A Nonconforming Finite Element Method for an Acoustic Fluid-Structure Interaction Problem

Susanne C. Brenner; Ayçıl Çeşmelioğlu; Jintao Cui; Li-Yeng Sung

doi:10.1515/cmam-2017-0050

Publicly Available Published by De Gruyter December 8, 2017

A Nonconforming Finite Element Method for an Acoustic Fluid-Structure Interaction Problem

Susanne C. Brenner , Ayçıl Çeşmelioğlu , Jintao Cui and Li-Yeng Sung

From the journal Computational Methods in Applied Mathematics

https://doi.org/10.1515/cmam-2017-0050

Abstract

We study a nonconforming finite element approximation of the vibration modes of an acoustic fluid-structure interaction. Displacement variables are used for both the fluid and the solid. The numerical scheme is based on an irrotational fluid displacement formulation and hence it is free of spurious eigenmodes. The method uses weakly continuous P1 vector fields for the fluid and classical piecewise linear elements for the solid, and it has O⁢(h2) convergence for the eigenvalues on properly graded meshes. The theoretical results are confirmed by numerical experiments.

Keywords: Nonconforming Finite Element Method; Fluid-Structure Interaction; Acoustic Fluid

MSC 2010: 65N25; 65N30; 74F10

1 Introduction

We consider the problem of determining the vibration modes of a linear elastic structure containing an acoustic (barotropic, inviscid and compressible) fluid.

Let ΩF and ΩS be the bounded polygonal domains in ℝ2 occupied by the fluid and the solid, respectively, as in Figure 1. We assume ΩF to be simply connected but not necessarily convex. Let ΓI denote the interface between the solid and the fluid and let 𝒏 be the unit normal vector pointing toward ΩS. The exterior boundary of the solid is the union of ΓD and ΓN: the structure is fixed along ΓD and free of stress along ΓN. Let 𝜼 denote the unit outward normal vector along ΓN.

Figure 1

Domains of fluid and solid.

We denote

Hs,γ⁢(div;ΩF)={𝒖∈[Hs⁢(ΩF)]2:∇⋅𝒖∈Hγ⁢(ΩF)}

and

∥𝒖∥Hs,γ⁢(div;ΩF)2=∥𝒖∥Hs⁢(ΩF)2+∥∇⋅𝒖∥Hγ⁢(ΩF)2.

The classical acoustic model for small-amplitude motions yields the following eigenvalue problem for the free vibration modes of the coupled system (see [21, 8]).

Problem.

Find λ≥0 and (𝒖,𝒘)∈H0,1⁢(div;ΩF)×[H1⁢(ΩS)]2, (𝒖,𝒘)≠(𝟎,𝟎), such that

(1.1)

(1.1a)c2⁢∇⁡(∇⋅𝒖)+λ⁢𝒖=𝟎in ΩF,

(1.1b)∇⋅𝝈⁢(𝒘)+λ⁢ρS⁢𝒘=𝟎in ΩS,

(1.1c)𝝈⁢(𝒘)⁢𝒏-(c2⁢ρF⁢∇⋅𝒖)⁢𝒏=𝟎on ΓI,

(1.1d)𝒘⋅𝒏-𝒖⋅𝒏=0on ΓI,

(1.1e)𝝈⁢(𝒘)⁢𝜼=𝟎on ΓN,

(1.1f)𝒘=𝟎on ΓD.

The physical meanings of the terms in equation (1.1) are as follows:

𝒖 (respectively, 𝒘) is the displacement of the fluid (respectively, solid) particle.
λ=ω is the frequency of the eigenmode.
ρF (respectively, ρS) is the density of fluid (respectively, solid).
ϵ⁢(𝒘)=12⁢(∇⁡𝒘+(∇⁡𝒘)T) is the strain tensor for the displacement 𝒘 of the solid.
𝝈⁢(𝒘)=2⁢μS⁢ϵ⁢(𝒘)+λS⁢(∇⋅𝒘)⁢𝜹 is the stress tensor, where 𝜹 is the 2×2 identity matrix. Here μS=M2⁢(1+ν) and λS=M⁢ν(1+ν)⁢(1-2⁢ν) are the Lamé constants, where M is the Young’s modulus and ν is the Poisson ratio of the solid.

Remark 1.1.

Equations (1.1a) and (1.1b) must be understood in the sense of distributions. Equations (1.1c) and (1.1d) hold in the sense of H-1/2⁢(ΓI). But since ∇⋅𝒖∈H1⁢(ΩF) and 𝒘⋅𝒏∈H1/2⁢(ΓI), both can be considered as equalities in L2⁢(ΓI).

Problem (1.1) is the displacement formulation for the acoustic fluid-structure interaction. An advantage of this formulation is that it is easy to maintain compatibility when displacement variables are used for both the fluid and the solid [23, 28]. On the other hand this is a non-elliptic formulation and hence the solution operator associated with the source problem is not a compact operator. Indeed it has an infinite-dimensional kernel (cf. Section 2.1). The functions in the infinite-dimensional eigenspace for λ=0 are pure rotations in the fluid that do not produce vibrations in the solid. Hence they are not physically relevant. But a naive discretization of the continuous problem would generate spurious eigenmodes that approximate these nonphysical eigenfunctions (cf. [21]). Such positive spurious eigenvalues pollute the approximation of the physical positive eigenvalues of (1.1).

Several approaches have been proposed to circumvent this drawback. A penalty method was introduced in [21]. It penalizes the curl-free condition so that the spurious eigenmodes are pushed towards higher frequencies and hence can be separated from the physical eigenvalues. An alternative approach [6, 2] uses standard piecewise linear elements for solid and the lowest order Raviart–Thomas elements for fluid. In this approach the discrete nonphysical 0 eigenvalue is isolated and all the positive discrete eigenvalues are spectrally correct. Optimal-order convergence on quasi-uniform meshes was established in [29]. This method has been adapted to deal with incompressible fluids in [29, 3], curved interfaces in [30] and to three dimensions in [5]. Related work can also be found in [18, 7, 4, 26, 25]. The convergence analysis in these papers requires sophisticated techniques due to the non-elliptic nature of the formulation.

In this paper, we introduce and analyze a nonconforming finite element method for (1.1) that is based on an elliptic formulation using only irrotational fluid displacement. Since all the pure rotational motions that are not physically relevant are excluded from the variational problem, the spurious eigenvalues of the discrete problem disappear. The method uses weakly continuous P1 vector fields for the fluid and classical piecewise linear elements for the solid. Furthermore, the convergence for the source problem is O⁢(h) and the convergence of the eigenvalues is O⁢(h2) (a well-known doubling phenomenon) on general domains, provided that three consistency terms involving the jumps of the vector fields across element boundaries in the fluid and on the fluid-solid interface are included in the discretization and properly graded meshes are used. The analysis of the numerical scheme is facilitated by its connections to the nonconforming finite element methods studied in [9, 11, 10] for the Maxwell’s equations. Furthermore, the compactness of the underlying operator greatly simplifies the analysis of the method as an eigensolver, which is similar to that in [12] for Maxwell’s eigenvalues.

The rest of the paper is organized as follows. In Section 2, we introduce a nonconforming finite element method for the source problem associated with (1.1). Optimal order convergence of the method is established in both the energy norm and the L2-norm. In Section 3, we present the convergence analysis for the nonconforming method as an eigensolver for (1.1). Results of a series of numerical experiments are reported in Section 4. We end the paper with a few concluding remarks in Section 5.

2 A Nonconforming Finite Element Method for the Source Problem

In this section, we introduce and analyze a nonconforming method on graded meshes for the source problem corresponding to (1.1).

2.1 The Source Problem

Let HΓD1⁢(ΩS) be the subspace of H1⁢(ΩS) whose members vanish on ΓD and

(2.1)𝒱={(𝒗,𝒘):𝒗∈H0,1⁢(div;ΩF),𝒘∈[HΓD1⁢(ΩS)]2,𝒗⋅𝒏=𝒘⋅𝒏⁢ on ⁢ΓI}.

Note that 𝒱 is a Hilbert space under the inner product defined by

((𝒗1,𝒘1),(𝒗2,𝒘2))𝒱=∫Ω[𝒗1⋅𝒗2+(∇⋅𝒗1)(∇⋅𝒗2)+𝒘1⋅𝒘2+∇𝒘1:∇𝒘2]dx.

The corresponding norm is denoted by ∥(𝒗,𝒘)∥. We also use |(𝒗,𝒘)| to denote the standard norm on [L2⁢(ΩF)]2×[L2⁢(ΩS)]2.

Given (𝒇,𝒈)∈[L2⁢(ΩF)]2×[L2⁢(ΩS)]2, the weak form of the source problem corresponding to (1.1) is to find (𝒖,𝒘)∈𝒱 such that

a⁢((𝒖,𝒘),(𝒗,𝒛))=b⁢((𝒇,𝒈),(𝒗,𝒛)) for all ⁢(𝒗,𝒛)∈𝒱,

where

(2.2)a((𝒖,𝒘),(𝒗,𝒛))=∫ΩF(ρFc2)(∇⋅𝒖)(∇⋅𝒗)dx+∫ΩS(𝝈(𝒘):ϵ(𝒛))dx,

(2.3)b⁢((𝒇,𝒈),(𝒗,𝒛))=∫ΩFρF⁢𝒇⋅𝒗⁢𝑑x+∫ΩSρS⁢𝒈⋅𝒛⁢𝑑x.

Note that

a⁢((𝒗1,𝒘1),(𝒗2,𝒘2))=0 for all ⁢(𝒗1,𝒘1)∈E0⁢ and ⁢(𝒗2,𝒘2)∈𝒱,

where E0 is the infinite-dimensional subspace of 𝒱 given by

E0=[∇×H01⁢(ΩF)]×{𝟎}.

Therefore this is a non-elliptic problem.

It is known (cf. [2, Lemma 2.3]) that the orthogonal complement of E0 in 𝒱 is the space 𝒱̊ defined by

𝒱̊={(𝒗,𝒘)∈𝒱:𝒗∈H⁢(curl0;ΩF)},

where

H⁢(curl0;ΩF)={𝒖=(u1u2)∈[L2⁢(ΩF)]2:∇×𝒖=∂⁡u2∂⁡x-∂⁡u1∂⁡y=0}.

We can avoid the complications of the non-elliptic weak problem by switching to the following source problem:

Problem.

Find (𝒖,𝒘)∈𝒱̊ such that

(2.4)a⁢((𝒖,𝒘),(𝒗,𝒛))=b⁢((𝒇,𝒈),(𝒗,𝒛)) for all ⁢(𝒗,𝒛)∈𝒱̊.

The well-posedness of (2.4) follows from Korn’s inequality, the compatibility condition (1.1d) and the following Friedrichs inequality [27, Section 4.4]:

∥𝒗∥L2⁢(ΩF)≤CΩF⁢(∥∇⋅𝒗∥L2⁢(ΩF)+∥𝒏⋅𝒗∥L2⁢(ΓI)) for all ⁢𝒗∈H⁢(div;ΩF)∩H⁢(curl0;ΩF).

Next we will show that the strong form of the reduced problem (2.4) is given by

(2.5)

(2.5a)-c2⁢∇⁡(∇⋅𝒖)=Q⁢𝒇in ΩF,

(2.5b)-∇⋅𝝈⁢(𝒘)=ρS⁢𝒈in ΩS,

(2.5c)𝝈⁢(𝒘)⁢𝒏-(ρF⁢c2⁢∇⋅𝒖)⁢𝒏=𝟎on ΓI,

(2.5d)𝒘⋅𝒏-𝒖⋅𝒏=0on ΓI,

(2.5e)𝝈⁢(𝒘)⁢𝜼=𝟎on ΓN,

(2.5f)𝒘=𝟎on ΓD,

where Q is the orthogonal projection from [L2⁢(ΩF)]2 onto H⁢(curl0;ΩF).

