergence of v, p, λ H to v, p, λ? 3 Te fictitious omain meto using te Q iv Q 0 element 3. Position of te problem For te volumic unknowns, we introuce a regular mes T of te rectangular omain C compose of square elements of lengt. In [4], we introuce for te problem witout obstacle new mixe finite elements, te so-calle Q iv k+ Q k elements, inspire by Néélec s secon family []. Tese elements are compatible wit mass lumping, an terefore allow for constructing an explicit sceme in time. A non stanar convergence analysis of tese Q iv k+ Q k elements as been carrie out, sowing te convergence witout te fictitious omain meto. Our first coice for te approximation spaces of te problem wit an obstacle was naturally te lowest orer element Q iv Q 0 for te velocity an te pressure fiels, X = {w X / K T, w K Q Q }, M = M 0 wit M 0 = {q M / K T, q K Q 0 }. Te egrees of freeom for te velocity are escribe in Figure. For more etails on tis element we refer to [4]. Notice tat te velocity approximation space X contains te lower orer Raviart Tomas element, X RT = {w X / K T, w K P 0 P 0 }. For te approximation of te Lagrange multiplier, we introuce a mes of Γ compose of N curvilinear segments S j of lengt H j, an we set H = sup j H j. We assume tat tis mes is uniformly regular, 3 ν, 0 < ν, suc tat : j, j N, H j νh. INRIA

11 CONVERGENCE OF THE FICTITIOUS DOMAIN METHOD 9 p v t x v l z v r z v b x Figure : Degrees of freeom for te Q iv Q 0 mixe finite element. We ten coose te space of continuous linear piecewise functions: 4 G H = { ν H G / S j, j =,..., N, ν H Sj P }. Te spaces X, M 0, G H clearly satisfy te approximation properties 0. Tis coice wic seeme to us natural, since te convergence was proven witout obstacle, is te one tat was use in [5] for te more complex elastoynamic case. However we ave not been able to prove te convergence of te fictitious omain meto wit tese spaces. Te convergence analysis of te fictitious omain meto applie to oter problems [, 3, 0] sows tat convergence ols if a compatibility conition between te step sizes of te two meses is satisfie, 5 H α. We will sow in wat follows some numerical illustrations wic seem to inicate tat for some special configurations of obstacles, te meto oes not converge. Before sowing tese numerical results, let us briefly recall te main ifficulty of te convergence analysis in te case witout object. Introucing te linear operators, B : X M w Bw : M R q < Bw, q >= bw, q B : X M w B w : M R it is easy to verify tat te inclusion q < B w, q >= bw, q 6 KerB KerB, is not satisfie an tat furtermore te bilinear form a.,. is not coercive on KerB even if it is on KerB, so tat our problem oes not fit te classical mixe finite element teory cf. [8, 4]. It was owever possible to overcome tis ifficulty wen ealing wit te problem witout fictitious omain meto. Wen couple wit te fictitious omain meto, te same tecnique cannot be applie. RR n 580

12 0 E. Bécace & J. Roríguez & C. Tsogka 3. Numerical illustrations Te computational omain is te square [0, 0]mm [0, 0]mm compose by an omogeneous isotropic material wit ρ = 000Kgr/m 3 an A = I 0 9 Pa. It is excite by an initial conition on te pressure centere on x c, z c = 5, 5mm, r px, z, t = 0 = 0. F r 0, were F r is supporte in [0, ] an given by for r [0, ] F r = A 0 A cosπr + A cos3πr A 3 cos6πr, wit r = x x c, z z c t, r = r, r 0 = mm an A 0 = , A = , A = 0.48, A 3 = We consier a uniform mes of squares using a iscretization step = 0.05mm. Te time iscretization is one using a leap frog sceme wit te time step t cosen in suc a way tat te ratio t/ is equal to te maximal value tat ensures te stability. Perfectly matce layers are use to simulate a non boune omain in all te bounaries. Horizontal obstacle. In te first experiment we consier a plane orizontal crack 7 x, z = 5 + t, 5 mm, t [0, ], tat we iscretize using a uniform mes of step H = R. Te meto converges an we obtain goo results for reasonable values for te parameter R in te interval [0.75, 3]. In te first column of figure 3 we sow te results for R =.. At te beginning, te wave is totally reflecte by te bounary. Wen te wave front reac te tips of te crack, two scattere waves are create. INRIA

13 CONVERGENCE OF THE FICTITIOUS DOMAIN METHOD a t = µs b t = µs c t = µs t = µs e t = µs f t = µs Figure 3: Q iv Q 0. Isotropic meium. H/ =. RR n 580

14 E. Bécace & J. Roríguez & C. Tsogka Diagonal obstacle. In te secon experiment we treat a plane iagonal efect given by 8 x, z = 5 + 4t, + 4tmm, t [0, ], tat is, te same obstacle consiere in te previous paragrap rotate by π/4 raians wit respect to x c, z c, te center of te initial conition. As te meium is isotropic, te solution of te continuous problem is also a rotation of te solution consiering te orizontal crack. We iscretize te Lagrange multiplier using again a uniform mes of step H = R wit several values for te parameter R. However, tis time, te approximate solution oes not seem to converge towars te pysical solution see for instance te secon column of te figure 3 for R =.. Te incient wave is not completely reflecte but also transmitte troug te interface. 4 Te moifie element Q iv P isc 4. Presentation of te moifie element In section 3., we ave conjecture tat te ifficulty of te convergence analysis comes from te fact tat inclusion 6 is not satisfie. In orer to avoi tis problem we propose to moify te space M in suc a way tat 9 iv X M, wic implies 6 an coul simplify te analysis. Tat is wy we ave cosen to iscretize te pressure in te space 0 M = M wit M = {q M / K T, q K P K}. Consequently, we will ave tree egrees of freeom per element on te unknown p as sown in figure 4. Since M 0 M we ave obviously inf p q q M ρ so tat te approximation properties 0 are still satisfie. inf q 0 M 0 p q 0 ρ, z p p x p v t x v l z v r z v b x Figure 4: Degrees of freeom for te Q iv P isc mixe finite element. INRIA

