Multiscale Sub-grid Correction Method for Time-Harmonic High-Frequency Elastodynamics with Wave Number Explicit Bounds

Letting f ∈ ( L 2 ⁢ ( Ω ) ) 3 , for the Newton potential (2.8), we have the estimate

where C > 0 is independent of 𝑘 and depends only on Ω , μ , λ .

Proof

According to representation (2.7) the Newton potential can be split as N k ⁢ ( f ) = N k E ⁢ ( f ) + N k H ⁢ ( f ) , where

The second part N k H ⁢ ( f ) is a vector version of the Newton potential of the acoustic Helmholtz equation, for which the bounds k 1 - m ⁢ ∥ N k H ⁢ ( f ) ∥ H m ⁢ ( Ω ) ≤ C ⁢ ∥ f ∥ L 2 ⁢ ( Ω ) for m = 0 , 1 , 2 have been established in [33]. We therefore only need to prove k 1 - m ⁢ ∥ N k E ⁢ ( f ) ∥ H m ⁢ ( Ω ) ≤ C ⁢ ∥ f ∥ L 2 ⁢ ( Ω ) , m = 0 , 1 , 2 , for the elastic part of the Newton potential.

We start by defining the auxiliary potential

and observe from representation (2.7b) that

The simplification in working with G ~ k E is that, in contrast to G k E , it only depends on the radial component. We proceed with a cut-off function argument and using Fourier techniques as in [32]. Suppose Ω ⊂ B R ⁢ ( 0 ) for some R > 0 . We extend 𝑓 to zero when considered outside of Ω into B R , but do not relabel. We define the cutoff function η ∈ C ∞ ⁢ ( R ≥ 0 ) such that supp ⁡ ( η ) ⊂ [ 0 , 4 ⁢ R ] ,

We set M ⁢ ( x ) := η ⁢ ( | x | ) and define an augmented Newton potential of (A.1) as

where G ~ k E is given by (A.1). For functions 𝑢 with compact support, recall the Fourier transform and the inverse transform are given for x , ξ ∈ R 3 by

For 𝑓 has support in B R , we write the truncated Newton potential componentwise, using the Einstein summation convention, as v i η ⁢ ( x ) = ( ( M ⁢ G ~ k E ) i ⁢ j * f j ) ⁢ ( x ) for i , j = 1 , 2 , 3 . Taking the Fourier transform, using the standard convolution identity, we obtain

For a multi-index α ∈ N 0 3 (non-negative integer vectors of dimension 3), we denote the corresponding multi-index derivatives as ∂ α in the standard way. For the corresponding derivatives in the Fourier variable, we denote the function P α : R 3 → R 3 , P α ⁢ ( ξ ) = ξ α . For 2 ≤ | α | ≤ 4 , we see using the Plancherel identity that

Thus, our estimate relies on the estimation of the supremum over 𝜉 on the last term. This will be estimated in Lemma A.2 below, where we prove that

This implies the asserted estimate. ∎

Let ( G ~ k E ) i ⁢ j be given by (A.1) and 𝑀 a cutoff function as above. Then there exists a C > 0 depending only on R , μ , λ , and not on 𝑘 so that, for | α | = 2 , 3 , 4 ,

Proof

We proceed as in [33, Lemma 3.7]. Since G k E from (A.1) depends only on the radial component, we can write

A key observation is that G ~ k E ⁢ ( r ) → 0 as r → 0 , which corresponds to the boundary terms of the following integration by parts vanishing, allowing for higher-order derivative estimates. We then have, from the definition of the Fourier transform and a change of variables to spherical coordinates,

where S 2 is the unit sphere in R 3 . The inner integral was explicitly computed in [33, equation ( 3.34 ) ] and equals 4 ⁢ π ⁢ sin ⁡ ( r ⁢ | ξ | ) / ( r ⁢ | ξ | ) . We thus obtain

We closely follow the arguments of [33, Lemma 3.7] and estimate s m ⁢ ι ⁢ ( s ) for m = 2 , 3 , 4 . For m = 2 , we use representation (A.2) and integration by parts and compute

By using the properties of 𝜂 and η ′ , the first term can be bounded by k - 2 ⁢ C and the second one by k - 1 ⁢ C ⁢ R so that | s 2 ⁢ ι ⁢ ( s ) | ≤ k - 1 ⁢ C . For m = 3 , in a similar fashion, we use integration by parts twice and obtain

With arguments analogous to above, we see that this is bounded by a constant 𝐶. Finally, for m = 4 , we compute

and see that this is bounded by C ⁢ k .

In summary, we have shown | s m ⁢ ι ⁢ ( s ) | ≤ C ⁢ k m - 3 . This implies for 2 ≤ | α | ≤ 4 that

which is the desired bound. ∎