Indeed, as (𝒖,𝒘)∈𝒱̊, conditions (2.5d) and (2.5f) are automatically satisfied. Let (𝟎,𝒛) with 𝒛∈[D⁢(ΩS)]2 be a test function. Then

∫ΩS(𝝈(𝒘):ϵ(𝒛))dx=∫ΩSρS𝒈⋅𝒛dx,

which implies (2.5b)

Let 𝒗∈[D⁢(ΩF)]2 be a test function. Then 𝒗-Q⁢𝒗∈∇×H01⁢(ΩF). Hence (𝒗-Q⁢𝒗,𝟎)∈𝒱, which implies (Q⁢𝒗,𝟎)∈𝒱̊. It follows from (2.4) that

∫ΩF(ρF⁢c2)⁢(∇⋅𝒖)⁢(∇⋅𝒗)⁢𝑑x=∫ΩF(ρF⁢c2)⁢(∇⋅𝒖)⁢(∇⋅[Q⁢𝒗+(𝒗-Q⁢𝒗)])⁢𝑑x

=∫ΩF(ρF⁢c2)⁢(∇⋅𝒖)⁢(∇⋅Q⁢𝒗)⁢𝑑x

=∫ΩFρF⁢𝒇⋅(Q⁢𝒗)⁢𝑑x=∫ΩFρF⁢(Q⁢𝒇)⋅𝒗⁢𝑑x,

which implies (2.5a). Now, for any (𝒗,𝒛)∈𝒱̊, integration by parts in (2.4), together with (2.5a), (2.5b), (2.5d) and (2.5f), gives

(2.6)∫ΓI[(ρF⁢c2⁢∇⋅𝒖)⁢𝒏-𝝈⁢(𝒘)⁢𝒏]⋅𝒛⁢𝑑s+∫ΓN[𝝈⁢(𝒘)⁢𝜼]⋅𝒛⁢𝑑s=0,

which implies (2.5c) and (2.5e) in the sense of H-1/2⁢(ΓI) and H-1/2⁢(ΓN), respectively. In fact, we can take any 𝒛∈[HΓD1⁢(ΩS)]2 as a test function in (2.6) since there exists 𝒗∈H0,1⁢(div;ΩF) such that (𝒗,𝒛)∈𝒱̊. For instance, we can take 𝒗=∇⁡q, where q∈H1⁢(ΩF) is a function satisfying ∂⁡q∂⁡𝒏=𝒛⋅𝒏 on ΓI.

2.2 Regularity of the Source Problem

The main difficulty arises at the interface where on one side we have the solid and on the other side we have the fluid. The regularity of the coupled problem is closely related to the regularity of the elasticity problem and because of the curl-free condition, the regularity of the Laplacian. Away from the interface, we can think of the problems separately and use the well-known regularity results [19, 20] for the solid and the fluid. However, on the interface the solutions from both sides are coupled through the interface conditions. In the following we focus on the singularities at a corner on the fluid-solid interface ΓI.

First consider the homogeneous source problem defined as follows:

-c2⁢∇⁡(∇⋅𝒖)=0in ΩF,

-∇⋅𝝈⁢(𝒘)=𝟎in ΩS,

𝝈⁢(𝒘)⁢𝒏-(ρF⁢c2⁢∇⋅𝒖)⁢𝒏=𝟎on ΓI,

𝒘⋅𝒏-𝒖⋅𝒏=0on ΓI,

𝝈⁢(𝒘)⁢𝜼=𝟎on ΓN,

𝒘=𝟎on ΓD.

For a given corner c of ΓI, we switch to the polar coordinates (r,θ) at c as the origin. We represent a vector function 𝒗⁢(r,θ) in polar coordinates as 𝒗⁢(r,θ)=vr⁢𝒓^+vθ⁢𝜽^, where

𝒓^=(cos⁡(θ)sin⁡(θ)) and 𝜽^=(-sin⁡(θ)cos⁡(θ)).

In other words, with respect to the basis 𝒓^,𝜽^,

𝒗⁢(r,θ)=(vrvθ).

Assume that c is at the origin and the edges of ΓI emanating from c are defined by θ=-ωS and θ=ωS (cf. Figure 2).

$Figure 2 Corner c with angle 2⁢ωs{2\omega_{s}}.$

Figure 2

Corner c with angle 2⁢ωs.

We apply separation of variables on the whole coupled problem. For the elasticity part we obtain the following general solution (cf. [24, Section 3.1.3]):

𝒘⁢(r,θ)=rγ⁢(A⁢cos⁡((γ+1)⁢θ)+B⁢sin⁡((γ+1)⁢θ)+C⁢cos⁡((γ-1)⁢θ)+D⁢sin⁡((γ-1)⁢θ)B⁢cos⁡((γ+1)⁢θ)-A⁢sin⁡((γ+1)⁢θ)+Θ⁢D⁢cos⁡((γ-1)⁢θ)-Θ⁢C⁢sin⁡((γ-1)⁢θ)),

where

Θ:=μSλS+2⁢μS⁢(γ-1)-(γ+1)μSλS+2⁢μS⁢(γ+1)-(γ-1)

and the constants A,B,C,D are arbitrary. The curl-free condition implies that the problem in the fluid part is equivalent to the Laplacian problem which has the general solution of the following form:

𝒖⁢(r,θ)=rγ⁢(E⁢cos⁡((γ+1)⁢θ)+F⁢sin⁡((γ+1)⁢θ)F⁢cos⁡((γ+1)⁢θ)-E⁢sin⁡((γ+1)⁢θ)),

where E,F are arbitrary constants. When γ≠1 (a value that is not important for our purposes), the equation ∇⁡(∇⋅𝒖)=0 for this type of 𝒖 implies that ∇⋅𝒖=0. Therefore the first interface condition is the same as the traction boundary condition for the elasticity problem, i.e.,

𝝈⁢(𝒘)⁢𝒏=𝟎 on ⁢ΓI.

This is equivalent to the following set of equations:

cos⁡((γ+1)⁢ωS)⁢B+(γ-1)⁢(μS+λS)(μS+λS)⁢γ-(3⁢μS+λS)⁢cos⁡((γ-1)⁢ωS)⁢D=0,

sin⁡((γ+1)⁢ωS)⁢A+(γ-1)⁢(μS+λS)(μS+λS)⁢γ-(3⁢μS+λS)⁢sin⁡((γ-1)⁢ωS)⁢C=0,

cos⁡((γ+1)⁢ωS)⁢A+(γ+1)⁢(μS+λS)(μS+λS)⁢γ-(3⁢μS+λS)⁢cos⁡((γ-1)⁢ωS)⁢C=0,

sin⁡((γ+1)⁢ωS)⁢B+(γ+1)⁢(μS+λS)(μS+λS)⁢γ-(3⁢μS+λS)⁢sin⁡((γ-1)⁢ωS)⁢D=0.

Therefore, the coupled problem has a nontrivial solution when γ satisfies

(2.7)γ2⁢sin2⁡(2⁢ωS)-sin2⁡(2⁢γ⁢ωS)=0,

where ωS∈(0,π).

The second interface condition yields

B⁢cos⁡((γ+1)⁢ωS)+Θ⁢D⁢cos⁡((γ-1)⁢ωS)+F⁢cos⁡((γ+1)⁢ωF)=0,

A⁢sin⁡((γ+1)⁢ωS)+Θ⁢C⁢sin⁡((γ-1)⁢ωS)+E⁢sin⁡((γ+1)⁢ωF)=0,

where ωF=π-ωS.

Assume B=D=F=0, sin⁡((γ+1)⁢ωF)≠0 and solve for A and E in terms of C. This choice corresponds to γ⁢sin⁡(2⁢ωS)+sin⁡(2⁢γ⁢ωS)=0, one of the two factors of equation (2.7). We have

A=-((γ-1)⁢(μS+λS)(μS+λS)⁢γ-(3⁢μS+λS))⁢sin⁡((γ-1)⁢ωS)sin⁡((γ+1)⁢ωS)⁢C,

E=-2⁢(2⁢μS+λS)(μS+λS)⁢γ-(3⁢μS+λS)⁢sin⁡((γ-1)⁢ωS)sin⁡((γ+1)⁢ωF)⁢C.

Therefore, letting C=1, we have

(2.8)𝒘⁢(r,θ)=rγ⁢(-((γ-1)⁢(μS+λS)(μS+λS)⁢γ-(3⁢μS+λS))⁢sin⁡((γ-1)⁢ωS)sin⁡((γ+1)⁢ωS)⁢cos⁡((γ+1)⁢θ)+cos⁡((γ-1)⁢θ)((γ-1)⁢(μS+λS)(μS+λS)⁢γ-(3⁢μS+λS))⁢sin⁡((γ-1)⁢ωS)sin⁡((γ+1)⁢ωS)⁢sin⁡((γ+1)⁢θ)-Θ⁢sin⁡((γ-1)⁢θ))

and

(2.9)𝒖⁢(r,θ)=-rγ⁢(2⁢(2⁢μS+λS)(μS+λS)⁢γ-(3⁢μS+λS)⁢sin⁡((γ-1)⁢ωS)sin⁡((γ+1)⁢ωF)⁢cos⁡((γ+1)⁢θ)-2⁢(2⁢μS+λS)(μS+λS)⁢γ-(3⁢μS+λS)⁢sin⁡((γ-1)⁢ωS)sin⁡((γ+1)⁢ωF)⁢sin⁡((γ+1)⁢θ)).

A similar solution can be obtained if we assume A=C=E=0, cos⁡((γ+1)⁢ωF)≠0 and solve for B,F in terms of D. This choice corresponds to γ⁢sin⁡(2⁢ωS)-sin⁡(2⁢γ⁢ωS)=0, the other factor of equation (2.7).

Remark 2.1.

Equation (2.7) is precisely the transcendental equation that determines the singularity of the elasticity problem with pure traction boundary condition (cf. [20, Section 4.2] and [24, Section 4.2]). Note that it is independent of the Lamé constants.

Consider the transcendental equation

(2.10)γ2⁢sin2⁡(ω)-sin2⁡(γ⁢ω)=0,ω∈(0,2⁢π).

Define the angle ω0∈(π,3⁢π2) by

ω0=tan⁡(ω0).

The following lemma defines the singularities from the solid side.

Lemma 2.2 ([20, Lemma 3.3.1, Lemma 3.3.2]).

In the strip 0<Re⁢(z)<1, equation (2.10) has no root when ω<π, has only one single real root γ1 when π<ω<ω0 and has two distinct simple real roots γ1<γ2 when ω0<ω<2⁢π. Moreover, when ω>π,

12<γ1<πω<γ2.

Remark 2.3.

One interesting observation is that the singularities of (2.4) are completely determined by the solid side and they are independent of the Lamé constants of the solid (cf. Remark 2.1). Note that for the fluid-structure interaction problem we take ω=2⁢ωS, ωS∈(0,π). For example, when ωS=3⁢π4, the singularity index corresponding to equation γ⁢sin⁡(3⁢π2)+sin⁡(γ⁢3⁢π2)=0 is γ=0.544483661651611.

We conclude this subsection by the following lemma. The proof, which is based on the a priori estimates for the Poisson problem and the linear elasticity problem, is similar to the proof of [2, Theorem 2.5].

Lemma 2.4.

The solution (𝐮,𝐰) of (2.4) belongs to H1+γ1,1⁢(div;ΩF)×[H1+γ1⁢(ΩS)]2 and

(2.11){∥𝒖∥H1+γ1,1⁢(div;ΩF)+∥𝒘∥[H1+γ1⁢(ΩS)]2}≤C|(𝒇,𝒈)|,

where γ1 is as in Lemma 2.2.

2.3 A Nonconforming Finite Element Method

Let 𝒯h be a family of triangulations of ΩF∪ΩS such that every triangle is completely contained either in ΩF or ΩS. We denote by h=maxT∈𝒯h⁡hT the mesh parameter of 𝒯h, where hT is the diameter of the triangle T. The triangulation 𝒯h is graded around the corners c1,…,cL on ΓI with property that

(2.12)C1⁢hT≤h⁢Φμ⁢(T)≤C2⁢hT,

where

(2.13)Φμ⁢(T)=∏ℓ=1L|cℓ-cT|1-μℓ.