15 CONVERGENCE OF THE FICTITIOUS DOMAIN METHOD 3 Remark Assuming 9 an tat te ensity is constant on eac element we ave tat w X = q := ρ ivw M. Introucing tis particular test function in te secon equation of we obtain p ivw x t ρ ivv ivw x = ρ fivw x. C C Deriving wit respect to time te first an tir equations of an using te last expression we euce tat our variational formulation implies te following secon orer formulation Fin v t, λ H t X G H suc tat w, µ H X G H, t Av w x + C v n µ H γ = 0, Γ v t = 0 = v,0, p t = 0 = p,0, C ρ ivw ivv x Γ C w n λ H γ = were λ H = λ H. Te nature of tis problem is close to te ones analyze in [3, 0]. t C ρ fivw x, 4. Some numerical illustrations of te fictitious omain meto using te moifie element Let us now sow some numerical illustrations of te beavior of te fictitious omain meto wit te new finite element space. Te numerical experiments tat we ave consiere are te same as in section 3. an will allow us to compare bot finite elements. Horizontal obstacle Once again we iscretize te orizontal crack efine by 7 using a uniform mes of step H = R. Te results obtaine wit te new mixe finite element Q iv P isc are similar to tose given by te Q iv Q 0 element. Te meto converges for reasonable values of te parameter R in te interval [0.75, 3]. In te first column of te figure 5 we can see te results for R =.. Diagonal obstacle We now consier te iagonal crack efine by te expression 8. We recall tat te continuous problem is a rotation of π/4 raians wit respect to te point x c, z c = 5, 5. Te Lagrange multiplier is again iscretize using an uniform mes of step H = R. Contrary to te results obtaine wit te element Q iv Q 0, te ones given by te moifie element Q iv P isc converge towars te pysical solution wen coosing reasonable values for te ratio H/. As we sow in te secon column of figure 5, tis time te incient wave is almost completely reflecte by te obstacle. Te scattere waves create by te tips of te crack are well approace. 4.3 Dispersion analysis of te moifie element It is useful an classical to perform a ispersion analysis in orer to specify te properties of a iscrete sceme in absence of obstacle e.g. [4, 9]. Tis consists in stuying te beavior of iscrete plane waves propagate wit te sceme. We will sow tat two spurious moes tat contaminate te iscrete solution appear, for te moifie element, because two new egrees of freeom per element were ae for te pressure. Assume tat our computational omain Ω = R is omogeneous an tat we use a uniform mes compose by squares of ege. In tis way, te egrees of freeom of te velocity fiel are place at te vertices of te squares, tat is, at te points x i, z j = i, j. We tus ave, v t x i,j, v b x i,j, v r z i,j, v l z i,j, i, j Z Z. RR n 580

16 4 E. Bécace & J. Roríguez & C. Tsogka a t = µs b t = µs c t = µs t = µs e t = µs f t = µs Figure 5: Q iv P isc. Isotropic meium. H/ =. INRIA

17 CONVERGENCE OF THE FICTITIOUS DOMAIN METHOD 5 Te egrees of freeom for te pressure are efine at te center of eac element, tat is, at te points x i+, z j+ = i + /, j + /. Te corresponing unknowns are enote by, p i+,j+, x p i+,j+, z p i+,j+, i, j Z Z. Te numerical sceme is tus given by, t p i+,j+ = ρ ρ t xp i+,j+ = t zp i+,j+ = {v t x i+,j v t x i,j + v b x i+,j+ v b x i,j+ } + {v r y i,j+ v r y i,j + v l y i+,j+ v l y i+,j }, { } vy l i+,j+ vy l i+,j vy r i,j+ + vy r i,j, ρ { } vx b i+,j+ vx b i,j+ vx t i+,j + vx t i,j, ρ were A 0 A 4 A 4 0 A A 4 A 4 A 4 A 4 A 0 A 4 A 4 0 A t vt x i,j t vb x i,j t vr z i,j t vl z i,j = B i,j B i,j B 3 i,j B 4 i,j B i,j = p i+,j+ p i,j+ z p i+,j+ z p i,j+, B i,j = B 3 i,j = B 4 i,j = p i+,j p i,j p i+,j+ p i+,j p i,j+ p i,j + z p i+,j z p i,j x p i+,j+ x p i+,j + x p i,j+ x p i,j,,,. We eliminate te unknowns associate to te velocity fiel using tese last expressions to write a system on p, x p, z p. Te next step of our analysis is to consier a plane wave solution for our problem at frequency ω an wit wave vector k = k x, k z, i.e., we assume tat 3 p i+,j+ x p i+,j+ z p i+,j+ [ = P exp i ωt k x x i+ P = [ p, x p, z p ] t, ] + k z z j+, Introucing tis expression in te numerical sceme we note tat te couple ω, P must be solution of te following eigenvalue problem 4 RR n 580 K P = ω k P, were k = k.

