Mass Lumping for Constant Density Acoustics

Lumping 1 Mass Lumping for Constant Density Acoustics William W. Symes ABSTRACT Mass lumping provides an avenue for efficient time-stepping of time-dependent problems wit conforming finite element spatial discretization. Typical justifications for mass lumping use quadrature error estimates wic do not old for nonsmoot coefficients. In tis paper, I sow tat te mass-lumped semidiscrete system for te constant-density acoustic wave equation wit Q 1 elements exibits optimal order convergence even wen te coefficient bulk modulus is merely bounded and measurable, provided tat te rigt-and side possesses some smootness in time. INTRODUCTION Mass lumping is te process of replacing te mass matrix, occurring in a finite element formulation of a time-dependent or eigenvalue problem, wit a diagonal matrix. Suc diagonal replacement is essential in time-domain finite element approximation of wave propagation: time steps must be fine enoug to resolve te oscillations caracteristic of wave problems, and solution of a nontrivial linear system at every time step is proibitively expensive for large-scale D or 3D simulations te reader can take tis expense as a definition of large-scale. Coen 001 describes many instances of tis construction for spectral C 0 element approximation of acoustic, elastic, and electromagnetic waves for example. According to standard teory for conforming spectral FEM Coen, 001, lumping is possible witout loss of accuracy troug use of a Gauss-Lobatto quadrature rule wit nodes coinciding wit tose of te nodal conforming basis of te metod. Tis approac to mass-lumping relies for its justification on quadrature error estimates suc as tat presented by?, problem 4.1.5: one obtains an error of order O k in te mass integral over a regular familly of elements of diameter for sufficiently regular trial solutions, provided tat te quadrature rule is exact for polynomials of degree k 1, and te multiplier mass density or oter coefficient belongs to te space W k,. For lowest order k = 1, energy errors of order follow so long as te quadrature rule is exact for linear functions and te coefficient in te mass integral is of class W 1,. Tis justification fails wen te coefficients factors in te mass matrix integrand lack sufficient regularity, for instance exibit discontinuities along piecewise smoot interfaces. Te purpose of tis note is to sketc a different argument for lowest order masslumping, wic olds for coefficients tat are merely bounded and measurable, and wic

Symes preserves convergence wit optimal order O in energy. Te essential ingredient is sufficient smootness of te solution to yield optimal order convergence in energy of te Galerkin approximation. Tese two caracteristics are somewat in tension: lack of regularity in te coefficients limits te regularity of te solution. In tis note I study te simplest special case, te constant-density acoustic wave equation, for wic te required degree of regularity is easily establised for solutions wic are smoot in time. Smootness in time is a natural feature of some problems, suc as simulation of seismograms, in wic wavefields ave fixed temporal bandwidt. Time regularity induces just enoug spatial regularity in te constant-density case to give bot optimal order convergence in energy and te same order of convergence for te mass-lumped approximation. A by-product of te argument is te observation tat wit additional coefficient regularity a global LIpsitz bound, for example, te error in te lumped-matrix approximation is te same, to leading order, as in te consistent approximation. In tis report, I demonstrate tis stricter approximation property only for 1D problems. Te constant density acoustic wave equation is a very special case. Even for te variable density wave equation, coefficient discontinuities generally destroy te regularity underlying te optimal order error estimate, even for lowest-order elements, along wit te corresponding convergence rate for te mass-lumped system. It is owever possible to coose special elements for wic optimal order convergence is restored. For isolated discontinuities along interfaces, tis approac is known as immersed finite elements IFE Li and Ito, 006. Owadi as recently sown ow to construct suitable elements for very general self-adjoint elliptic second order problems, vastly generalizing te IFE metod Owadi and Zang, 007; Owadi and Zang, 008. It seems possible to base a mass-lumping strategy for more general wave problems on tese concepts. Tis report begins wit a review of te properties of weak solutions of te acoustic wave equation. In te next two sections I establis general properties of te Galerkin approximation, and derive an error estimate for for mass lumping wit Q 1 elements on a regular mes. Te more refined estimate for Lipsitz coefficients in 1D is te topic of te next section. I end wit a brief discussion of prospects for generalization. An Appendix reviews te standard error estimate for Galerkin metods applied to te wave equation. WEAK SOLUTIONS Te constant density acoustic wave equation relates te fields ux, t, ηx, and fx, t troug η u t u = f. 1 In te acoustic setting, u is excess pressure; η is related to te sound velocity c, bulk modulus κ, and density ρ by η = c = ρ/κ. Te rigt-and side f is eiter te divergence of a specific body force or te time rate of cange of a constitutive law defect. I assume tat u and f are causal, tat is, vanis for large negative times, and tat te wave equation 1 olds in te domain Ω R, Ω R d. Te field u vanises on te