Here cT is the center of T and the positive constants C1 and C2 are independent of h. The vector μ contains the grading parameters μ1,…,μL chosen according to

(2.14)

μℓ=1if ⁢ωℓ≤π,

12<μℓ<γ1(<πωℓ)if ⁢ωℓ>π,

where ω1,…,ωL are the interior angles at the corners c1,…,cL of ΩS. (An example of such graded meshes for the domain in Figure 1 is given in Figure 4.)

Note that

(2.15)hT≲hfor all ⁢T∈𝒯h,

(2.16)hT≈h1/μℓif the corner cℓ is a vertex of T∈𝒯h.

Let ℰh,ΩF be the set of the edges in 𝒯h∩ΩF. Let ℰh,ΩFi denote set of the interior edges in 𝒯h∩ΩF and let ℰhΓI (respectively, ℰhΓN, ℰhΓD) denote the set of the edges on ΓI (respectively, ΓN, ΓD). We define

Nh⁢(ΩF)={𝒗∈[L2⁢(ΩF)]2:𝒗T=𝒗|T∈[P1⁢(T)]2,𝒗⁢ is continuous at the midpoint of any ⁢e∈ℰh,ΩFi}

and

Lh⁢(ΩS)={𝒛∈[H1⁢(ΩS)]2:𝒛T=𝒛|T∈[P1⁢(T)]2⁢ for all T∈𝒯h, T∈ΩS, and 𝒛=0 on any ⁢e∈ℰhΓD}.

Then we define

Vh={(𝒗,𝒛)∈Nh⁢(ΩF)×Lh⁢(ΩS):∫e(𝒗-𝒛)⋅𝒏⁢𝑑s=0⁢ on any ⁢e∈ℰhΓI}.

Let ∇h⋅ and ∇h be the piecewise div and grad operators defined by

(∇h⋅𝒗)T=∇⋅(𝒗T)for all ⁢T∈𝒯h,

(∇h⁡𝒗)T=∇⁡(𝒗T)for all ⁢T∈𝒯h.

Let e∈ℰhi be shared by the two triangles Te,1,Te,2∈𝒯h and let 𝒏1 (respectively, 𝒏2) be the unit normal of e pointing towards the outside of Te,1 (respectively, Te,2). We define, on e,

(2.17)

(2.17a)[[𝒏×𝒗]]=𝒏1×(𝒗Te,1|e)+𝒏2×(𝒗Te,2|e),

(2.17b)[[𝒏⋅𝒗]]=𝒏1⋅(𝒗Te,1|e)+𝒏2⋅(𝒗Te,2|e).

The nonconforming finite element method for (2.4) is:

Problem.

Find (𝒖h,𝒘h)∈Vh such that

(2.18)ah⁢((𝒖h,𝒘h),(𝒗,𝒛))=b⁢((𝒇,𝒈),(𝒗,𝒛)) for all ⁢(𝒗,𝒛)∈Vh,

where

ah⁢((𝒖,𝒘),(𝒗,𝒛))=∫ΩF(ρF⁢c2)⁢(∇h⋅𝒖)⁢(∇h⋅𝒗)⁢𝑑x+h-2⁢∫ΩF(ρF⁢c2)⁢(∇h×𝒖)⁢(∇h×𝒗)⁢𝑑x

+∫ΩS(𝝈(𝒘):ϵ(𝒛))dx+∑e∈ℰh,ΩFi1|e|∫e(ρFc2)[[𝒏⋅𝒖]][[𝒏⋅𝒗]]ds

+∑e∈ℰh,ΩFi1|e|⁢∫e(ρF⁢c2)⁢[[𝒏×𝒖]]⁢[[𝒏×𝒗]]⁢𝑑s

(2.19)+∑e∈ℰhΓI1|e|⁢∫e(ρF⁢c2)⁢(𝒏⋅(𝒖-𝒘))⁢(𝒏⋅(𝒗-𝒛))⁢𝑑s.

Here ϵh⁢(𝒛)=12⁢(∇h⁡𝒛+(∇h⁡𝒛)T) and 𝝈h⁢(𝝍)=2⁢μS⁢ϵh⁢(𝒘)+λS⁢(∇h⋅𝒘)⁢𝜹 are the discrete versions of the strain and stress tensors defined in Section 1. From now on we will no longer keep track of the dependence on the constants ρF, ρS, c, μS and λS.

For any s>12, we define a weak interpolation operator ΠT:[Hs⁢(T)]2→[P1⁢(T)]2 as follows:

(2.20)(ΠT⁢𝜻)⁢(mej)=1|ej|⁢∫ej𝜻⁢𝑑s for ⁢1≤j≤3,

where e1, e2 and e3 are the edges of T, and me and |e| denote the midpoint and the length of the edge e. It follows immediately from (2.20), the midpoint rule and Green’s theorem that

(2.21)∫T∇×(ΠT⁢𝜻)⁢𝑑x=∫T∇×𝜻⁢𝑑x,

(2.22)∫T∇⋅(ΠT⁢𝜻)⁢𝑑x=∫T∇⋅𝜻⁢𝑑x.

Furthermore, the operator ΠT satisfies a standard error estimate [16]:

(2.23)∥𝜻-ΠT⁢𝜻∥L2⁢(T)+hTmin⁡(s,1)⁢|𝜻-ΠT⁢𝜻|Hmin⁡(s,1)⁢(T)≤CT⁢hTs⁢|𝜻|Hs⁢(T)

for all 𝜻∈[Hs⁢(T)]2 and s∈(12,2], where the positive constant CT depends on the minimum angle of T.

We can define a global interpolation operator Πh,ΩF:Hs,1⁢(div;ΩF)→Nh by piecing together the local interpolation operators:

(Πh,ΩF⁢𝒗)T=ΠT⁢𝒗T for all ⁢T∈𝒯h,ΩF.

A suitable Vh-interpolation operator 𝐈h:{H1+α,1⁢(div;ΩF)×[H1+α⁢(ΩS)]2}∩𝒱→Vh(α>12) is defined in the following way:

(2.24)𝐈h⁢(𝒖,𝒘)|T={(Lh,ΩS⁢𝒘)|Tif ⁢T⊂ΩS,(Πh,ΩF⁢𝒖)|Tif ⁢T⊂ΩF⁢and⁢∂⁡T∩ΓI=∅,(Πh,ΩF^⁢𝒖)|Tif ⁢T⊂ΩF⁢and⁢∂⁡T∩ΓI≠∅,

where Lh,ΩS⁢𝒘 is the Lagrange interpolant of 𝒘 in Lh⁢(ΩS), and (Πh,ΩF^⁢𝒖)|T is the function in Nh⁢(ΩF) such that

(2.25)((Πh,ΩF^⁢𝒖)×𝒏)|e=((Πh,ΩF⁢𝒖)×𝒏)|e for all ⁢e∈ℰh,ΩF

and

(2.26)(Πh,ΩF^⁢𝒖⋅𝒏)⁢(me)={((Πh,ΩF⁢𝒖)⋅𝒏)⁢(me)if ⁢e⊄ΓI,1|e|⁢∫e((Lh,ΩS⁢𝒘)⋅𝒏)|Te⁢d⁢sif ⁢e⊂ΓI,

with Te the triangle contained in ΩS such that ∂⁡T∩∂⁡Te=e, and me is the midpoint of e.

2.4 Preliminary Error Estimates

We will measure the discretization error in the mesh-dependent energy norm ∥(⋅,⋅)∥h defined by

∥(𝒗,𝒛)∥h2=∥∇h⋅𝒗∥L2⁢(ΩF)2+h-2⁢∥∇h×𝒗∥L2⁢(ΩF)2+∥∇h⁡𝒛∥L2⁢(ΩS)2+∑e∈ℰh,ΩFi1|e|⁢∥[[𝒏⋅𝒗]]∥L2⁢(e)2

(2.27)+∑e∈ℰh,ΩFi1|e|⁢∥[[𝒏×𝒗]]∥L2⁢(e)2+∑e∈ℰhΓI1|e|⁢∥𝒏⋅(𝒗-𝒛)∥L2⁢(e)2.

Observe that ah⁢((⋅,⋅),(⋅,⋅)) is bounded by the energy norm, i.e.,

(2.28)|ah⁢((ϕ,𝝍),(𝒗,𝒛))|≤C⁢∥(ϕ,𝝍)∥h⁢∥(𝒗,𝒛)∥h

for all (ϕ,𝝍),(𝒗,𝒛)∈{H1+α,1⁢(div;ΩF)×[H1+α⁢(ΩS)]2}∩𝒱̊+Vh.

Due to Korn’s inequality, ah⁢((⋅,⋅),(⋅,⋅)) is also coercive with respect to ∥(⋅,⋅)∥h, i.e.,

(2.29)ah⁢((𝒗,𝒛),(𝒗,𝒛))≥γ⁢∥(𝒗,𝒛)∥h2

for all (𝒗,𝒛)∈{H1+α,1⁢(div;ΩF)×[H1+α⁢(ΩS)]2}∩𝒱̊+Vh.

Lemma 2.5.

Let (𝐮,𝐰)∈V̊ be the solution of (2.4), and let (𝐮h,𝐰h) satisfy the discrete problem (2.18). It holds that

∥(𝒖,𝒘)-(𝒖h,𝒘h)∥h≤(1+Cγ)⁢inf(𝒗,𝒛)∈Vh⁡∥(𝒖,𝒘)-(𝒗,𝒛)∥h+1γ⁢sup(ϕ,𝝍)∈Vh∖{(𝟎,𝟎)}⁡ah⁢((𝒖,𝒘)-(𝒖h,𝒘h),(ϕ,𝝍))∥(ϕ,𝝍)∥h.

Proof.

Let (𝒗,𝒛)∈Vh be arbitrary. It follows from (2.28) and (2.29) that

∥(𝒖,𝒘)-(𝒖h,𝒘h)∥h≤∥(𝒖,𝒘)-(𝒗,𝒛)∥h+∥(𝒗,𝒛)-(𝒖h,𝒘h)∥h

≤∥(𝒖,𝒘)-(𝒗,𝒛)∥h+1γ⁢sup(ϕ,𝝍)∈Vh∖{(𝟎,𝟎)}⁡ah⁢((𝒗,𝒛)-(𝒖h,𝒘h),(ϕ,𝝍))∥(ϕ,𝝍)∥h

≤(1+Cγ)⁢∥(𝒖,𝒘)-(𝒗,𝒛)∥h+1γ⁢sup(ϕ,𝝍)∈Vh∖{(𝟎,𝟎)}⁡ah⁢((𝒖,𝒘)-(𝒖h,𝒘h),(ϕ,𝝍))∥(ϕ,𝝍)∥h,

as desired. ∎

The following lemma is useful for the error analysis.

Lemma 2.6.

For any (𝐮,𝐰)∈{H1+α,1⁢(div;ΩF)×[H1+α⁢(ΩS)]2}∩V, T∈ΩF such that ∂⁡T∩ΓI≠∅ and Te∈ΩS with ∂⁡T∩∂⁡Te=e, there exists a positive constant C (depending on the minimum angle of T and Te) such that

(2.30)∑T∈ΩF∂⁡T∩ΓI≠∅∥ΠT⁢𝒖-(Πh,ΩF^⁢𝒖)|T∥H⁢(div;T)2≤C⁢h2⁢|(𝒇,𝒈)|2.

Proof.

Let ϕe,n be the basis vectors of 𝒩h⁢(ΩF) corresponding to the normal vectors on e. It follows from (2.1), (2.20) and (2.26) that

(2.31)(Πh,ΩF⁢𝒖-Πh,ΩF^⁢𝒖)|T=|1|e|⁢∫e(𝒖-Lh,ΩS⁢𝒘)⋅𝒏⁢𝑑s|⁢ϕe,n=|1|e|⁢∫e(𝒘-Lh,ΩS⁢𝒘)⋅𝒏⁢𝑑s|⁢ϕe,n.