18 6 E. Bécace & J. Roríguez & C. Tsogka Te ermitian matrix K is given by K = K = K 3 = K = K 3 = K 33 = ρk IE ρk IE ρk ρk 4E 3ρk sin ρk A sin kx + C sin kz B cosk z sin kx sink z sin kx 3 sink x sin kz 3 C kz 3 sin A 3 sin kx kx + E sink x sink z + + D cosk x sin kz + ID IB sink x sin kz 3 sink z sin kx 3, D 3 cosk x sin kz sin kz, B 3 cosk z sin kx,,,, were 5 A = C C C C, C = C C C C, B = C C, D = C C, E = C. an te tensor C = C ij enotes te inverse of te tensor A, C = A. Since te matrix K is ermitian, tere are tree real eigenvalues wit tree ortogonal eigenvectors. Performing a Taylor expansion we obtain te following results. For te first couple enote ω pys, P pys we ave, c,pys = ω pys k = cos θc + sin θc + C sinθ cosθ + Ok, ρ V pys = [, 0, 0 ] t + Ok. Te pase velocity of te pysical numerical wave is a secon orer approximation of te pase velocity of te continuous wave. Te oter two solutions are spurious moes prouce by te introuction of te aitional egrees of freeom an are given by, c,spur = ω spur k = sin θ C C C + Ok, 3 ρ C V spur = [ 0,, 0 ] t + Ok, c,spur = ω spur k = cos θ C C C + Ok, 3 ρ C V spur = [ 0, 0, ] t + Ok. INRIA

19 CONVERGENCE OF THE FICTITIOUS DOMAIN METHOD Damping of te spurious moes As we ave seen in te previous section, te moifie element gives rise to some spurious moes. In tis section we propose a way to amp te amplitue of tese moes witout amping te pysical part, so tat tey o not perturb too muc te approximate solution. Te moifie space M can be ecompose as 6 M = M 0 M r, were M 0 is te space of piecewise constants an M r is its ortogonal complement for te L scalar prouct. Te space M r is compose of P iscontinuous functions wit vanising mean value per element. From te ispersion analysis, we observe tat te main components of te spurious moes te O part belong to M r. In orer to amp tis main part, we introuce te L ortogonal projection on M r, tat we enote by P M r, efine for any p M as, P M r p M r an P M r p, q = p, q, q M r. Te approximate problem wit amping consists in fining p, v M X suc tat 7 t av, w + bw, p < w n, λ H > Γ = 0, w X, t p, q ρ + P M r p, q β bv, q = f, q, q M, < v n, µ H > Γ = 0, µ H G H. In tis system β represents a amping parameter, wic is cosen as a positive constant in te applications. Te case β = 0 gives back te non-ampe problem, wile a strictly positive β correspons to a issipative problem. From te numerical point of view, it remains to efine a proceure in orer to coose tis parameter in an appropriate way. 5 Convergence analysis In tis section we sow te convergence of te fictitious omain meto using te moifie element wit amping. Te proof of convergence is compose of two main steps. One step consists in relating, using energy tecniques, error estimates for te evolution problem in terms of te ifference between te exact solution an its elliptic projection tat as to be cleverly efine. Te secon step amounts to analyzing te elliptic projection error an we will start wit tis point. 5. Elliptic projection error We efine te elliptic projection operator in te following way: v, p, λ X M G Π v, p, λ = v, p, λ H X M G H, were v, p, λ H X M G H is solution of p p, q b v v, q = 0, q M, 8 a v v, w + bw, p p < w n, λ H λ > Γ = 0, w X, RR n 580 < v v n, µ H > Γ = 0, µ H G H.

20 8 E. Bécace & J. Roríguez & C. Tsogka It is easy to sow tat tis problem is equivalent to efine first te couple v, λ H X G H satisfying 9 { a v v, w + iv v v, ivw < w n, λ λ H > Γ = 0, w X, an ten p M satisfying 30 p p, q = b v v, q, q M. < v v n, µ H > Γ = 0, µ H G H, Tis follows from te fact tat iv X M, so tat we can coose q = iv w. It is well known tat te convergence of v, λ H to v, λ is relate to te uniform iscrete inf-sup conition, 3 C > 0 inepenent of suc tat µ H G H, w X, < w n, µ H > Γ C w X µ H G. Teorem 5. If assumption 3 is satisfie, ten tere exists a constant α > 0 suc tat if H α, te uniform iscrete inf-sup conition 3 is satisfie for te couple of spaces X, G H. Proof. Te result as been sown in [0] for te couple of spaces X RT, G H. Te space X consiere ere clearly contains X RT cf. [4], wic sows tat te inf-sup conition is still true for te couple X, G H. Once te inf-sup conition is satisfie, tere is no ifficulty in applying te classical Babuska-Brezzi [8] teory an we obtain te elliptic projection estimates, Teorem 5. We assume tat H α were α is te constant given in teorem 5.. Te problem 8 as a unique solution p, v, λ H M X G H wic satisfies 3 v v X + λ λ G H C inf v w w X + inf λ µ H G, X µ H G H 33 p p M C inf p q M + inf v w q M w X + inf λ µ H G. X µ H G H Proof. Te error estimates for v v, λ λ H follow from te classical teory [8]. For p, we use 30 wic implies tat p q M p q M + iv v v, q M, an ten p p M inf p q q M M + v v X. Te following teorem sows tat only te projection of te pressure on M 0 tens to p, since te oter part tens to zero. Teorem 5.3 We assume tat H α were α is te constant given in teorem 5.. If p 0, pr = P M 0p, P M r p M 0 M r enote te ortogonal projections of p on M 0 an M r, we ave: 34 p 0 p M + p r M C w inf v w X X + inf λ µ H µ H G G + inf p q 0 M. H q 0 M 0 INRIA