Lumping 3 boundary Ω. Te weak form of 1 is: dt ηu, d dt φ + au, φ = dt f, φ, φ C 0R, H 1 0Ω, in wic, is te inner product in L Ω and a, is te usual energy form: av, w = v w 3 Weak finite energy causal solutions of 1 are known to exist and be uniquely determined by teir data in oter words, f, under very weak ypoteses on η and f: log η sould be bounded and measurable, and f sould be locally square integrable as a function of time wit values in L Ω. A standard argument using energy estimates and duality Lions, 197; Stolk, 000 establises te existence of a causal weak solution u C 1 R, H0Ω 1 of, satisfying for any α > 0 e[u]t 1 η u u t, t t t Ω + aut, ut 1 dτ e αt τ fτ. 4 4α Te symbol on te RHS of te preceding equation denotes te norm in L Ω: g = g, g. In tis generality, very little can be said about te convergence of finite element approximations, except tat it takes place: te cited references establis existence of weak solutions precisely by demonstrating convergence of a generic Galerkin approximation. Tis report exploits te improved approximation possible wen f is smooter in t. For instance, if f Hloc k R, L Ω, ten te same energy estimates sow tat te weak solution possesses additional regularity: u H k+1 loc R, L Ω. For k = 1, te strong form of te equation olds and it follows tat u C 0 R, H Ω. Te solution is tus smoot enoug for to establis te usual optimal order error estimates for P 1 or Q 1 Galerkin approximations. SEMIDISCRETE GALERKIN METHOD Coose finite dimensional subspaces V H 1 0Ω, define Galerkin approximation u C R, V by d dt ηu, φ + au, φ = f, φ, φ V, 5 plus appropriate initial conditions. Enumerate a basis {v j : 1 j N } of V, write N ux, t = Uj tv j x. 6 j=1

4 Symes Define mass, stiffness, and load matrices M, K, F by M i,j = ηv i, v j, K i,j = av i, v j, F j t = f, t, v j. 7 Ten te Galerkin system 5 is equivalent to te system of ODEs M d U dt + K U = F 8 I presume tat te family of subspaces V H 1 0Ω as te standard approximation properties: tere exists C 0, 0 < C C independent of so tat d C I M d C I, and a similar inequality olds for te matrix M 1 in wic η is replaces by 1; a similar elementwise bound olds: for any i, j, 0 M i,j C d ; for all g H Ω H 1 0Ω, tere is v V for wic g v H 1 0 Ω C g H Ω; for all v V, and v H Ω C 1 v H 1 0 Ω, v H 1 0 Ω C 1 v L Ω. Q 1 elements on a regular rectangular grid, for example, ave tese properties REF. Note tat te first of tese properties implies tat u = i U i v i C d U T U u L Ω C d U T U. 9 Also du e[u ] = 1 T M du + U T K U E[U ]. 10 dt dt An argument similar to tat used to establis 4 yields E[u ]t 1 4α dτ e αt τ F τ T M 1 F τ. 11 As remarked at te end of te last section, if f C 1 R, L Ω and f = 0 for t << 0, ten te solution u is smoot enoug tat te usual optimal order error estimate in energy olds: e[u u ]t C f t. 1 See te appendix for a derivation. If f as yet more square-integrable derivatives, corresponding estimates old for iger time derivatives of u u.