Therefore,

Note that

∫T|∇⋅ϕe,n|2⁢𝑑x=1|T|⁢|∫T∇⋅ϕe,n⁢𝑑x|2=1|T|⁢|∫∂⁡Tϕe,n⋅𝒏T⁢𝑑s|2=|e|2|T|.

Hence ∥ϕe,n∥H⁢(div;T)2≤C for a constant C that depends on the minimum angle of T.

If Te is away from the reentrant corners on ΓI, by using (2.11), the trace theorem (with scaling) and standard interpolation results [15, 13], we have

|1|e|⁢∫e(𝒘-Lh,ΩS⁢𝒘)⋅𝒏⁢𝑑s|2≤1|e|⁢∥𝒘-Lh,ΩS⁢𝒘∥[L2⁢(e)]22

≤C⁢{hT-2⁢∥𝒘-Lh,ΩS⁢𝒘∥[L2⁢(Te)]22+|𝒘-Lh,ΩS⁢𝒘|[H1⁢(Te)]22}

(2.33)≤C⁢hT2⁢|𝒘|[H2⁢(Te)]22.

For triangles Te inside the neighborhood of a reentrant corner but not touching the corner, it follows from (2.12) that

|1|e|∫e[(𝒘-Lh,ΩS𝒘)⋅𝒏ds|2≤1|e|∥𝒘-Lh,ΩS𝒘∥[L2⁢(e)]22

≤C⁢{hT-2⁢∥𝒘-Lh,ΩS⁢𝒘∥[L2⁢(Te)]22+|𝒘-Lh,ΩS⁢𝒘|[H1⁢(Te)]22}

≤C⁢hT2⁢|𝒘|[H2⁢(Te)]22

≤C⁢h2⁢[Φμ⁢(T)]2⁢|𝒘|[H2⁢(Te)]22

(2.34)≤C⁢h2⁢|𝒘|[H2⁢(Te)]22,

where we have applied the fact that

(2.35)∫01r2⁢(1-μℓ)⁢r2⁢(γ1-2)⁢r⁢𝑑r<∞ if ⁢μℓ<γ1.

For triangles Te touching a reentrant corner, we can apply an interpolation error estimate for the fractional order Sobolev spaces [17] together with (2.11), (2.16), (2.14) and the trace theorem with scaling to obtain

|1|e|∫e[(𝒘-Lh,ΩS𝒘)⋅𝒏ds|2≤1|e|∥𝒘-Lh,ΩS𝒘∥[L2⁢(e)]22

≤C⁢{hT-2⁢∥𝒘-Lh,ΩS⁢𝒘∥[L2⁢(Te)]22+|𝒘-Lh,ΩS⁢𝒘|[H1⁢(Te)]22}

(2.36)≤C⁢hT2⁢μℓ⁢|𝒘|[H1+μℓ⁢(Te)]22.

Estimate (2.30) then follows from the regularity result (2.11) and the summation of (2.32)–(2.36) over T∈ΩF with ∂⁡T∩ΓI≠∅. ∎

Lemma 2.7.

Let (𝐮,𝐰)∈V̊ be the solution of (2.4). We have

(2.37)∑T∈ΩF∂⁡T∩ΓI=∅∥∇⋅(𝒖-ΠT𝒖)∥L2⁢(T)2+∑T∈ΩF∂⁡T∩ΓI≠∅∥∇⋅(𝒖-(Πh,ΩF^𝒖)|T)∥L2⁢(T)2≤Ch2|(𝒇,𝒈)|.

Proof.

Observe that (2.22) implies

∇⋅(ΠT⁢𝒖)=ΠT0⁢(∇⋅𝒖),

where ΠT0 is the orthogonal projection from L2⁢(Ω) onto the space of piecewise constant functions with respect to T∈𝒯h,ΩF. Hence, by using a standard interpolation error estimate [15, 13], we have

(2.38)∑T∈ΩF∂⁡T∩ΓI=∅∥∇⋅(𝒖-ΠT⁢𝒖)∥L2⁢(T)2=∑T∈ΩF∂⁡T∩ΓI=∅∥∇⋅𝒖-Πh0⁢(∇⋅𝒖)∥L2⁢(T)2≤C⁢h2⁢|∇⋅𝒖|H1⁢(ΩF)2.

It then follows from (2.38), Lemma 2.6 and the regularity result (2.11) that

∑T∈ΩF∂⁡T∩ΓI≠∅∥∇⋅(𝒖-(Πh,ΩF^𝒖)|T)∥L2⁢(T)2≤∑T∈ΩF∂⁡T∩ΓI≠∅∥∇⋅(𝒖-ΠT𝒖)∥L2⁢(T)2+∑T∈ΩF∂⁡T∩ΓI≠∅∥∇⋅(ΠT𝒖-(Πh,ΩF^𝒖)|T)∥L2⁢(T)2

(2.39)≤C⁢{h2⁢|∇⋅𝒖|H1⁢(ΩF)2+h2⁢|(𝒇,𝒈)|2}.

We conclude the proof of (2.37) by combining (2.11), (2.38) and (2.39). ∎

Note that the interpolation operator Πh,ΩF defined on ℰh,ΩFi is identical with the one employed in [9]. The following result can be proved similarly as in [9, Lemma 5.2]:

Lemma 2.8.

Let (𝐮,𝐰)∈V̊ be the solution of (2.4). We have

(2.40)∑e∈ℰh,ΩFi1|e|⁢∥[[𝒖-Πh,ΩF⁢𝒖]]∥L2⁢(e)2≤C⁢h2⁢|(𝒇,𝒈)|2.

Proof.

Let e∈ℰh,ΩFi and let 𝒯e be the set of the triangles in 𝒯h having e as an edge. We have

(2.41)1|e|⁢∥[[𝒖-Πh,ΩF⁢𝒖]]∥L2⁢(e)2≲∑T∈𝒯e|e|-1⁢∥𝒖-ΠT⁢𝒖∥L2⁢(e)2.

If T∈𝒯e is away from the reentrant corners on ΓI, then we have, by the trace theorem (with scaling), (2.15) and (2.23) (with s=2),

(2.42)|e|-1⁢∥𝒖-ΠT⁢𝒖∥L2⁢(e)2≲hT-2⁢∥𝒖-ΠT⁢𝒖∥L2⁢(T)2+|𝒖-ΠT⁢𝒖|H1⁢(T)2≤C⁢hT2⁢|𝒖|H2⁢(T)2.

Note that if T∈𝒯e is inside the neighborhood of a reentrant corner but not touching the corner, estimate (2.42) also holds in view of (2.35). On the other hand, if T∈𝒯e has a reentrant corner cℓ as one of its vertices, we can use (2.16) and (2.23) (with s=1+μℓ) to obtain

(2.43)|e|-1⁢∥𝒖-ΠT⁢𝒖∥L2⁢(e)2≲hT-2⁢∥𝒖-ΠT⁢𝒖∥L2⁢(T)2+|𝒖-ΠT⁢𝒖|H1⁢(T)2≲hT2⁢μℓ⁢|𝒖|H1+μℓ⁢(T)2.

Estimate (2.40) follows from the regularity result (2.11) and the summation of (2.41)–(2.43) over e∈ℰh,ΩFi. This completes the proof of the lemma. ∎

The following lemma, which is identical with [9, Lemma 5.3], is useful for estimating terms involving the jumps of the weakly continuous P1 vector fields across the edges. The proof is based on the trace theorem (with scaling) and a standard interpolation error estimate [15, 13].

Lemma 2.9.

It holds that

∑e∈ℰh,ΩF|e|⁢∥η-η¯Te∥L2⁢(e)2≤C⁢h2⁢|η|H1⁢(ΩF)2 for all ⁢η∈H1⁢(ΩF),

where

(2.44)η¯Te=1|Te|⁢∫Teη⁢𝑑x

is the mean of η over Te, one of the triangles in Th that has e as an edge.

Recall that Q is the L2-orthogonal projection onto H⁢(curl0;ΩF). The following result is useful in addressing the consistency error caused by Q in (2.5a).

Lemma 2.10.

The following estimate holds:

(2.45)∥𝒗-Q⁢𝒗∥[L2⁢(ΩF)]2≤C⁢h⁢∥(𝒗,𝒘)∥h

for all (𝐯,𝐰)∈[H1+α,1⁢(div;ΩF)×[H1+α⁢(ΩS)]2]+Vh.

Proof.

For any 𝒗∈H1+α,1⁢(div;ΩF), we have (𝒗-Q⁢𝒗)∈∇×H01⁢(ΩF), the orthogonal complement of H⁢(curl0;ΩF). Hence by duality,

(2.46)∥𝒗-Q⁢𝒗∥[L2⁢(ΩF)]2=supη∈H01⁢(ΩF)∖{0}⁡(𝒗-Q⁢𝒗,∇×η)∥∇×η∥L2⁢(ΩF)=supη∈H01⁢(ΩF)∖{0}⁡(𝒗,∇×η)∥∇×η∥L2⁢(ΩF).

It follows from integration by parts that

(𝒗,∇×η)=(∇h×𝒗,η)+∑e∈ℰh,ΩFi∫eη⁢[[𝒏×𝒗]]⁢𝑑s.

In view of (2.27) and the Poincaré–Friedrichs inequality, we have

(2.47)(∇h×𝒗,η)≤∥∇h×𝒗∥L2⁢(ΩF)⁢∥η∥L2⁢(ΩF)≤C⁢h⁢∥(𝒗,𝒘)∥h⁢∥∇×η∥L2⁢(ΩF).

Since [[𝒏×𝒗]] vanishes at the midpoints of the interior edges, using the midpoint rule we can write

∑e∈ℰh,ΩFi∫eη⁢[[𝒏×𝒗]]⁢𝑑s=∑e∈ℰh,ΩFi∫e(η-η¯Te)⁢[[𝒏×𝒗]]⁢𝑑s

≤C⁢{∑e∈ℰh,ΩFi|e|⁢∥η-η¯Te∥L2⁢(e)2}1/2⁢{∑e∈ℰh,ΩFi1|e|⁢∥[[𝒏×𝒗]]∥L2⁢(e)2}1/2

(2.48)≤C⁢h⁢|η|H1⁢(ΩF)⁢∥(𝒗,𝒘)∥h,

where η¯Te is defined by (2.44) and we used Lemma 2.9 for the last inequality.

Estimate (2.45) follows from (2.46), (2.47) and (2.48). ∎

2.5 Convergence Analysis

We begin with the approximation property of Vh:

Lemma 2.11.

Let (𝐮,𝐰)∈V̊ be the solution of (2.4). We have

(2.49)inf(𝒗,𝒛)∈Vh⁡∥(𝒖,𝒘)-(𝒗,𝒛)∥h≤∥(𝒖,𝒘)-𝐈h⁢(𝒖,𝒘)∥h≤C⁢h⁢|(𝒇,𝒈)|.

Proof.

Since ∇×𝒖=0 implies ∇h×(Πh,ΩF⁢𝒖)=0, and because of (2.21), (2.24) and (2.25), we have

∥(𝒖,𝒘)-𝐈h(𝒖,𝒘)∥h2=∑T∈ΩF∂⁡T∩ΓI=∅∥∇⋅(𝒖-ΠT𝒖)∥L2⁢(T)2+∑T∈ΩF∂⁡T∩ΓI≠∅∥∇⋅(𝒖-(Πh,ΩF^𝒖)|T)∥L2⁢(T)2

+∥∇h⁡(𝒘-Lh,ΩS⁢𝒘)∥L2⁢(ΩS)2+∑e∈ℰh,ΩFi1|e|⁢∥[[𝒏⋅(𝒖-Πh,ΩF⁢𝒖)]]∥L2⁢(e)2

+∑e∈ℰh,ΩFi1|e|⁢∥[[𝒏×(𝒖-Πh,ΩF⁢𝒖)]]∥L2⁢(e)2

(2.50)+∑e∈ℰhΓI1|e|⁢∥𝒏⋅((𝒖-Πh,ΩF^⁢𝒖)-(𝒘-Lh,ΩS⁢𝒘))∥L2⁢(e)2.