21 CONVERGENCE OF THE FICTITIOUS DOMAIN METHOD 9 Proof. Using equation 30 for q 0 M 0, all te terms in M r isappear, p 0 p, q0 = b v v, q 0, q0. Wit te same arguments as previously we obtain, 0 p p M C iv v v + inf q 0 M 0 p q 0 M. Since p r = p p + p p 0, it suffices to combine bot estimates 33 an te first estimate of 34 to obtain te estimate on p r. Remark 3 Te elliptic projection of time epenent functions v, p, λ epens also on time an it is easy to ceck tat if v, p, λ C k [0, T ]; X M G, ten an Π v, p, λ C k [0, T ]; X M G H, k k t k Π v v, p, λ = Π t k, k p t k, k λ t k. We will also nee in te following error estimates on te time erivative of te elliptic projection, Corollaire 5. We assume tat H α were α is te constant given in teorem 5. an tat v, p, λ epen on time t an v, p, λ C k [0, T ]; X M G. Ten 35 t k v v X + t k λ H λ G C inf w t k w v X + inf µ H t k λ G, X µ H G H 36 k t p p M C inf w w t k v X + X inf q t k p M + inf µ H t k λ G q M µ H G H, 37 P M 0 k t p k t p M + P M r k t p M inf q 0 M 0 C inf w t k w v X+ X q 0 k t p M + inf µ H t k λ G µ H G H. Finally, classical approximation properties for finite elements give estimates wit respect to, for more regularity, i.e., for ε > 0: 38 inf v w w X C / ε v X H ε, v H ε iv, C, iv inf q M p q M C / ε p H ε, p H ε C, inf µ H G H λ µ H G CH / ε λ H ε Γ, λ H ε Γ. an tis finally implies te following error estimates wit respect to. RR n 580

22 0 E. Bécace & J. Roríguez & C. Tsogka Corollaire 5. Assume tat v, p, λ C k [0, T ]; H ε iv, C H ε C H ε Γ, ε > 0 an tat H α were α is te constant given in teorem 5.. Ten we ave te estimates 39 t k v v X + t k λ H λ G C ε t kv H + H ε ε t kλ H ε Γ, iv 40 P M 0 k t p t k p M + P M r t k p M C ε t k v + H ε iv C k t p + H H ε ε k C t λ H ε Γ. 5. Error estimates Te error estimates for te evolution problem are ten quite stanar. Tey follow from energy estimates. We efine te iscrete energy of te error as 4 E = p p ρ + a v v, v v. We first prove te energy ientity: Teorem 5.4 Te iscrete energy of te error satisfies te ientity, 4 were E t + β p r pr = F, C 43 F = t p p, p p ρ + a v v, v v p p, p p ρ a v v, v v + C β p r pr pr. Proof. Te ifference between te continuous problem 5 an te iscrete one 7 gives a problem satisfie by te error v v, p p, λ H λ = v v, p p, λ H λ + v v, p p, λ H λ H. Using te efinition of te elliptic projection operator, all te embarrassing terms isappear te terms wic woul give rise to ifficulty in obtaining te energy estimate, essentially tose not equivalent to L norms. It remains; q, w, µ H M X G H : t p p, q ρ βp r, q b v v, q = t p p, q ρ p p, q ρ, t a v v, w + bw, p p < ṽ n, λ H λ H > Γ = < v v n, µ H > Γ = 0. = t a v v, w a v v, w, INRIA

23 CONVERGENCE OF THE FICTITIOUS DOMAIN METHOD For q = p p an w = v v we obtain t p p ρ pr, p p β b v v, p p = = t p p, p p ρ p p, p p ρ, t a v v, v v + b v v, p p < v v n, λ H λ H > Γ = = t a v v, v v a v v, v v, < v v n, µ H > Γ = 0, µ H G H. Aing te first two equations an using te tir one gives 4. Te following proposition gives a boun of te iscrete energy of te error in terms of te elliptic projection error. Proposition 5. Te iscrete energy of te error satisfies te following estimate: 44 sup E t C E T 0 + C t p p M + t v v L C+ t T 0 were C is a constant inepenent of an β. C [ T p p M + v v L C s + C ] βp M r p M s, 0 Proof. Te proof is base on equality 4. Due to Young s inequality, te last term in 43 can be boune by: βpm r p P M r p p x β PM r p p β x + PM r C 4 p x. Simple computations ten give t E t C p p M t p p M + p p M + C C v v L C t v v L C + v v L C + 4 βp M r p M CE t t p p M + p p M + t v v L C + v v L C + 4 βp M r p M Integrating in time, we obtain t T RR n 580 E t E 0 + C sup E T t t p p M + t v v L C t T 0 T + p p M + v v L C s + βp M r 4 p M s. 0

