# An interior penalty discontinuous Galerkin method for the time-domain acoustic-elastic wave interaction problem

In this paper, we consider numerical solutions of a time domain acoustic-elastic wave interaction problem which occurs between a bounded penetrable elastic body and a compressible inviscid fluid. It is also called the fluid-solid interaction problem. Firstly, an artificial boundary is introduced to transform the original transmission problem into a problem in a bounded region, and then its well-posed results are presented. We then employ a symmetric interior penalty discontinuous Galerkin method for the solution of the reduced interaction problem consisting of second-order wave equations. A priori error estimate in the energy norm is presented, and numerical results confirm the accuracy and validity of the proposed method.

## Authors

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• ### Discontinuous Galerkin methods for a dispersive wave hydro-sediment-morphodynamic model

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## 1 Introduction

In a time-domain fluid-solid interaction (FSI) problem, an incident acoustic wave is scattered by a bounded elastic obstacle immersed in a homogeneous, compressible and inviscid fluid. The problem of determining the scattered wave field plays prominent roles in many scientific and engineering areas, such as detecting and identifying submerged objects, geophysical exploration, medical imaging, seismology, and oceanography (such as [30]). Because of the difficulties of dealing with the time dependence, the FSI problem is usually studied based on a time-harmonic setting. Various efficient and accurate numerical methods have been developed to solve the time-harmonic FSI problem. Some of them provide proper guidance also for time-domain problems. Popular methods include the boundary integral equation (BIE) method, see [33, 46] and coupling methods such as the so-called coupled FEM-BEM method, see [47]. An artificial boundary or an absorbing layer ([43]) can be introduced to reduce the original unbounded problem to a bounded problem which could be solved using field equation solvers such as the finite element method.

The studies of this time-domain scattering problems now gain more and more attention since the time-domain model gives more information about the wave, more general material, and nonlinearity, see [7, 39]. There are relatively fewer mathematical analysis and numerical studies for the time-domain wave scattering problems. As to the numerical studies towards the time-domain wave scattering problems, the main challenge is how to handle the problem defined on the unbounded domain. Many approaches attempted to solve the time-domain problems numerically are developed, such as coupling of boundary element and finite element with different time quadratures, see [17, 21], time-domain boundary elemment methods  [26, 25], and to name a few. The authors of [27] gave the exact non-reflective boundary condition for the three dimensional time-domain wave scattering problem in 1995. For basic isotropy wave equation with constant coefficients, a planar PML method in one space direction designed for some particular domain is considered in [29]. In [8], the mathematical analysis of a time-domain DtN operator and the convergence analysis of the PML method for acoustic wave scattering were given. In [39], the time-domain exact nonreflective boundary conditions in both two and three dimensions were computed and analyzed. For the time-domain wave FSI problem, the rigorous mathematical study is still an open challenge, and related work include [19, 31, 22, 6]. Concerning its numerical solutions, the authors of  [32] applied time-domain boundary integral equations to time-domain FSI problem and analyzed the resulting nonlocal initial-boundary problem, motivated by the time-harmonic FSI problems. Coupling methods are also utilized in  [17, 18, 20, 23, 24] for solving the time-domain FSI problems.

Instead of numerical methods mentioned above, in this paper, we focus on using discontinuous Galerkin (DG) methods, which has natural advantages for dealing with the time-domain FSI problem, see [16]. The original DG methods were proposed for the numerical solution of hyperbolic neutron transport equations, as well as for discontinuities in some elliptic and parabolic problems. For second order wave equations, various DG methods have been proposed, which first reformulate the original wave problem to a first-order hyperbolic system, such as the local discontinuous Galerkin (LDG) methods in [13, 42, 9], and we refer to [10, 11, 34, 40] for a review of some other DG methods for the first order wave equations. The first DG method for the original second-order formulation of acoustic wave equation was proposed in [36], which is based on a nonsymmetric formulation. Here, we propose and analyze a symmetric interior penalty DG (SIPDG) method to solve the time-domain FSI problem. It is the same method used in [28] for the spatial discretization of the second-order scalar wave equation. Compared to the nonsymmetric formulation in [36], the symmetric discretization of the second-order form wave equation offers extra benefits such as a positive definite stiffness matrix and hence is free of any (unnecessary) damping. One can refer to [3] for details of the DG methods for second order equations. Finally we note that another DG formulation for wave equations in second order for is the energy based method proposed in [4] and extended to the coupled acoustic-elastic problem in [5].