The first two terms on the right-hand side of (2.50) have been estimated by Lemma 2.7. The fourth and fifth terms are estimated by Lemma 2.8. The following estimate for the third term on the right-hand side of (2.50) can be obtained by a similar argument as in the proof of Lemma 2.6:

(2.51)∥∇h⁡(𝒘-Lh,ΩS⁢𝒘)∥L2⁢(ΩS)2=∑T∈ΩS|𝒘-Lh,ΩS⁢𝒘|[H1⁢(T)]2≤C⁢h2⁢|(𝒇,𝒈)|2.

For the last term, we have

∑e∈ℰhΓI1|e|⁢∥𝒏⋅((𝒖-Πh,ΩF^⁢𝒖)-(𝒘-Lh,ΩS⁢𝒘))∥L2⁢(e)2

(2.52)≤C⁢∑e∈ℰhΓI{1|e|⁢∥𝒏⋅(𝒖-Πh,ΩF^⁢𝒖)∥L2⁢(e)2+1|e|⁢∥𝒏⋅(𝒘-Lh,ΩS⁢𝒘)∥L2⁢(e)2}.

The second term on the right-hand side of (2.52) can be bounded by C⁢h2⁢|(𝒇,𝒈)|2, by a similar argument as in Lemma 2.6.

It only remains to estimate the first term on the right-hand side of (2.52). Note that

(2.53)1|e|⁢∥𝒏⋅(𝒖-Πh,ΩF^⁢𝒖)∥[L2⁢(e)]22≤1|e|⁢∥𝒏⋅(𝒖-Πh,ΩF⁢𝒖)∥[L2⁢(e)]22+1|e|⁢∥𝒏⋅(Πh,ΩF⁢𝒖-Πh,ΩF^⁢𝒖)∥[L2⁢(e)]22.

The first term on the right-hand side of (2.53) can be estimated by a trace theorem as in Lemma 2.8. Moreover,

(Πh,ΩF⁢𝒖-Πh,ΩF^⁢𝒖)|T=|1|e|⁢∫e[(𝒘-Lh,ΩS⁢𝒘)⋅𝒏]⁢𝑑s|⁢ϕe,n,

where T∈ΩF has e as an edge and ϕe,n is defined as in (2.31). Therefore, the second term on the right-hand side of (2.53) becomes

(2.54)1|e|⁢∥𝒏⋅(Πh,ΩF⁢𝒖-Πh,ΩF^⁢𝒖)∥[L2⁢(e)]22=|1|e|⁢∫e[(𝒘-Lh,ΩS⁢𝒘)⋅𝒏]⁢𝑑s|2⁢(1|e|⁢∥𝒏⋅ϕe,n∥L2⁢(e)2).

Note that 1|e|⁢∥𝒏⋅ϕe,n∥L2⁢(e)2≤C, where C is a constant that depends on the minimum angle of T. Therefore, we can derive from (2.52), the summation of (2.53)–(2.54) over e∈ΓI, and similar arguments as in Lemma 2.6 and Lemma 2.8 that

(2.55)∑e∈ℰhΓI1|e|⁢∥𝒏⋅((𝒖-Πh,ΩF^⁢𝒖)-(𝒘-Lh,ΩS⁢𝒘))∥L2⁢(e)2≤C⁢h2⁢|(𝒇,𝒈)|2.

Estimate (2.49) then follows from (2.51), (2.55) and Lemmas 2.7–2.8. ∎

Lemma 2.12.

Let (𝐮,𝐰)∈V̊ be the solution of (2.4), and (𝐮h,𝐰h) satisfy the discrete problem (2.18). It holds that

(2.56)sup(ϕ,𝝍)∈Vh∖{(𝟎,𝟎)}⁡ah⁢((𝒖,𝒘)-(𝒖h,𝒘h),(ϕ,𝝍))∥(ϕ,𝝍)∥h≤C⁢h⁢|(𝒇,𝒈)|.

Proof.

Let (ϕ,𝝍)∈Vh be arbitrary. Using (2.5), integration by parts, the fact that ∇×𝒖=0⁢in⁢ΩF, and 𝝈⁢(𝒘)⁢𝒏=[(ρF⁢c2)⁢(∇⋅𝒖)]⁢𝒏, we find

ah((𝒖,𝒘),(ϕ,𝝍))=∑T∈ΩF∫T(ρFc2)(∇⋅𝒖)(∇⋅ϕ)dx+∫ΩS(𝝈h(𝒘):ϵh(𝝍))dx

=b⁢((Q⁢𝒇,𝒈),(ϕ,𝝍))+∑e∈ℰh,ΩF∫e(ρF⁢c2)⁢(∇⋅𝒖)⁢[[𝒏⋅ϕ]]⁢𝑑s-∑e∈ℰhΓI∫e(𝝈h⁢(𝒘)⁢𝒏)⋅𝝍⁢𝑑s

=b⁢((Q⁢𝒇,𝒈),(ϕ,𝝍))+∑e∈ℰh,ΩFi∫e(ρF⁢c2)⁢(∇⋅𝒖)⁢[[𝒏⋅ϕ]]⁢𝑑s

(2.57)+∑e∈ℰhΓI∫e(ρF⁢c2)⁢(∇⋅𝒖)⁢[𝒏⋅(ϕ-𝝍)]⁢𝑑s.

Therefore, by (2.18),

ah⁢((𝒖,𝒘)-(𝒖h,𝒘h),(ϕ,𝝍))=b⁢((Q⁢𝒇-𝒇,𝟎),(ϕ,𝝍))+∑e∈ℰh,ΩFi∫e(ρF⁢c2)⁢(∇⋅𝒖)⁢[[𝒏⋅ϕ]]⁢𝑑s

+∑e∈ℰhΓI∫e(ρF⁢c2)⁢(∇⋅𝒖)⁢[𝒏⋅(ϕ-𝝍)]⁢𝑑s

=∫ΩFρF⁢𝒇⋅(Q⁢ϕ-ϕ)⁢𝑑x+∑e∈ℰh,ΩFi∫e(ρF⁢c2)⁢(∇⋅𝒖)⁢[[𝒏⋅ϕ]]⁢𝑑s

(2.58)+∑e∈ℰhΓI∫e(ρF⁢c2)⁢(∇⋅𝒖)⁢[𝒏⋅(ϕ-𝝍)]⁢𝑑s.

We observe from (2.58) that there are three sources for the consistency error of scheme (2.18), namely the projection Q, the discontinuity of the vector fields Vh inside ΩF, and on ΓI.

In view of Lemma 2.10, the first term on the right-hand side of equation (2.58) satisfies the estimate

(2.59)∫ΩF𝒇⋅(Q⁢ϕ-ϕ)⁢𝑑x≤C⁢∥𝒇∥[L2⁢(ΩF)]2⁢∥Q⁢ϕ-ϕ∥[L2⁢(ΩF)]2≤C⁢h⁢|(𝒇,𝒈)|⁢∥(ϕ,𝝍)∥h.

By the definition of Vh, the Cauchy–Schwarz inequality, (2.27) and Lemma 2.9, we can estimate the second term on the right-hand side of (2.58) as follows:

∑e∈ℰh,ΩFi∫e(∇⋅𝒖)⁢[[𝒏⋅ϕ]]⁢𝑑s=∑e∈ℰh,ΩFi∫e(∇⋅𝒖-(∇⋅𝒖)¯Te)⁢[[𝒏⋅ϕ]]⁢𝑑s

≤{∑e∈ℰh,ΩFi|e|⁢∥∇⋅𝒖-(∇⋅𝒖)¯Te∥L2⁢(e)2}1/2⁢{∑e∈ℰh,ΩFi1|e|⁢∥[[𝒏⋅ϕ]]∥L2⁢(e)2}1/2

≤C⁢h⁢|∇⋅𝒖|H1⁢(ΩF)⁢∥(ϕ,𝝍)∥h

(2.60)≤C⁢h⁢|(𝒇,𝒈)|⁢∥(ϕ,𝝍)∥h.

Now we turn to estimate the third term on the right-hand side of (2.58). For any e∈ℰhΓI, note that

∫e(ϕ-𝝍)⋅𝒏⁢𝑑s=0.

Let TeF∈ΩF and TeS∈ΩS be the triangles such that TeF∩TeS=e. By the Cauchy–Schwarz inequality,

|∫e(∇⋅𝒖)⁢[(ϕ-𝝍)⋅𝒏]⁢𝑑s|=|∫e(∇⋅𝒖-(∇⋅𝒖)¯TeF)⁢(ϕ⋅𝒏-𝝍⋅𝒏)⁢𝑑s|

(2.61)≤(|e|1/2⁢∥∇⋅𝒖-(∇⋅𝒖)¯TeF∥L2⁢(e))⁢(|e|-1/2⁢∥ϕ⋅𝒏-𝝍⋅𝒏∥L2⁢(e)).

In view of Lemma 2.9, we have

(2.62)∑e∈ℰhΓI|e|⁢∥∇⋅𝒖-(∇⋅𝒖)¯TeF∥L2⁢(e)2≤C⁢h2⁢|∇⋅𝒖|H1⁢(ΩF)2≤C⁢h2⁢|(𝒇,𝒈)|2.

Combining (2.27), (2.5), and (2.62), we obtain that

(2.63)∑e∈ℰhΓI∫e(ρF⁢c2)⁢(∇⋅𝒖)⁢[𝒏⋅(ϕ-𝝍)]⁢𝑑s≤C⁢h⁢|(𝒇,𝒈)|⁢∥(ϕ,𝝍)∥h.

Finally, estimate (2.56) follows from (2.58), (2.59), (2.60) and (2.63). This completes the proof of the lemma. ∎

The following theorem is an immediate consequence of Lemma 2.5, Lemma 2.11 and Lemma 2.12.

Theorem 2.13.

The following discretization error estimates hold for the solution (𝐮h,𝐰h)∈Vh of (2.18):

∥(𝒖,𝒘)-(𝒖h,𝒘h)∥h≤C⁢h⁢|(𝒇,𝒈)|.

In the rest of this section, we derive the error estimate in the standard L2-norm |(⋅,⋅)| on [L2⁢(ΩF)]2×[L2⁢(ΩS)]2.

Theorem 2.14.

Let (𝐮,𝐰)∈V̊ be the solution of (2.4) and (𝐮h,𝐰h)∈Vh satisfy (2.18). Then we have

(2.64)|(𝒖,𝒘)-(𝒖h,𝒘h)|≤C⁢(h2⁢|(𝒇,𝒈)|+h⁢∥(𝒖,𝒘)-(𝒖h,𝒘h)∥h).

Proof.

Let (𝒑,𝒒)∈𝒱̊ satisfy

(2.65)a⁢((𝒗,𝒛),(𝒑,𝒒))=b⁢((𝒗,𝒛),(𝒖,𝒘)-(𝒖h,𝒘h))

for all (𝒗,𝒛)∈𝒱̊. Here a⁢(⋅,⋅) and b⁢(⋅,⋅) are defined by (2.2) and (2.3).

Note that the strong form of (2.65) is

(2.66)-c2⁢∇⁡(∇⋅𝒑)=Q⁢(𝒖-𝒖h)in ΩF,

(2.67)-∇⋅𝝈⁢(𝒒)=ρS⁢(𝒘-𝒘h)in ΩS,

and we have the following estimate:

(2.68)|∇⋅𝒑|H1⁢(ΩF)≤C⁢|(𝒖,𝒘)-(𝒖h,𝒘h)|.

Furthermore, we can rewrite (2.65) as

(2.69)ah⁢((𝒗,𝒛),(𝒑,𝒒))=b⁢((𝒗,𝒛),(𝒖,𝒘)-(𝒖h,𝒘h))

for all (𝒗,𝒛)∈𝒱̊. Here ah⁢(⋅,⋅) is defined by (2.19).