24 E. Bécace & J. Roríguez & C. Tsogka We ten take te maximum on t T an apply Young s inequality to te first integral term : [ T sup E t C E 0 + C t p p M + t v v L C t T 0 wic easily implies 44. We can now give error estimates: + p p M + v v L C s ] + C T 0 βp M r p M s, Teorem 5.5 Let f C 0 [0, T ], M, v 0 X, p 0 M satisfying 3 an v, p, λ te solution of problem 5 efine in teorem.. Let v, p, λ H te solution of 7 wit initial ata v,0, p,0 te two first components of Π v 0, p 0, 0. Ten, we ave te error estimates 45 v v C 0 [0,T ];L C + p p C 0 [0,T ];M C + T v v C [0,T ];L C + p p C [0,T ];M + C T β L C P M r p C 0 [0,T ];M. Furtermore, if v, p C [0, T ]; L C M ten 46 v v C 0 [0,T ];X C v v C 0 [0,T ];X + C + T + β L C v v C [0,T ];L C + p p C [0,T ];M + β L C + T β L P M r p C [0,T ];M, 47 λ H λ C 0 [0,T ];G C λ H λ C 0 [0,T ];G + C + T v v C [0,T ];L C + p p C [0,T ];M + C T β L P M r p C [0,T ];M. Proof. First, notice tat te coice one for te approximate initial ata implies E 0 = 0. Ten inequality 44 easily implies 45. Tis gives an error estimate for v in te L norm. In orer to obtain an estimate in te X norm, we first state a similar result for te time erivative of te solution. Assuming tat te solution is one orer more regular, ten t v v C 0 [0,T ];L C + t p p C 0 [0,T ];M 48 C + T t v v C [0,T ];L C + t p p C [0,T ];M + C T β L C P M r t p C 0 [0,T ];M. Making te ifference between te secon equation of 7 an te secon equation of 5 we obtain ivv v, q = t p p, q ρ + βp M r p, q, q M. Tis implies ivv v M ivv v L C C t p p M + iv v v L C + βp M r p M, INRIA

25 CONVERGENCE OF THE FICTITIOUS DOMAIN METHOD 3 an terefore ivv v L C C t p p C 0 [0,T ];M + v v C 0 [0,T ];X + C β L C p p M + p p M + P M r p C 0 [0,T ];M. Ten using 45 an 48 we obtain 46. It remains to obtain te estimate for te Lagrange multiplier. Due to te uniform iscrete inf-sup conition 3 applie to λ H λ H, tere exists w X suc tat C λ H λ H G w X < w n, λ H λ H > Γ Tis implies tat λ H λ G C an terefore, using 45 an 48 we sow estimate 47. = < w n, λ H λ > Γ + < w n, λ λ H > Γ = a t v v, w + bw, p p + < w n, λ λ H > Γ. λ λ H GΓ + t v v L C + p p M, Finally, te following teorem gives te orer of convergence of te meto. Teorem 5.6 We make te same assumptions as in teorem 5.5. Ten we ave if v, p, λ C [0, T ]; H ε iv, C H ε C H ε Γ 49 v v C 0 [0,T ];L C + p p C 0 [0,T ];M C ε + H ε [ + T v C [0,T ];H + p ε iv C C [0,T ];H + λ ε C C [0,T ];H ε Γ + ] T β L C v C 0 [0,T ];H + p ε iv C C 0 [0,T ];H + λ ε C C 0 [0,T ];H ε Γ, if v, p, λ C [0, T ]; H ε iv, C H ε C H ε Γ 50 v v C 0 [0,T ];X C ε + H ε + T + β 3 L C v C [0,T ];H + p ε iv C C [0,T ];H + λ ε C C [0,T ];H ε Γ, 5 λ H λ C 0 [0,T ];G [ + T T β L C C ε + H ε v C [0,T ];H ε iv + p C C [0,T ];H + λ ε C C [0,T ];H ε Γ + ]. v C [0,T ];H + p ε iv C C [0,T ];H + λ ε C C [0,T ];H ε Γ Proof. Tis is a consequence of estimates on te evolution problem obtaine in 5.5 combine wit te estimates obtaine on te elliptic projection error in corollary 5.. RR n 580

26 4 E. Bécace & J. Roríguez & C. Tsogka 6 Numerical error estimates In tis section we are intereste in estimating numerically te orer of convergence of te meto. To o so, we consier solving te wave equation on a isk Ω IR wit omogeneous Neumann bounary conitions on its bounary Γ = Ω. Te geometry of te problem is presente in Figure 6. Γ Ω Σ Γ C Figure 6: Te geometry of te problem. On te left te initial omain of propagation Ω an on te rigt te extene omain, C, introuce by te fictitious omain formulation of te problem. To compute te solution we exten te unknowns in te omain of simple geometry C see Figure 6 an use te fictitious omain formulation 5 wit a zero force term f = 0 an te initial conitions given in section 3.. Te center of te initial conition, x c, z c = 5, 5mm, coincies wit te center of te isk Ω wose raius is R = 4mm. Te pysical properties of te material an size of te computational omain are te same as in section 3.. In practice to truncate te extene omain C, we surroun te computational omain by a perfectly matce absorbing layer moel PML, [7, ]. We remark tat te solution of tis problem is rotationally invariant because of te symmetry in te geometry an te initial conitions. We use tis symmetry in orer to compute a reference solution by solving an one imensional problem. More precisely, wen expresse in cylinrical coorinates, it is easy to see tat te solution of te two imensional problem, Ω being [0, R] [0, π], an were ϱ = 000Kgr/m 3 an a = 0 9 Pa, wit initial conitions, a v r t p r = 0, a v θ t p θ = 0, ϱ p t v r r r v r = 0, v r = 0, in [0, R] [0, π], in [0, R] [0, π], in [0, R] [0, π], on [r = R] [0, π], p 0 r, θ = 0.F r/r 0, v r = v θ = 0, epens only on r, i.e., v r r, θ = v r r, v θ = 0, pr, θ = pr. Tus, it can be euce by solving te following one-imensional problem, a v r t p r = 0, in [0, R], 5 ϱ p t v r r r v r = 0, in [0, R], v r = 0, for r = 0 an r = R, wit initial conitions, pr, t = 0 = 0.F r/r 0, v r t = 0, r = 0. INRIA