The remainder of the paper is organized as follows. We first describe the original time-domain FSI problem in Section 2. Then in Section 3, the unbounded problem is reduced to a bounded initial-boundary value problem. In Section 4, we establish a priori error estimates for the IPDG solution of the reduced problem. Numerical experiments are presented in Section 5 to confirm the theoretical results, and finally, a conclusion is made in Section 6. For the sake of completeness, we provide the mathematical analysis towards well-posedness of the reduced problem introduced in Section 3 in the appendix.

## 2 Model

Here we study the same model as in [31], where the authors gave mathematical analysis from the aspect of integral equation method. The statement of the model is as follows. Suppose is a bounded domain of a elastic body with boundary , which is enclosed by the unbounded homogenous compressible inviscid fluid domain (see Figure 1), and a finite time interval . Given an incident wave , the scattered wave is generated by the induced solid in the fluid domain . In the fluid domain , the governing equations are the linearized Euler equation and the linearized equation of continuity

 ρ1∂v∂t+∇p=0,∂p∂t+c2ρ1div v=0,(x,t)∈Ω+×J, (2.1)

where is the velocity field, , and are the pressure, the density and the speed of sound in the fluid respectively.

For an irrational flow, the velocity potential s be the velocity potential satisfies:

 v=−∇φ,andp=ρ1∂φ∂t

Then, the wave equation for takes the form:

 1c2∂2φ∂t2−Δφ=0,(x,t)∈Ω+×J. (2.2)

The elastic wave in is described by the displacement field , which satisfies the dynamic linear elastic equation:

 ρ2∂2u∂t2−div(σ(u))=0,(x,t)∈Ω×Jσ(u)=λtr(ε(u))I+2με(u)ε(u)=12(∇u+(∇u)T). (2.3)

Here is the constant density of the elastic body, which is assumed to be homogeneous and isotropic with the Lamé constants and such that and .

The velocity potential in equation (2.2) and the elastic displacement field in equation (2.3) are coupled by the transmission conditions on , together with the homogeneous initial conditions, we can get an initial-boundary value problem:

 ρ2∂2u∂t2−div(σ(u)) =0, in~{}Ω×J, (2.4) 1c2∂2φ∂t2−Δφ =0, in~{}Ω+×J, (2.5) σ(u)n =−ρ1(∂φ∂t+∂φi∂t)n, on~{}Γ×J, (2.6) ∂φ∂n =−∂φi∂n−∂u∂t⋅n, on~{}Γ×J, (2.7) r12(∂φ∂n+∂φ∂t) →0as~{}r=|x|→∞, a.e.~{}t∈J, (2.8) u|t=0=∂u∂t∣∣∣t=0=0, x∈Ω~{}and~{}φ|t=0=∂φ∂t∣∣∣t=0=0, x∈Ω+. (2.9)

Here

is the unit outward vector on

from towards , is the given incident field and equation (2.8) is the radiation condition for .

Before the discussion, we first introduce the definitions of some relevant Sobolev spaces and norms. Let be the function space consisting of all square integrable functions over equipped with the norm

 ∥u∥0,Ω=(∫Ω|u(x,t)|2dx)1/2.

For , the standard Sobolev space is denoted by

 Hs(Ω)={Dαu∈L2(Ω)~{}for all~{}|α|≤s}

with the norm

 ∥u∥s,Ω=⎛⎝∫Ω∑|α|

and the trace functional space for under the inner produce

 ⟨u,v⟩Γ=∫Γuvγ.

It is clear to note that and are the dual space of and . We also denote and . For any , the Frobenius norm is defined as:

 ∥∇u∥F(Ω)=(2∑j=1∫Ω|∇uj|2dx)1/2.

And a simple calculation gives:

 ∥∇u∥2F(Ω)+∥div~{}u∥20,Ω≤C∥u∥21,Ω, (2.10)

where is a positive constant.