It follows from (2.65), (2.66), (2.67), and integration by parts that the following analog of (2.57) holds:

ah((𝒖h,𝒘h),(𝒑,𝒒))=∑T∈ΩF∫T(ρFc2)(∇⋅𝒒)(∇⋅𝒖h)dx+∫ΩS(𝝈h(𝒒):ϵh(𝒘h))dx

=b⁢((𝒖h,𝒘h),(Q⁢(𝒖-𝒖h),(𝒘-𝒘h)))+∑e∈ℰh,ΩF∫e(ρF⁢c2)⁢(∇⋅𝒑)⁢[[𝒏⋅𝒖h]]⁢𝑑s

-∑e∈ℰhΓI∫e(𝝈h⁢(𝒒)⁢𝒏)⋅𝒘h⁢𝑑s

=b⁢((𝒖h,𝒘h),(Q⁢(𝒖-𝒖h),(𝒘-𝒘h)))+∑e∈ℰh,ΩFi∫e(ρF⁢c2)⁢(∇⋅𝒑)⁢[[𝒏⋅𝒖h]]⁢𝑑s

(2.70)+∑e∈ℰhΓI∫e(ρF⁢c2)⁢(∇⋅𝒑)⁢[𝒏⋅(𝒖h-𝒘h)]⁢𝑑s.

Combining (2.69) and (2.70), we have

|(𝒖,𝒘)-(𝒖h,𝒘h)|2≈b⁢((𝒖,𝒘)-(𝒖h,𝒘h),(𝒖,𝒘)-(𝒖h,𝒘h))

=b⁢((𝒖,𝒘),(𝒖,𝒘)-(𝒖h,𝒘h))-b⁢((𝒖h,𝒘h),(𝒖,𝒘)-(𝒖h,𝒘h))

=ah⁢((𝒖,𝒘)-(𝒖h,𝒘h),(𝒑,𝒒))-b⁢((𝒖h,𝒘h),((I-Q)⁢(𝒖-𝒖h),𝟎))

(2.71)+∑e∈ℰh,ΩFi∫e(ρF⁢c2)⁢(∇⋅𝒑)⁢[[𝒏⋅𝒖h]]⁢𝑑s+∑e∈ℰhΓI∫e(ρF⁢c2)⁢(∇⋅𝒑)⁢[𝒏⋅(𝒖h-𝒘h)]⁢𝑑s.

We will estimate the four terms on the right-hand side of (2.71) separately.

We can rewrite the first term as

(2.72)ah⁢((𝒖,𝒘)-(𝒖h,𝒘h),(𝒑,𝒒))=ah⁢((𝒖,𝒘)-(𝒖h,𝒘h),(𝒑,𝒒)-𝐈h⁢(𝒑,𝒒))+ah⁢((𝒖,𝒘)-(𝒖h,𝒘h),𝐈h⁢(𝒑,𝒒)).

Then from (2.28) and Lemma 2.11 (applied to (𝒑,𝒒)) we immediately have

ah⁢((𝒖,𝒘)-(𝒖h,𝒘h),(𝒑,𝒒)-𝐈h⁢(𝒑,𝒒))≤C⁢∥(𝒖,𝒘)-(𝒖h,𝒘h)∥h⁢∥(𝒑,𝒒)-𝐈h⁢(𝒑,𝒒)∥h

(2.73)≤C⁢h⁢∥(𝒖,𝒘)-(𝒖h,𝒘h)∥h⁢|(𝒖,𝒘)-(𝒖h,𝒘h)|.

We can rewrite the second term on the right-hand side of (2.72) as

ah⁢((𝒖,𝒘)-(𝒖h,𝒘h),𝐈h⁢(𝒑,𝒒))=b⁢((Q⁢𝒇-𝒇,𝟎),𝐈h⁢(𝒑,𝒒))+∑e∈ℰh,ΩFi∫e(ρF⁢c2)⁢(∇⋅𝒖)⁢[[𝒏⋅(Πh,ΩF⁢𝒑)]]⁢𝑑s

(2.74)+∑e∈ℰhΓI∫e(ρF⁢c2)⁢(∇⋅𝒖)⁢[𝒏⋅(Πh,ΩF^⁢𝒑-Lh,ΩS⁢𝒒)]⁢𝑑s.

Since 𝒑∈H⁢(curl0;ΩF), we have Q⁢𝒑=𝒑. By Lemma 2.10 (applied to 𝒑) and Lemma 2.11 (applied to (𝒑,𝒒)), we have

b⁢((Q⁢𝒇-𝒇,𝟎),𝐈h⁢(𝒑,𝒒))=ρF⁢(Q⁢𝒇-𝒇,Πh,ΩF¯⁢𝒑)ΩF

=ρF⁢(𝒇,Q⁢(Πh,ΩF¯⁢𝒑)-Πh,ΩF¯⁢𝒑)ΩF

=ρF⁢(𝒇,Q⁢(Πh,ΩF¯⁢𝒑-𝒑)-(Πh,ΩF¯⁢𝒑-𝒑))ΩF

≤C⁢h⁢∥𝒇∥L2⁢(ΩF)⁢∥(𝒑,𝒒)-𝐈h⁢(𝒑,𝒒)∥h

(2.75)≤C⁢h2⁢|(𝒇,𝒈)|⁢|(𝒖,𝒘)-(𝒖h,𝒘h)|.

Here

Πh,ΩF¯⁢𝒑|T={(Πh,ΩF⁢𝒑)|Tif ⁢T⊂ΩF⁢and⁢∂⁡T∩ΓI=∅,(Πh,ΩF^⁢𝒑)|Tif ⁢T⊂ΩF⁢and⁢∂⁡T∩ΓI≠∅.

We can rewrite the second term on the right-hand side of (2.74) using the notation introduced in (2.44) as

∑e∈ℰh,ΩFi∫e(ρF⁢c2)⁢(∇⋅𝒖)⁢[[𝒏⋅(Πh,ΩF⁢𝒑)]]⁢𝑑s=∑e∈ℰh,ΩFi∫e(ρF⁢c2)⁢(∇⋅𝒖-(∇⋅𝒖)Te¯)⁢[[𝒏⋅(Πh,ΩF⁢𝒑)]]⁢𝑑s

(2.76)=∑e∈ℰh,ΩFi∫e(ρF⁢c2)⁢(∇⋅𝒖-(∇⋅𝒖)Te¯)⁢[[𝒏⋅(Πh,ΩF⁢𝒑-𝒑)]]⁢𝑑s,

since 𝒏⋅𝒑 is continuous at the midpoints of any edge e∈ℰh,ΩFi. It then follows from the Cauchy–Schwarz inequality, (2.11), Lemma 2.8 (applied to (𝒑,𝒒)) and Lemma 2.9 that

∑e∈ℰh,ΩFi∫e(ρF⁢c2)⁢(∇⋅𝒖)⁢[[𝒏⋅(Πh,ΩF⁢𝒑)]]⁢𝑑s≤C⁢[∑e∈ℰh,ΩFi|e|⁢∥∇⋅𝒖-(∇⋅𝒖)Te¯∥L2⁢(e)2]1/2

×[∑e∈ℰh,ΩFi|e|-1⁢∥[[𝒏⋅(Πh,ΩF⁢𝒑-𝒑)]]∥L2⁢(e)2]1/2

≤C⁢(h⁢|∇⋅𝒖|H1⁢(ΩF))⁢(h⁢|(𝒖,𝒘)-(𝒖h,𝒘h)|)

(2.77)≤C⁢h2⁢|(𝒇,𝒈)|⁢|(𝒖,𝒘)-(𝒖h,𝒘h)|.

Using the definition of Πh,ΩF^ and the fact that 𝒏⋅𝒑=𝒏⋅𝒒 on ΓI, we can rewrite the third term on the right-hand side of (2.74) as

∑e∈ℰhΓI∫e(ρF⁢c2)⁢(∇⋅𝒖)⁢[𝒏⋅(Πh,ΩF^⁢𝒑-Lh,ΩS⁢𝒒)]⁢𝑑s

=∑e∈ℰhΓI∫e(ρF⁢c2)⁢(∇⋅𝒖-(∇⋅𝒖)Te¯)⁢[𝒏⋅(Πh,ΩF^⁢𝒑-Lh,ΩS⁢𝒒)]⁢𝑑s

(2.78)=∑e∈ℰhΓI∫e(ρF⁢c2)⁢(∇⋅𝒖-(∇⋅𝒖)Te¯)⁢[𝒏⋅(𝒑-Πh,ΩF^⁢𝒑)-𝒏⋅(𝒒-Lh,ΩS⁢𝒒)]⁢𝑑s.

It follows from the Cauchy–Schwarz inequality, (2.11), Lemma 2.9 and similar arguments as in Lemma 2.11 that

∑e∈ℰhΓI∫e(ρF⁢c2)⁢(∇⋅𝒖)⁢[𝒏⋅(Πh,ΩF^⁢𝒑-Lh,ΩS⁢𝒒)]⁢𝑑s≤C⁢[∑e∈ℰhΓI|e|⁢∥∇⋅𝒖-(∇⋅𝒖)Te¯∥L2⁢(e)2]1/2

×[∑e∈ℰhΓI|e|-1⁢∥𝒏⋅(𝒑-Πh,ΩF^⁢𝒑)-𝒏⋅(𝒒-Lh,ΩS⁢𝒒)∥L2⁢(e)2]1/2

≤C⁢(h⁢|∇⋅𝒖|H1⁢(ΩF))⁢(h⁢|(𝒖,𝒘)-(𝒖h,𝒘h)|)

(2.79)≤C⁢h2⁢|(𝒇,𝒈)|⁢|(𝒖,𝒘)-(𝒖h,𝒘h)|.

Combining (2.72)–(2.79), we obtain

(2.80)ah⁢((𝒖,𝒘)-(𝒖h,𝒘h),(𝒑,𝒒))≤C⁢(h2⁢|(𝒇,𝒈)|+h⁢∥(𝒖,𝒘)-(𝒖h,𝒘h)∥h)⁢|(𝒖,𝒘)-(𝒖h,𝒘h)|.

Next we estimate the second term on the right-hand side of (2.71). By Lemma 2.10 and the fact that (I-Q)⁢𝒖=0, we have

-b⁢((𝒖h,𝒘h),((I-Q)⁢(𝒖-𝒖h),𝟎))=b⁢((𝒖-𝒖h,𝒘-𝒘h),((I-Q)⁢(𝒖-𝒖h),𝟎))

(2.81)≤C⁢h⁢|(𝒖-𝒖h,𝒘-𝒘h)|⁢∥(𝒖-𝒖h,𝒘-𝒘h)∥h.

We then estimate the third term on the right-hand side of (2.71). Since 𝒏⋅𝒖h is continuous at the midpoints of the interior edges and [[𝒏⋅𝒖]]=0, we get, using the notation introduced in (2.44),

∑e∈ℰh,ΩFi∫e(ρF⁢c2)⁢(∇⋅𝒑)⁢[[𝒏⋅𝒖h]]⁢𝑑s=∑e∈ℰh,ΩFi∫e(ρF⁢c2)⁢(∇⋅𝒑-(∇⋅𝒑)Te¯)⁢[[𝒏⋅𝒖h]]⁢𝑑s

(2.82)=∑e∈ℰh,ΩFi∫e(ρF⁢c2)⁢(∇⋅𝒑-(∇⋅𝒑)Te¯)⁢[[𝒏⋅(𝒖h-𝒖)]]⁢𝑑s.