27 CONVERGENCE OF THE FICTITIOUS DOMAIN METHOD 5 To solve numerically te one imensional problem 5, we use piecewise constant functions for te iscretization of p r an continuous piecewise linear functions for v r. For te time iscretization a secon orer leap frog sceme is employe. In figure 7 we isplay te results of te numerical convergence analysis. Te reference solution in D, is obtaine on a fine gri wit a space iscretization step = /60mm. Te two imensional problem is solve wit four ifferent iscretizations using x = z = = /0, /0, /40 an /80mm. For eac iscretization we compute te ifference between te obtaine solution an te reference one. In figure 7 we isplay te logaritm of te error as a function of te logaritm of te iscretization step. Te rate of convergence is euce from te slope s of te line. We can remark tat te results obtaine numerically are sligtly better tan our teoretical preictions. Note owever, tat te estimate obtaine on te L [0, T ], Hiv norm of v is 0.48 wic inicates tat te teoretical estimations are optimal a sup v v X, s = t T b sup p p M, s = t T c sup λ λ G, s =.. t T Figure 7: Numerical error on v, p an λ versus te iscretization step. In figure 8 we isplay te same results but wit te norm of te error now compute in C = C/B b Γ, i.e., te omain C restricte from B b Γ, efine by { } 53 B b Γ = x C s.t. min x y b. y Γ We observe tat te convergence rate of te meto is iger, actually one approximately recovers te orer of convergence of te meto witout obstacle, ere O. Furtermore, we remarke numerically tat b = is te critical value, i.e., te convergence rate oes not cange for bigger values of b an it ecreases for b <. Tis agrees wit our intuition in te sense tat te elements tat we nee to remove are te ones in wic te solution as less regularity see remark, i.e., te elements tat ave non-zero intersection wit te bounary Γ. Finally, notice tat te rate of convergence on λ approximately is iger tan expecte /. We conjecture tat tis is ue to te close bounary an tat tis rate woul be lower for an open bounary see remark. 7 Extension to te elastoynamic case We consier in tis section te problem of elastic wave scattering by a crack. Te generalization of te fictitious omain meto to tis case was presente in [5]. For te space iscretization of te volume unknowns, wic are in tis case te stress tensor an te velocity fiel, an original finite element meto was propose an analyze in [6]. Te lower orer elements of tis family couple wit piecewise linear continuous elements for te surface unknown were use in [5]. Tis coice correspons to te vectorial analogue of te Q iv Q 0 element couple wit P continuous elements on te crack tat was iscusse in section 3. Also ere, te same questions an ifficulties arise. Namely, from te teoretical point of view, te convergence of te meto RR n 580

28 6 E. Bécace & J. Roríguez & C. Tsogka a sup t T v v Hiv C e. s = 0.8. b sup p p L C e. s =. 6 t T Figure 8: Numerical error on v, p an λ versus te iscretization step. Here we compute te norm of te error in te omain C wic is C restricte from B b Γ, i.e., Γ an its vicinity 53. was not prove an numerical examples inicate tat for some crack geometries te meto oes not converge. Te solution we propose is to use instea te moifie Q iv P isc element. As for te case witout crack, te teoretical convergence of te meto is not a straigtforwar generalization of te scalar case. Te main ifficulty in generalizing te results presente in section 5 is te proof of te inf-sup conition 3. Altoug tis conition was not sown teoretically we will sow numerical results tat inicate te convergence of te meto. Let us remark tat te inf-sup conition is sufficient an not necessary for te convergence of te meto. See [4] or [6] for examples were te convergence ols uner weaker conitions. We briefly present in te following te continuous elastoynamic problem, te finite elements use for te space iscretization an te numerical results obtaine in tis case. 7. Te continuous problem Consier te geometry given in Fig. an assume now tat te material filling Ω is an elastic soli. In tis case, an uner te assumption of small eformations, wave propagation is governe by te linear elastic wave equations, 54 Fin σ, v : x, t Ω [0, T ] σx, t, vx, t R 4 R satisfying, ρ v t iv σ = f, in Ω, a A σ t εv = 0, in Ω, b σn = 0, on Γ, c togeter wit te initial conitions, v = 0, on Σ, 55 { vt = 0 = v0, σt = 0 = σ 0. In 54, v is te velocity fiel an σ te stress tensor. Tis formulation is preferre to te classical isplacement formulation because te bounary conition on te crack is natural of Neumann type for te isplacement wile it becomes essential on σ an te fictitious omain approac can ten be followe. Note tat te couple σ, v plays te same role ere as te couple v, p in te scalar case. Te matrix A becomes now a fourt-orer symmetric efinite positive tensor. INRIA