Furthermore, for we will make use of the Bochner space , endowed with the norm:

 ∥w(x,t)∥Lq(J;H1(Ω))=⎧⎪ ⎪⎨⎪ ⎪⎩(∫J∥w∥q1,Ω)1/q1≤q<∞,esssup0≤t≤T∥w∥1,Ωdtq=∞,

and the Bochner space , endowed with norm:

 ∥w(x,t)∥H1(J;Hs(Ω))=(∫J(∥w∥2s,Ω+∥wt∥2s,Ωdt))1/2.

## 3 The reduced problem in bounded domain

To reduce this exterior problem to a problem in a bounded domain, we impose the first order approximate boundary condition on the artificial boundary (see Figure 2):

 ∂φ∂n=−∂φ∂t,~{}on~{}ΓR×J,

where is the outward unit vector from on . Then the reduced problem on the bounded domain reads:

 ρ2∂2u∂t2−div(σ(u)) =0, ~{}in~{}Ω×J, (3.1) 1c2∂2φ∂t2−Δφ =0, ~{}in~{}ΩR×J, (3.2) σ(u)n =−ρ1(∂φ∂t+∂φi∂t)n, ~{}on~{}Γ×J, (3.3) ∂φ∂n =−∂φi∂n−∂u∂t⋅n, ~{}on~{}Γ×J, (3.4) ∂φ∂n =−∂φ∂t, ~{}on~{}ΓR×J, (3.5) u|t=0=∂u∂t∣∣∣t=0 =0,x∈Ω,and φ|t=0=∂φ∂t∣∣∣t=0=0,x∈ΩR. (3.6)

Then, we present the well-posedness and stability of the reduced problem in the follows.

### 3.1 Well-posedness and stability

First of all, we need to show that the reduced interaction problem in the bounded domain is well-posed and stable.

###### Theorem 3.1.

Suppose the incident wave , and . Then (3.1)-(3.6) has a unique solution satisfying:

 u∈L2(J;H1(Ω))∩H1(J;L2(Ω)),
 φ∈L2(J;H1(ΩR))∩H1(J;L2(ΩR)),

and we have the following stability estimate:

 maxt∈[0,T](∥ut∥0,Ω+∥∇u∥F(Ω)+∥∇⋅u∥0,Ω) ≤ C(∥∥φit∥∥2L1(J;H−1/2(Γ))+∥∥∇φi∥∥2L1(J;H−1/2(Γ))+maxt∈[0,T]∥∥φitt∥∥2−12,Γ +maxt∈[0,T]∥∥∇φit∥∥2−12,Γ+∥φittt∥2L1(J;H−1/2(Γ))+∥∇φitt∥2L1(J;H−1/2(Γ))), (3.7)

and

 maxt∈[0,T](∥φt∥0,ΩR+∥∇φ∥0,ΩR) ≤ C(∥∥φit∥∥2L1(J;H−1/2(Γ))+∥∥∇φi∥∥2L1(J;H−1/2(Γ))+maxt∈[0,T]∥∥φitt∥∥2−12,Γ (3.8) +maxt∈[0,T]∥∥∇φit∥∥2−12,Γ+∥φittt∥2L1(J;H−1/2(Γ))+∥∇φitt∥2L1(J;H−1/2(Γ))). (3.9)

Although different boundary conditions are considered to develop the reduced problem, the proof of above theorem 3.1 is analogous to the proof of Theorem 3.2 in [6]. We omit to present the proof of Theorem 3.1 in this section, and report the detailed proof of Theorem 3.1 in the appendix for the sake of completeness.

### 3.2 Variational formulation

The standard variational formulation of problem (3.1)-(3.6) is as follows: For given incident wave , find

 (u,φ)∈L2(J;H1(Ω))×L2(J;H1(ΩR)),

with

such that, :

 ρ2ρ1⟨utt,v⟩+1c2⟨φtt,ϕ⟩+a(u,φ;v,ϕ)=L(v,ϕ). (3.10)

with the initial conditions

 u|t=0=ut|t=0=0,φ|t=0=φt|t=0=0.

Here denotes the duality pairing between spaces and with the associated domain, and and are defined as:

 a(u,φ;v,ϕ) = (3.11) +(∇φ,∇ϕ)0,ΩR−(ut⋅n,ϕ)0,Γ+(φt,ϕ)0,ΓR, L(v,ϕ) = (∂φi∂n,ϕ)0,Γ−(φitn,v)0,Γ (3.12)

with the standard inner product in -space.