Lumping 5 MASS LUMPING FOR RECTANGULAR Q 1 ELEMENTS In tis section I suppose tat Ω is a rectangle, wic after translation could be taken to be Ω = [0, l 1 ] [0, l ]... [0, l d ]. Te usual reflection construction makes H 1 0Ω isomorpic to te odd subspace of H 1 perr d, were subscript per signifies multiple periodicity wit multiperiod l 1,..., l d. I will use tis identification witout comment in wat follows. Also note tat te odd multiperiodic extenstion of H 1 0Ω H Ω is a subspace of H perr d. Let = 1,..., d be te vector of cell side lengts, and V H 1 0Ω te corresponding space of Q 1 elements. For a multiindex α = α 1,...α d, let v α be te element of V satisfying v α α 1 1,..., α d d = 1; v α β 1 1,..., β d d = 0, α β N d. Set = {β Z d : β i 1, i = 1,..., d}. Ten M U α = β M α,α+βu α+β = β M α,α+βu α+β U α + M α U α, in wic M α = β M α,α+β is te diagonal α, α entry of te usual row-sum lumped mass matrix. Let Ũ be te solution of te mass-lumped Galerkin system M d Ũ dt + K Ũ = F. 13 For any solution of suc a system, an energy inequality similar to 11 olds: for any α > 0, du Ẽ[u ]t 1 T du M + U T K U t dt dt 1 dτ e αt τ F τ T M 1 F τ. 14 4α A little algebra sows tat te difference between Ũ and te Galerkin trajectory U satisfies [ ] M d dt U Ũ + K U Ũ = d U M α+β α,α+β d Uα. 15 α dt dt β According to 14, Ẽ[U Ũ ] 1 dτ e αt τ 4α

6 Symes [ β M, +β d U +β dt ] T [ d U τ M 1 M dt, +β β C d β dτ d U +β dt d U dt d U +β dt ] d U τ. dt, 16 in wic = max 1,..., d and C from ere on will stand for a constant depending on te coefficients and te geometry of te problem but not on. Define te unitary translation operator T β Since v α+β = Tβ v α, d Uα+β dt α d U α dt T β ux = ux + diagβ. v α = α d U α dt : L perr d L perr d by v α β d U α dt v α = T β I u t. Using te bounds on mass matrices described in te last section, obtain d U +β d U d U C d α+β d U α v dt dt dt dt α = C d T β I u t α L Ω L Ω. 17 Combining 16 and 17 gives Ẽ[U Ũ ] C β dτ T β I u t τ L Ω. 18 Suppose tat in addition to te previous ypoteses, f Hloc 3 R, L Ω. Ten 4 applied to te u/ t gives u u t t t C dτ f t τ, wence Ẽ[U Ũ ] C β dτ L Ω T β I u t τ L Ω L Ω + C dτ f t τ As noted earlier, under tese ypoteses on f, u/ t C 0 R, H Ω, and even C 0 R, HperR d, wit u t t t C dτ 3 f t τ 3. H Ω L Ω L Ω

Lumping 7 A straigtforward argument based on te trace teorem sows tat T β I u t t L Ω C dτ 3 f t τ 3 L Ω. Combining te last few inequalities, obtain e[ũ u] CẼ[U Ũ ] + e[u u] C dτ 3 f t τ 3 L Ω. 19 Tat is, te error in te lumped mass Galerkin approximation is te same order as te error in te consistent mass approximation. MASS LUMPING FOR P 1 ELEMENTS FOR DIMENSION 1 If te coefficient η as a bit more regularity, ten a more precise estimate tan 19 is possible. In tis section I describe te case d = 1; it seems likely tat similar estimates old for d > 1. In tis section, d = 1, Ω = [0, 1]. Define te element mes x i = i, i = 0,...N + 1 = 1. 0 Coose for V te piecewise linear functions wit nodes {x i }, tat is, te P1 or equally well Q1 element space. Te usual nodal basis is given by v j x = x x j 1, x j 1 x x j x j+1 x, x j x x j+1 0, else Ten K i,i = /, K i,i±1 = 1/, and K i,j = 0 for i j > 1, wit appropriate modifications at te endpoints. Te usual prescription for mass lumping is to replace M by te diagonal matrix M of its row sums. Note tat M is also tridiagonal. Tus it is always possible to write 1 M i, = M i, + ak i, + bd i,. D is te centered difference matrix, te tridiagonal matrix wit A little algebra sows tat D i,i±1 = ±1, D i,i = 0, i =,...N 1. a = M i,i+1 + M i,i 1 P +, b = M i,i+1 M i,i 1 P.