29 CONVERGENCE OF THE FICTITIOUS DOMAIN METHOD 7 56 Te fictitious omain formulation, analogous to 5 for te scalar problem, is Fin σt, vt, λt X sym M G satisfying, t aσ, τ + bτ, v < τ n, λ > Γ = 0, τ X sym, t v, w ρ bσ, w = f, w, w M, < σn, µ > Γ = 0, µ G, σ, v /t=0 = σ 0, v 0, were te bilinear forms are efine by 6 wit te obvious canges. Te functional spaces are efine by X= XC = X X, M = M M, G = G G, te stress tensor belonging to te subspace of symmetric tensors in X, X sym = { σ X / asσ = 0 }, wit asσ efine in D by, asσ = σ σ. Te generalization of te existence an uniqueness results presente in section to tis problem is straigtforwar. 7. Te approximate problem Te semi-iscrete formulation For te approximation in space of tis problem, we introuce finite imensional spaces X sym X sym, M M an G H G satisfying te usual approximation properties, 57 inf inf τ τ sym X = 0, τ X sym, 0 τ X inf inf v v 0 v M M = 0, v M, inf H 0 inf µ µ H G = 0, µ G. µ H G H Te semi-iscrete problem is ten, 58 Fin σ t, v t, λ H t X sym M G H suc tat, t aσ, τ + bτ, v < τ n, λ H > Γ = 0, τ X sym, t v, w ρ bσ, w = f, w, w M, σ t = 0 = σ,0, v t = 0 = v,0, < σ n, µ H > Γ = 0, µ H G H, an were σ,0, p,0 X sym M is an approximation of te exact initial conition. RR n 580

30 8 E. Bécace & J. Roríguez & C. Tsogka Te two families of mixe finite elements. Following te same ieas as for te scalar problem, an original finite element for te elastoynamic system wic is compatible wit mass lumping was introuce in [6]. Te ifference wit te Q iv k+ Q k elements is ue to te symmetry of te stress tensor. Namely te lowest orer element in tis case is, X = { τ X / K T, τ K Q Q }, 59 { } X sym = τ X / asτ = 0, M = M 0 = M 0, M 0 being efine as for te scalar case. Anoter caracterization X sym is 60 X sym = { σ H Ω/σ K Q, K T an σ, σ Hiv, Ω/σ, σ K Q, K T }. Te egrees of freeom of te lowest orer element for te elastic problem are illustrate in Figure 9. We v σ t xx σ l zz σ xz σ r zz σ b xx Figure 9: Degrees of freeom for te mixe finite element efine by 59 obviously o not ave X sym = X X. Tis implies in particular tat te approximation space X sym oes not a priori contain te lowest orer Raviart Tomas element. Moreover te remarks mae in te scalar case on te coercivity an te non- inclusion property 6 remain true. Terefore, once again te assumptions of te classical mixe finite element teory cf. [8, 4] are not satisfie. However, convergence results an error estimates for te problem witout Lagrange multiplier were obtaine in [6]. We recall ere te approximation properties for te space X sym. Let τ X sym wit τ, τ H,0 H 0, see [6] for te efinition of tese spaces, an τ = τ H ten lim inf τ τ sym X = 0. 0 τ X Moreover, if τ, τ H, H, an τ H ten 6 inf τ X sym τ τ X C τ H, + τ H, + τ H. Te approximation of te Lagrange multiplier is one in te space G H = G, G being efine by 4. Te spaces M 0, G H satisfy te usual approximation properties 0. INRIA

31 CONVERGENCE OF THE FICTITIOUS DOMAIN METHOD 9 Te coice of te above approximation spaces seeme once again reasonable. However, as for te scalar case no teoretical convergence results were obtaine for te fictitious omain formulation. Moreover, numerical examples inicate tat for some crack geometries te meto oes not converge. As a response to tis rawback, te same approac as for te scalar case was followe. Te moifie element consists in tis case to iscretize te velocity in M = M. From te numerical point of view tis coice introuces spurious moes in te velocity wose amplitue is more important tan in te acoustic case. Te selective amping of te spurious moes is acieve using te same tools as in section 4.4. Namely, te secon equation of 58 is replace by t v, w ρ + P M rv, w β bσ, w = f, w, w M, were M r = M r an β is te amping parameter. Convergence issues. From te numerical point of view we observe tat te meto converges uner a compatibility conition of te form 5 between te two iscretization meses. From te teoretical point of view te convergence proof in tis case is not a straigtforwar generalization of te results in section 5. Te main sym ifficulty comes from te non stanar regularity require to obtain te approximation properties for X see 6. Inee, te maximal regularity in space of te stress tensor in te case of a omain wit a crack is σ H ε iv, C, an σ is symmetric. Tis regularity is not sufficient to obtain 6 an tus we cannot conclue. 8 Numerical illustrations We present in wat follows some numerical results tat illustrate te ifficulties relate wit te convergence of te meto tat we iscusse in te previous section. Te computational omain is again te square [0, 0] [0, 0] mm compose by an omogeneous isotropic material wit ensity an Lame coefficients given by 6 ρ = 000 Kgr/m 3, λ = Pa, µ = Pa. We introuce an initial conition on te velocity fiel centere on x c, z c = 5, 5mm, r vx, z, t = 0 = 0. F were F, r an r ave been efine in section 3. an r 0 =.5mm. We consier te iagonal crack parameterize by 8 on wic we impose a free surface bounary conition. We use a mes compose by squares wit a iscretization step = 0.05mm. For te time iscretization we use again a leap frog sceme wit a time step t suc tat te ratio t/ is equal to te maximal value tat guarantees te stability. Te crack is also iscretize using a regular mes wit H =.. Perfectly matce layers are use to boun te computational omain. r 0 r r, Results wit te Q iv Q 0 element. Wen we use te original finite element, te incient wave wic is ere a pressure wave is not completely reflecte by te obstacle but also transmitte as it can be clearly seen in figure 0-a. Similar results are also obtaine wen using oter ratios between H an an wen refining te meses. Tis inicates a lack of convergence as in te scalar case. RR n 580