## 4 The IPDG method

### 4.1 Spaces, jumps and averages

Assume that with Lipschitz boundary is regularly divided into disjoint elements by mesh such that , where is a triangle or quadrilateral in 2D, or a tetrahedron or hexahedron in 3D. Similarly, The regular meshes partitions the fluid domain into disjoint elements such that . The diameter of element is denoted by , and is the mesh size given by .

The boundary of the elastic body , i.e., is approximated by the boundary edges of the subdivision: The set of boundary edges of is , since , the edges in are also the boundary edges of , so the sets of boundary edges of are , . We denote by and the set of interior edges of the subdivision and respectively.

For any real number , the broken Sobolev spaces and are defined as :

 Hm(Eh)={v∈L2(Ω):∀E∈Eh,v|E∈Hm(E)},

and

 Hm(Kh)={φ∈L2(ΩR):∀K∈Kh,φ|K∈Hm(K)},

with the broken Sobolev norms:

 ∥v∥m,Eh=⎛⎝∑E∈Eh∥v∥2m,E⎞⎠12,|φ∥m,Kh=⎛⎝∑K∈Kh∥φ∥2m,K⎞⎠12.

In particular, we also define the broken gradient seminorm:

 ∥∇v∥0,Eh=⎛⎝∑E∈Eh∥∇v∥20,E⎞⎠12, ∥∇φ∥0,Kh=⎛⎝∑K∈Kh∥∇φ∥20,K⎞⎠12.

If , the trace of along any side of each element is well defined. Let be the edge between the elements and , then the jump and average of a scalar function on are given by:

 {φ}=12(φ|K1+φ|K2),and[φ]=φ|K1nK1+φ|K2nK2,

respectively, where is the outward unit normal from to and likewise for . On the boundary edge or , we extend the definition: , where is the element that or .

Similarly, for a vector valued function , the jump and average of a vector function on are given by:

 {v}=12(v|E1+v|E2),%and[v]=v|E1⋅nE1+v|E2⋅nE2,

respectively, where is outward unit normal from to and likewise for . Similarly, on the boundary edge : , where is the element that .

### 4.2 Spatial discretization

For given partitions of , of and an approximation order , we can approximate the solution of (3.1)-(3.6) in the finite element subspaces

 Dk(Eh):={v∈L2(Ω): ∀E∈Eh, vi|E∈Pk(E),i=1,2},

and

 Dk(Kh):={φ∈L2(ΩR): ∀K∈Kh, φ|K∈Pk(K)},

of the broken Sobolev space and for . Here denotes the space of polynomials of degree at most .

Then we consider the following semidiscrete DG approximation of (3.1)-(3.6): find and such that

 ρ2ρ1⟨uhtt,v⟩+1c2⟨φhtt,ϕ⟩+ah(uh,φh;v,ϕ)=Lh(v,ϕ), (4.1)

holds for any and . Here, the discrete bilinear form and linear form are given by the IPDG discretization as:

 ah(uh,φh;v,ϕ) = ∑E∈Eh(λρ1(∇⋅uh,∇⋅v)0,E+2μρ1(ε(uh):ε(v))F(E)) (4.2) −∑e∈Γ1I∫e{σ(uh)⋅n}[v]dS−∑e∈Γ1I∫e{σ(v)⋅ν}[uh]dS +∑K∈Kh(∇φh,∇ϕ)0,K−∑e∈Γ2I∫e{∂φh∂n}[ϕ]dS−∑e∈Γ2I∫e{∂ϕ∂n}[φh]dS +∑e∈Γ2I∫eα|e|β[φh][ϕ]dS+∑e∈Γ1I∫eα|e|β[uh][v]dS+∑e∈Γh∫eφhtnvdS −∑e∈Γh∫euht⋅nϕdS+∑e∈ΓRh∫eφhtϕdS Lh(v,ϕ) = ∑e∈Γh∫e∂φi∂nϕdS−∑e∈Γh∫eφitnvdS. (4.3)

The interior penalty stabilization function penalizes the jumps over the edges of and , where is a positive parameter independent of the local mesh sizes. Here simply means the length of and we have

 |e|≤hE/K≤h,∀e∈∂E~{}or~{}∂K,

with , and when the method is superpenalized.