Using the Cauchy–Schwarz inequality, (2.27), Lemma 2.9 (applied to 𝒑) and (2.68), we can obtain that

∑e∈ℰh,ΩFi∫e(ρF⁢c2)⁢(∇⋅𝒑)⁢[[𝒏⋅𝒖h]]⁢𝑑s≤C⁢[∑e∈ℰh,ΩFi|e|⁢∥∇⋅𝒑-(∇⋅𝒑)Te¯∥L2⁢(e)2]1/2⁢[∑e∈ℰhi|e|-1⁢∥[[𝒖h-𝒖]]∥L2⁢(e)2]1/2

≤C⁢h⁢|∇⋅𝒑|H1⁢(ΩF)⁢∥(𝒖,𝒘)-(𝒖h,𝒘h)∥h

(2.83)≤C⁢h⁢|(𝒖,𝒘)-(𝒖h,𝒘h)|⁢∥(𝒖,𝒘)-(𝒖h,𝒘h)∥h.

Finally, we estimate the fourth term on the right-hand side of (2.71). Using the notation introduced in (2.44) and the fact that on any e∈ΓI, 𝒏⋅𝒖=𝒏⋅𝒘 and ∫e𝒏⋅(𝒖h-𝒘h)⁢𝑑s=0, we have

∑e∈ℰhΓI∫e(ρF⁢c2)⁢(∇⋅𝒑)⁢[𝒏⋅(𝒖h-𝒘h)]⁢𝑑s=∑e∈ℰhΓI∫e(ρF⁢c2)⁢(∇⋅𝒑-(∇⋅𝒑)Te¯)⁢[𝒏⋅(𝒖h-𝒘h)]⁢𝑑s

(2.84)=∑e∈ℰhΓI∫e(ρF⁢c2)⁢(∇⋅𝒑-(∇⋅𝒑)Te¯)⁢[𝒏⋅(𝒖h-𝒖)-𝒏⋅(𝒘h-𝒘)]⁢𝑑s.

Using the Cauchy–Schwarz inequality, (2.27), Lemma 2.9 (applied to 𝒑) and (2.68), we can obtain that

∑e∈ℰhΓI∫e(ρF⁢c2)⁢(∇⋅𝒑)⁢[𝒏⋅(𝒖h-𝒘h)]⁢𝑑s≤C⁢[∑e∈ℰh,ΩFΓI|e|⁢∥∇⋅𝒑-(∇⋅𝒑)Te¯∥L2⁢(e)2]1/2

×[∑e∈ℰhΓI|e|-1⁢∥𝒏⋅(𝒖h-𝒖)-𝒏⋅(𝒘h-𝒘)∥L2⁢(e)2]1/2

≤C⁢h⁢|∇⋅𝒑|H1⁢(ΩF)⁢∥(𝒖,𝒘)-(𝒖h,𝒘h)∥h

(2.85)≤C⁢h⁢|(𝒖,𝒘)-(𝒖h,𝒘h)|⁢∥(𝒖,𝒘)-(𝒖h,𝒘h)∥h.

Estimate (2.64) follows from (2.71), (2.80), (2.81)–(2.85). ∎

Combining Theorem 2.13 and Theorem 2.14, we have the following corollary for the L2-error estimate.

Corollary 2.15.

The following estimate holds for the solution (𝐮h,𝐰h)∈Vh of (2.18):

|(𝒖,𝒘)-(𝒖h,𝒘h)|≤C⁢h2⁢|(𝒇,𝒈)|.

3 A Nonconforming Finite Element Method for the Eigenproblem

The nonconforming method studied in Section 2 can be applied to (1.1) as an eigensolver. In this section, we summarize the convergence property of the nonconforming eigensolver by using the results from Section 2.

We consider the following weak problem:

Problem.

Find λ∈ℝ and (𝒖,𝒘)∈𝒱̊ such that

(3.1)a⁢((𝒖,𝒘),(𝒗,𝒛))=λ⁢b⁢((𝒖,𝒘),(𝒗,𝒛)) for all ⁢(𝒗,𝒛)∈𝒱̊,

where a⁢(⋅,⋅) and b⁢(⋅,⋅) are defined by (2.2) and (2.3).

Given (𝒇,𝒈)∈[L2⁢(ΩF)]2×[L2⁢(ΩS)]2, we define T⁢(𝒇,𝒈)∈𝒱̊ by

a⁢(T⁢(𝒇,𝒈),(𝒗,𝒛))=b⁢((𝒇,𝒈),(𝒗,𝒛)) for all ⁢(𝒗,𝒛)∈𝒱̊.

It is clear that (λ,(𝒖,𝒘)) is a solution of (3.1) if and only if (1λ,(𝒖,𝒘)) is an eigenpair of T, i.e.,

T⁢(𝒖,𝒘)=1λ⁢(𝒖,𝒘).

By Lemma 2.4, T is a bounded linear operator from [L2⁢(ΩF)]2×[L2⁢(ΩS)]2 to H1+γ,1⁢(div;ΩF)×[H1+γ⁢(ΩS)]2 with γ>12. Therefore, the operator T:[L2⁢(ΩF)]2×[L2⁢(ΩS)]2→[L2⁢(ΩF)]2×[L2⁢(ΩS)]2 is symmetric, positive and compact. Hence the spectrum of T consists of a sequence of finite-multiplicity eigenvalues μn>0, n∈ℕ, converging to 0.

It was shown in [4] that λ is a positive eigenvalue of (1.1) if and only if μ=1λ is a positive eigenvalue of the operator T, and the corresponding associated eigenfunctions coincide.

Next, we consider a nonconforming approximation of T. The nonconforming eigensolver for (3.1) is defined as follows:

Problem.

Find λh∈ℝ and (𝒖h,𝒘h)∈Vh such that

(3.2)ah⁢((𝒖h,𝒘h),(𝒗,𝒛))=λh⁢b⁢((𝒖h,𝒘h),(𝒗,𝒛)) for all ⁢(𝒗,𝒛)∈Vh,

where ah⁢(⋅,⋅) is defined by (2.19).

The discrete analog of T is the operator Th:[L2⁢(ΩF)]2×[L2⁢(ΩS)]2→Vh⊂[L2⁢(ΩF)]2×[L2⁢(ΩS)]2 defined by

ah⁢(Th⁢(𝒇,𝒈),(𝒗,𝒛))=b⁢((𝒇,𝒈),(𝒗,𝒛)) for all ⁢(𝒗,𝒛)∈Vh.

In other words Th⁢(𝒇,𝒈)∈Vh is the nonconforming finite element approximation of the solution T⁢(𝒇,𝒈) of the source problem. Moreover,

Th⁢(𝒖,𝒘)=1λh⁢(𝒖,𝒘)

is equivalent to (3.2).

The following discretization error estimates for the source problem have been derived in Corollary 2.15:

(3.3)|(T-Th)⁢(𝒇,𝒈)|≤C⁢h2⁢|(𝒇,𝒈)|

for all (𝒇,𝒈)∈[L2⁢(ΩF)]2×[L2⁢(ΩS)]2. The following theorem can be obtained by applying the classical theory of spectral approximation [22, 14, 1] to the nonconforming eigensolver (3.2). The proof, which is based on estimate (3.3), is identical with that of [12, Theorem 3.1] for the Maxwell eigenvalues.

Theorem 3.1.

Let 0≤λ1≤λ2≤⋯ be the eigenvalues of (3.1), and let λ=λj=λj+1=λj+m-1 be an eigenvalue with multiplicity m. Let 0≤λh,1≤λh,2≤⋯ be the eigenvalues obtained by (3.2). Then as h↓0, we have

|λh,ℓ-λ|≤Cλ⁢h2 ℓ=j,j+1,…,j+m-1.

Furthermore, if Vλ is the eigenspace for λ and Vh,λ is the space spanned by the discrete eigenfunctions corresponding to λh,1,…,λh,j+m-1, then the gap between Vh,λ and Vλ is O⁢(h2) in the L2-norm and O⁢(h) in the norm ∥⋅∥h.

4 Numerical Experiments

In this section we report the results of a series of numerical experiments that corroborate the theoretical results obtained in Section 2 and Section 3.

4.1 Numerical Results for the Source Problem

We first examine the convergence behavior of the numerical scheme (2.18) for the source problem (2.4) on graded meshes. The computational domain is depicted in Figure 3, where ΩF=(1,3)2 and ΩS=(0,4)2∖Ω¯F. We take ρS, ρF, c, μS and λS all to be 1 in the experiment.

Note that at the corner c=(3,3) of ΩF, ωF=π4 and ωS=3⁢π4. The corresponding singularity index is γ=0.544483661651611.

Figure 3

Domains of fluid and solid.

Let (r,θ) be the polar coordinates at the corner c=(3,3) of the interface ΓI. In view of (2.8) and (2.9), we take

𝒘^⁢(r,θ)=rγ⁢(-A⁢cos⁡((γ+1)⁢θ)+cos⁡((γ-1)⁢θ)A⁢sin⁡((γ+1)⁢θ)-Θ⁢sin⁡((γ-1)⁢θ)) and 𝒖~⁢(r,θ)=rγ⁢(E⁢cos⁡((γ+1)⁢θ)-E⁢sin⁡((γ+1)⁢θ)),

where

A:=-((γ-1)⁢(μ+λ)(μ+λ)⁢γ-(3⁢μ+λ))⁢sin⁡((γ-1)⁢ωS)sin⁡((γ+1)⁢ωS),

E:=2⁢(2⁢μ+λ)(μ+λ)⁢γ-(3⁢μ+λ)⁢sin⁡((γ-1)⁢ωS)sin⁡((γ+1)⁢ωF),

Θ:=μλ+2⁢μ⁢(γ-1)-(γ+1)μλ+2⁢μ⁢(γ+1)-(γ-1).

It can easily be checked that 𝒘^ and 𝒖~ satisfy the compatibility conditions (2.5c) and (2.5d). Moreover, we have 𝒖~=∇⁡p, where p⁢(r,θ)=E⁢rγ+1γ+1⁢cos⁡(γ+1)⁢θ.

Define the cut-off function around the corner c=(3,3) by

ϕ⁢(r)={1,r≤0.25,-16⁢(r-0.75)3⁢[5+15⁢(r-0.75)+12⁢(r-0.75)2],0.25≤r≤0.75,0,r≥0.75.

We take 𝒘~=𝒘^⁢ϕ⁢(r) and 𝒖=∇⁡(p⁢(r,θ)⁢ϕ⁢(r)). It is clear that 𝝈⁢(𝒘~)⁢𝜼=𝟎⁢on ΓN, 𝒘~=𝟎⁢on ΓD and ∇×𝒖=0⁢in ΩF. Since [∇⁡ϕ⁢(r)]⋅𝒏=0, we still have 𝒘~⋅𝒏=𝒖⋅𝒏⁢on ΓI. However, 𝝈⁢(𝒘~)⁢𝒏≠(ρF⁢c2⁢∇⋅𝒖)⁢𝒏 on the part of ΓI where 0<ϕ⁢(r)<1. To fix this problem, we introduce a vector field 𝒛~ in ΩS and define 𝒘=𝒘~+𝒛~.

We take 𝒛~=𝒛~a+𝒛~b, where 𝒛~a (respectively, 𝒛~b) is associated with the edge Γa (respectively, Γb) that has (3,1) and (3,3) (respectively, (1,3) and (3,3)) as endpoints. The vector field 𝒛~a is given by

𝒛~a=(v1⁢(x2)⁢(x1-3)2⁢μ+λv2⁢(x2)⁢(x1-3)μ)⁢ρ⁢(x),

where

𝒗⁢(x2)=(v1⁢(x2)v2⁢(x2))=(ρF⁢c2⁢∇⋅𝒖)⁢𝒏-𝝈⁢(𝒘~)⁢𝒏 on Γa

and ρ⁢(x) is the following C2-function defined on the interval [3,4]:

ρ⁢(x)={-192⁢x5+3120⁢x4-20240⁢x3+65520⁢x2-105840⁢x+68257,3≤x≤3.5,0,x≥3.5.

Note that ρ⁢(3)=1.

The vector field 𝒛~b on ΩS is defined by symmetry. Then (𝒖,𝒘) satisfies all the boundary conditions and compatibility conditions in (2.4) and we can take it to be an exact solution for the source problem (2.4).