8 Symes Te lumped version of te Galerkin ODE system is M d Ũ dt + K Ũ = F. 3 A little more algebra sows tat te difference between its solution Ũ and te Galerkin trajectory U satisfies Since M d dt U Ũ + K U Ũ = P + K d U + P dt D d U dt. 4 M i,i = 1 te first of te following two inequalities follows: were So te lumped energy form 0 ηv i, η min M i,i η max i = 1,..., N, 5 η min I M η max I 6 η min = inf 0 x 1 ηx, η max = sup 0 x 1 ηx. Ẽ U = 1 du T dt M du dt + U T KU is positive definite, as is te consistent energy E U = 1 du T M du dt dt + U T K U. Note tat u = i U i v i E U = e[u], 7 and for suitable positive constants C and C. C E [U] Ẽ [U] C E [U] 8 Te standard energy estimate for causal solutions of system 8 see Appendix is E U t 1 e αt τ F τ T M 1 F τ 9 4α for any α > 0. A similar estimate olds for solutions of te lumped system; in particular, Ẽ U Ũ 1 [ t e αt τ P + K d U T α dt M P 1 + K d U dt

Lumping 9 + P D d U T dt M P ] 1 D d U 30 dt To estimate te first term on te RHS of 30, note first tat te equation satisfied by te 3rd time derivative of U, implies via 9 tat d 4 U dt 4 M d5 U dt 5, M d4 U 1 dt 4 4α On te oter and, te second time derivative satisfies M d4 U dt 4 + K d3 U dt 3 = d3 F dt 3, 31 d dτe αt τ 3 F dt, M 1 d3 F 3 dt 3. 3 + K d U dt = d F dt, 33 wence d 4 U dt, M d4 U = K d U d F, M 1 K d U d F 4 dt 4 dt dt dt dt 1 K d U dt, M 1 K d U d F dt dt, M 1 d F dt From tis point on, I will use C to denote a constant depending on upper and lower bounds for η and on an upper bound for te time t, but independent of. It follows from te matrix bounds 5 tat P ± is bounded by C in any weigted l norm wit diagonal weigt matrix. Tus P + K d U dt t, M 1 P + K d U dt t C 4 K d U dt t, M 1 K d U dt t C 4 K d U dt t, M 1 K d U dt t d C 4 F dt tt, M 1 d F d 3 dt t F + dt, M 1 d3 F. 35 3 dt 3 To bound te second term in 30, note tat except in te first and last rows, K satisfies K = D + D = D D +, D ± U k = ± 1 U k±1 U k, 34

10 Symes wereas Also note tat D = D+ + D. D ± T = D. Denote by Γ te operator norm of P in te weigted l norm wit weigt matrix M 1. According to 5, Γ = O in general. Ten P D d U T M P 1 D d U dt dt Γ D + + D d U T dt M D 1 + + D d U dt [ Γ D + d U T D + d U + D d U T D ] d U η min dt dt dt dt [ d Γ U T D D + d U d U T + D ] + D d U dt dt dt dt η min 4Γ d U T d U K 8Γ d U E η min dt dt η min dt d C Γ T F d M 1 F dt dt Combine 30, 35, and 36 to yield Ẽ [U Ũ ] C 4 + Γ Define ũ = i Ũ i v i ; ten 3 i=0 e[u ũ ] = E [U Ũ ] CẼ [U Ũ ] C 4 + Γ On te oter and, te rigt and side is C 4 + Γ i=0. d i T F d M 1 i F dt i 3 i=0 3 i f t i i=0. 36 dt i. 37 d i T F d M 1 i F dt i dt i 38 We distinguis two cases. First, suppose tat we know only tat log η is bounded and measurable. Ten Γ = O, and tis is sarp if η as discontinuities. Accordingly, te Galerkin error bound 1 combines wit 38 and 39 to yield 3 e[u ũ ] C i f t i. 40 39.

Lumping 11 Second, suppose tat η is Lipsitz continuous. Ten Γ = O 3, and we get instead e[u ũ ] = e[u u ] + O 4, 41 tat is, lumping te mass matrix does not cange te leading beaviour of te error at all. REFERENCES Ciarlet, P., 00, Te finite element metod for elliptic problems: SIAM. Coen, G. C., 001, Higer order numerical metods for transient wave equations: Springer. Li, Z. and K. Ito, 006, Te immersed interface metod: Numerical solutions of pdes involving interfaces and irregular domains: SIAM. Lions, J., 197, Nonomogeneous boundary value problems and applications: Springer Verlag. Owadi, H. and L. Zang, 007, Metric based upscaling: Communications in Pure and Applied Matematics, 60, 675 73. Owadi, H. and L. Zang, 008, Homogenization of te acoustic wave equation wit a continuum of scales: Computer Metods in Applied Mecanics and Engineering, 198 3-4, 397 406. Stolk, C., 000, On te modeling and inversion of seismic data: PD tesis, Universiteit Utrect..