We define the space and . On and , we define the DG energy norm as

and

 ∥ϕ∥2h:=∑K∈Kh∥∇φ∥20,K+1h∑e∈Γh∪Γ2I∥[ϕ]∥20,e.

The consistency of the scheme is straightforward since the jumps at element boundaries vanishes when and for . Next we will discuss the property of the bilinear form . For more details about the IPDG method, one can refer to [28]

###### Lemma 4.1 (Coercivity).

There exists a positive constant independent of such that for all and

 ah(v,ϕ;v,ϕ) ≥ Ccoer⎛⎜⎝∑E∈Eh∥ε(v)∥2F(E)+∑e∈Γh∪Γ1I∥[v]∥20,e (4.4) +∑K∈Kh∥∇ϕ∥20,K+∑e∈Γh∪Γ2I∥[ϕ]∥20,e⎞⎟⎠.

Proof: By Cauchy-Schwarz inequality,

 ∑e∈Γ1I∫e{σ(v)⋅n}[v]dS ≤ ∑e∈Γ1I∥{σ(v)⋅n}∥0,e∥[v]∥0,e ≤ ∑e∈Γ1I∥{σ(v)⋅n}∥0,e(1|e|β)12−12∥[v]∥0,e.

Consider the average of the fluxes for an interior edge shared by and and apply the trace theorem, we have:

 ∥{σ(v)⋅n}∥0,e ≤ 12∥(σ(v)⋅n)|Ee1∥0,e+12∥(σ(v)⋅n)|Ee2∥0,e ≤ C+2h−12Ee1∥σ(v)∥F(Ee1)+C+2h−12Ee1∥σ(v)∥F(Ee2).

Assume and , we have:

 ∫e{σ(v)⋅n}[v]dS ≤ C+2|e|β2(h−12Ee1∥σ(v)∥F(Ee1)+h−12Ee1∥σ(v)∥F(Ee2))(1|e|β)12∥[v]∥0,e ≤ C+2(hβ2−12Ee1+hβ2−12Ee1)(∥σ(v)∥2F(Ee1)+∥σ(v)∥2F(Ee2))12(1|e|β)12∥[v]∥0,e ≤ C+(∥σ(v)∥2F(Ee1)+∥σ(v)∥2F(Ee2))12(1|e|β0)12∥[v]∥0,e.

We denote the maximum number of neighbors an element can have.

 ∑e∈Γ1I∫e{σ(v)⋅n}[v]dS ≤ ≤ C+√n0(∑E∈Eh∥σ(v)∥2F(E))12(∑e∈Γ1I1|e|β∥[v]∥20,e)12.

Using Young’s inequality, we have for

 ∑e∈Γ1I∫e{σ(v)⋅n}[v]dS ≤

Following a similar steps, we get:

 ∑e∈Γ2I∫e{∂ϕ∂n}[ϕ]dS ≤ δ2∑K∈Kh∥∇ϕ∥20,K+C2+n02δ∑e∈Γ2I1|e|β∥[ϕ]∥20,e.

Thus we obtain a lower bound for :

 ah(v,ϕ;v,ϕ) ≥ ∑E∈Eh(λρ1∥∇⋅v∥20,E+2μρ1∥ε(v)∥2F(E)−δ∥σ(v)∥2F(E)) +∑e∈Γ1Iαδ−C2+n0δ|e|β∥[v]∥20,e+∑K∈Kh(1−δ)∥∇ϕ∥20,K+∑e∈Γ2Iαδ−C2+n0δ|e|β∥[ϕ]∥20,e −C∑e∈Γh(∥ϕt∥20,e+∥v∥20,e−∥ϕ∥20,e−∥vt∥20,e)−C∑e∈ΓRh(∥ϕt∥20,e+∥ϕ∥20,e).

For any , using the Young’s inequality, similar as (38) we have

 ∥v∥20,e≤C∥vt∥20,e∥ϕ∥20,e≤C∥ϕt∥20,e.

Then, we have

 ah(v,ϕ;