32 30 E. Bécace & J. Roríguez & C. Tsogka a Q iv Q 0 b Q iv P isc Figure 0: Moulus of te velocity fiel at t =.5965 µs. Results wit te Q iv P isc element. Te solution obtaine wit te new finite element seems to converge towars te solution of te continuous problem. Te incient wave is completely reflecte by te obstacle an te scattere waves generate by te extremities of te crack are well approximate see figure 0-b. As in te scalar case, te enricment of te M approximation space introuces spurious moes in te solution. Altoug te amplitue of tese non-pysical waves goes to zero wit te size of te iscretization step, it is still significant for a usual coice of te iscretization parameters, typically corresponing to 0 points per wavelengt. Tese spurious moes are for example visible in te results presente in figure -a were we ave amplifie by a factor eigt te results of figure 0-b. In orer to stuy in more etail tese penomena we represent in figure te evolution in time of te moulus of te velocity fiel at tree points: x = x, z = 6.5, 3.5mm, x = x, z = 7.5,.5mm an x 3 = x 3, z 3 = 5, 0.5mm. Te first two points are centere wit respect to te crack, one bein an te oter in front of it. Te tir point is locate near te lower tip of te crack, were te spurious waves seem stronger. To etermine te spee of convergence of te meto, we use tree ifferent meses wit a space iscretization step of = 0.05, 0.05 an mm. Te results obtaine at te first point in front of te crack are alreay goo wit te coarse mes see figure -top left. Te amplitue of te spurious waves is very small wit respect to te amplitue of te incient t [0.5, ] µs, reflecte t [0.75, ] µs an first scattere wave t [.75, 4.5] µs. Te secon point is in te saow region were te amplitue of te pysical waves is about 5% smaller tan te amplitue of te incient wave see figure -mile left. Consequently te error is more visible. In te time interval t [0, ] µs we can see te part of te incient wave tat as been transmitte across te obstacle. Te amplitue of tose waves goes to zero wen we refine te mes. From t = to t = 4.5 µs we can see te first group of scattere waves. At te beginning an at te en of tis time interval we can see some oscillations tat come from te spurious moes introuce by te enricment of te velocity fiel iscretization space. Te error on tose amplitues is about % of te amplitue of te incient wave. At te tir point, te spurious penomena are muc stronger see figure -bottom left. Here, te amplitue of te incient an scattere waves is about 40% of te amplitue of te actual incient wave. As we can INRIA

33 CONVERGENCE OF THE FICTITIOUS DOMAIN METHOD 3 a Q iv P isc, β = 0 b Q iv P isc, β = 5ϱ0 6 Figure : Moulus of te velocity fiel 8 at t =.5965 µs. observe, comparing te solutions obtaine wit te ifferent meses, te meto converges very slowly see te time interval t [, 5.5] µs. Tat is te effect of te spurious moes create by te singularity on te tips of te crack see also figure -a. Tese spurious penomena can be reuce using a positive value of β, te amping coefficient. Let us analyze te results obtaine wit β = 5ϱ0 6. As we can see in figure -b te results seem to be better tan tose obtaine wit β = 0 see figure -a. Te signal obtaine at te first point x is very similar for bot coices see figure -top rigt. Te results for te secon point are also comparable. We remark tat some oscillations on te time intervals t [, 3] an t [4.5, 6] are remove wit te amping. It is in te signal recore on te tir point were te effect of te amping of te spurious moes is more efficient see figure -bottom rigt. Te oscillations observe wit β = 0 are completely remove an te meto gives a goo solution even wit te coarsest mes. Influence of te amping parameter on te solution. To illustrate te effect of te amping parameter β on te solution we present in figure 3 results obtaine on te coarsest gri for ifferent values of β. We ave mae te following observations: wen we o not use any amping, te solution is pollute by spurious moes. On te oter an, te amplitue of te transmitte non-pysical waves troug te crack increases as te value of te amping β increases. Tis is expecte because te limit case β + correspons to seeking te velocity in Q 0 an we know tat in tis case, te meto oes not converge. Tere is tus an optimal value for te amping parameter β to be etermine so as te spurious moes are ampe wile te transmitte non-pysical wave remains small. In te next section we etermine numerically te rate of convergence of te meto for a particular geometry an we iscuss a proceure to coose te value of β. RR n 580