We solve the source problem by the numerical scheme (2.18) on graded meshes, where the grading parameter μ=γ≈0.54448366 at corners (1,1), (1,3), (3,1) and (3,3). The first three levels of triangulations are depicted in Figure 4. The errors in the L2-norm |(⋅,⋅)| on [L2⁢(ΩF)]2×[L2⁢(ΩS)]2 and the energy norm ∥(⋅,⋅)∥h are tabulated in Table 1. The benefit of the graded meshes is observed.

Figure 4 The first three levels of triangulations
with graded meshes.

Figure 4

The first three levels of triangulations with graded meshes.

Table 1

Convergence of the scheme with graded meshes.

h	∥(𝒖~,𝒘^)-(𝒖h,𝒘h)∥h\|(𝒇,𝒈)\|	Order	\|(𝒖~,𝒘^)-(𝒖h,𝒘h)\|\|(𝒇,𝒈)\|	Order
18	1.82 E-1	–	6.86 E-2	–
116	9.56 E-2	0.93	1.36 E-2	2.32
132	5.16 E-2	0.88	3.71 E-3	1.88
164	3.37 E-2	0.81	1.02 E-3	1.86

4.2 Numerical Results for the Eigenproblem

In this subsection we examine the convergence behavior of the numerical scheme (3.2) for the eigenproblem (3.1). The first three numerical experiments are conducted on the domain depicted in Figure 1, where ΩF=(0.25,1.25)2 and ΩS=(0,1.5)2∖Ω¯F. The same set-up was used in [6].

In the first numerical experiment, we consider the case where ΩS is made with steel and ΩF is filled with air. In this case, the physical parameters in (1.1) are taken to be ρF=1⁢kg/m3, ρS=7700⁢kg/m3, c=340⁢m/s, M=1.44×1011⁢Pa and ν=0.35. They are identical to the parameters in the third numerical experiment in [6]. We use uniform meshes and compute the eigenfrequencies by (3.2). The computed eigenvalues are reported in Table 2. Since we do not know the analytical eigenvalues, the order of convergence is computed in terms of the “exact” eigenvalues obtained by extrapolating the approximate solutions for h=1128 and 1256. The results are very close to the ones reported in [6, Table 3]. Since the density of air is much smaller than the density of steel, this is a small perturbation of an uncoupled problem with a rigid cavity where the eigenfunctions are smooth (cf. [6]). Hence the order of convergence for eigenvalues is close to 2 with uniform meshes.

Table 2

Air in a rectangular steel cavity: the eigenvalues computed with uniform meshes.

Mode	h=164	h=1128	h=1256	Orders		“Exact”
1	695.115	675.990	669.280	1.58	1.65	2.00	667.044
2	1067.768	1068.055	1068.118	2.07	2.13	2.00	1068.139
3	1068.188	1068.168	1068.162	1.88	1.91	2.00	1068.160
4	1509.982	1510.439	1510.549	2.11	2.04	2.00	1510.585
5	2134.651	2135.673	2135.911	2.13	2.07	2.00	2135.991
6	2134.843	2135.927	2136.186	2.08	2.03	2.00	2136.274
7	2350.401	2304.084	2290.394	0.44	1.82	2.00	2285.831
8	2384.996	2387.605	2388.235	1.52	2.04	2.00	2388.445
9	2387.387	2388.197	2388.388	1.52	2.07	2.00	2388.451

In the second experiment, we replace air with water, which corresponds to the choices of ρF=1000⁢kg/m3 and c=1430⁢m/s. They are identical to the choices in the fourth experiment in [6]. The results obtained on uniform meshes are tabulated in Table 3, where the extrapolated eigenvalues obtained on graded meshes in Table 4 are used as “exact” solutions. In this case the order of magnitude of solid and fluid densities are similar so the interaction is much stronger. Therefore the order of convergence of the computed eigenvalues is lower than the case of a steel-air interaction in Table 2. Again they are very close to the results reported in [6, Table 4].

Table 3

Water in a rectangular steel cavity: the eigenvalues computed with uniform meshes.

Mode	h=164	h=1128	h=1256	Orders		“Exact”
1	671.958	653.327	646.805	1.46	1.38	642.733
2	2203.156	2159.534	2146.480	1.72	1.66	2140.444
3	3511.318	3439.433	3413.737	1.45	1.38	3397.769
4	3951.314	3900.298	3880.075	1.31	1.27	3865.819
5	4239.285	4219.354	4213.419	1.77	1.81	4211.056
6	4726.272	4705.872	4697.966	1.34	1.30	4692.539
7	5189.372	5165.067	5157.320	1.60	1.50	5153.047
8	5522.207	5449.036	5426.561	1.67	1.61	5415.586
9	6291.773	6275.419	6268.471	1.25	1.28	6263.591

In the third experiment, we investigate the case of water in a steel cavity as in the second set of experiments, but with graded meshes. The meshes are graded around the corners (0.25,0.25), (0.25,1.25), (1.25,0.25), (1.25,1.25) with grading parameter μ=γ≈0.54448366. The results are tabulated in Table 4, where the “exact” eigenvalues are obtained by extrapolating the approximate solutions for h=1128 and 1256. The benefit of graded meshes is observed, but the asymptotic order of convergence in Theorem 3.1 has not yet been reached.

Table 4

Water in a rectangular steel cavity: the eigenvalues computed with graded meshes.

Mode	h=164	h=1128	h=1256	Orders		“Exact”
1	664.078	648.593	644.198	1.74	1.86	2.00	642.733
2	2199.476	2156.226	2144.390	1.78	1.90	2.00	2140.444
3	3479.186	3420.264	3403.393	1.67	1.86	2.00	3397.769
4	3919.545	3880.839	3869.574	1.68	1.84	2.00	3865.819
5	4242.217	4219.668	4213.209	1.74	1.86	2.00	4211.056
6	4713.047	4698.381	4693.999	1.62	1.81	2.00	4692.539
7	5182.508	5160.915	5155.014	1.85	1.90	2.00	5153.047
8	5505.496	5439.868	5421.657	1.80	1.89	2.00	5415.586
9	6283.721	6269.711	6265.121	1.30	1.72	2.00	6263.591

Remark 4.1.

Since the extrapolated eigenvalues are computed by using the approximate solutions corresponding to h=1128 and 1256, the last order of convergence in Tables 2 and 4 is exactly 2.

The set-up for the last two experiments is depicted in Figure 5, where ΩF is an L-shaped domain with vertices (0.25,0.25), (1.25,0.25), (0.75,0.75), (1.25,0.75), (0.25,1.25), (0.75,1.25) and ΩS=[0,1.5]2∖ΩF.

$Figure 5 The domain of fluid-structure interaction. The outer dimensions are 1.5×1.5⁢m2{1.5\times 1.5\,\text{m}^{2}} and the inner domain is L-shaped.$

Figure 5

The domain of fluid-structure interaction. The outer dimensions are 1.5×1.5⁢m2 and the inner domain is L-shaped.

In the fourth experiment the solid is made with steel and ΩF is filled with water so that ρF=1000⁢kg/m3, ρS=7700⁢kg/m3, c=1430⁢m/s, M=1.44×1011⁢Pa and ν=0.35. The results obtained on uniform meshes are tabulated in Table 5. The extrapolated eigenvalues obtained on graded meshes in Table 6 are used as “exact” solutions in Table 5.

Table 5

Water in an L-shaped steel cavity: the eigenvalues computed with uniform meshes.

Mode	h=164	h=1128	h=1256	Orders		“Exact”
1	860.551	806.439	786.546	1.47	1.52	775.913
2	2284.585	2185.225	2152.371	1.62	1.66	2137.153
3	3816.121	3685.666	3639.322	1.50	1.52	3614.502
4	4125.524	3974.467	3924.566	1.63	1.61	3900.743
5	4811.590	4732.613	4707.209	1.70	1.66	4695.887
6	4533.367	5027.816	5264.461	1.19	1.36	5413.846
7	6899.529	6795.979	6744.889	1.13	1.28	6709.262
8	7657.389	7348.713	7241.233	1.52	1.51	7183.215
9	8283.759	8148.368	8098.794	1.45	1.47	8070.865

In the final experiment we replace the uniform meshes in the fourth experiment with meshes graded around the corners (0.25,0.25), (0.25,1.25), (1.25,0.25), (0.75,1.25), (1.25,0.75) with grading parameter μ=γ≈0.54448366. The results are tabulated in Table 6. The benefit of graded meshes is again observed.

Table 6

Water in an L-shaped steel cavity: the eigenvalues computed with graded meshes.

Mode	h=164	h=1128	h=1256	Orders		“Exact”
1	796.096	781.464	777.301	1.73	1.86	2.00	775.913
2	2179.638	2147.821	2139.820	1.75	1.99	2.00	2137.153
3	3657.074	3623.638	3616.786	1.89	2.22	2.00	3614.502
4	3961.971	3917.148	3904.844	1.84	1.90	2.00	3900.743
5	4730.336	4705.406	4698.267	1.71	1.86	2.00	4695.887
6	5345.490	5396.282	5409.456	1.83	1.96	2.00	5413.846
7	5779.461	6447.883	6643.917	1.33	1.83	2.00	6709.262
8	7334.446	7222.159	7192.952	1.60	1.95	2.00	7183.215
9	8131.221	8086.949	8074.885	1.76	1.90	2.00	8070.865

5 Conclusions

In this paper, we introduce a nonconforming finite element method for the acoustic fluid-structure interaction problem. The approximation spaces are weakly continuous P1 vector fields for the fluid and standard piecewise linear polynomials for the solid. Optimal estimates in both the energy and the L2-norms are obtained and validated by our numerical experiments. By excluding the pure rotational motions from the weak formulation of the problem, we guarantee that our discrete solution does not possess any spurious eigenvalues. Furthermore, this exclusion simplifies our analysis.

We note that it is also possible to use locally curl-free vector fields in the construction of the finite element space on the fluid side, in which case the penalty term for the curl of the fluid displacement in (2.19) is not needed. Such an approach was carried out for the Maxwell equations in [9, 10].

The approach in this paper can also be applied to the acoustic fluid-structure interaction problem in three dimensions. The convergence analysis in the case of quasi-uniform meshes can be carried out as in Sections 2.4 and 2.5. On the other hand the construction of graded meshes in three dimensions with the desired properties would be much more challenging.

Funding source: National Science Foundation

Award Identifier / Grant number: DMS-10-16332

Award Identifier / Grant number: DMS-16-20273

Funding statement: The work of S. C. Brenner and L.-Y. Sung was supported in part by the National Science Foundation under Grant No. DMS-10-16332 and Grant No. DMS-16-20273.

Acknowledgements

We would like to thank two anonymous referees for their many helpful comments.

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Received: 2017-05-01

Accepted: 2017-10-25

Published Online: 2017-12-08

Published in Print: 2018-07-01

A Nonconforming Finite Element Method for an Acoustic Fluid-Structure Interaction Problem

Abstract

1 Introduction

Problem.

Remark 1.1.

2 A Nonconforming Finite Element Method for the Source Problem

2.1 The Source Problem

Problem.

2.2 Regularity of the Source Problem

Remark 2.1.

Lemma 2.2 ([20, Lemma 3.3.1, Lemma 3.3.2]).

Remark 2.3.

Lemma 2.4.

2.3 A Nonconforming Finite Element Method

Problem.

2.4 Preliminary Error Estimates

Lemma 2.5.

Proof.

Lemma 2.6.

Proof.

Lemma 2.7.

Proof.

Lemma 2.8.

Proof.

Lemma 2.9.

Lemma 2.10.

Proof.

2.5 Convergence Analysis

Lemma 2.11.

Proof.

Lemma 2.12.

Proof.

Theorem 2.13.

Theorem 2.14.

Proof.

Corollary 2.15.

3 A Nonconforming Finite Element Method for the Eigenproblem

Problem.

Problem.

Theorem 3.1.

4 Numerical Experiments

4.1 Numerical Results for the Source Problem

4.2 Numerical Results for the Eigenproblem

Remark 4.1.

5 Conclusions

Acknowledgements

References

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