Semi- and Fully-Discrete Analysis of Lowest-Order Nonstandard Finite Element Methods for the Biharmonic Wave Problem

A.1 Error Analysis of Explicit Scheme Using Semidiscrete Error Bounds

In this subsection error estimates for the explicit scheme are obtained by using semidiscrete estimates rather than a direct approach used in Subsection 3.2. We split the error as

Recall τ ~ n is defined in Lemma 3.2. A combination of (2.3) and (3.4) yields the error equation in χ n as

Recall that ϕ 1 2 = ( ϕ 0 + ϕ 1 ) / 2 . The next lemma establishes the bounds on initial error χ 1 2 := u h 1 2 - U 1 2 .

The next theorem gives the error bounds under the CFL condition on mesh ratio discussed in Theorem 3.6.

A.2 Regularity

This subsection is devoted to the proof of Lemma 1.1 following the approach from [16, Theorem 12.3]. First, we recall that { ψ n } n = 1 ∞ is an orthonormal basis of L 2 ⁢ ( Ω ) , and hence (cf. [20, Chapter 1, Theorem 4.13])

where d n ⁢ ( t ) := ( u ⁢ ( t ) , ψ n ) and d n ⁢ t ⁢ t ⁢ ( t ) = ( u t ⁢ t ⁢ ( t ) , ψ n ) . A combination of (1.1) and (A.5) and the fact that the ψ n ’s are smooth functions satisfying (1.4) reveal that for every j ≥ 1 ,

The orthonormality of the ψ n ’s simplifies the above equation to a second-order linear ODE

with

for all t ∈ [ 0 , T ] . For n ≥ 1 , the solution of this ODE is

where I ⁢ ( t ) := ∫ 0 t sin ⁡ ( λ n ⁢ ( t - s ) ) ⁢ f n ⁢ ( s ) ⁢ d s . A successive differentiation with respect to t shows that

Then we can apply integration by parts thrice to the term I t ⁢ t ⁢ ( t ) to observe

(A.7)

The expressions in (A.7) are utilized appropriately to control u t ⁢ t in the L 2 ⁢ ( Ω ) and H 2 + γ 0 ⁢ ( Ω ) norms. In addition, the integration by parts is aimed at reducing the spatial regularity of f and its time derivatives. Next, we proceed to differentiate (A.6) twice and use (1.5) (with D ⁢ ( A 0 ) = L 2 ⁢ ( Ω ) ), which leads to

A combination of this with (A.7a) and a use of the definitions of f n , f n ⁢ s with elementary manipulations show

Since f t ∈ L 2 ⁢ ( Ω ) , an application of the monotone convergence theorem and (1.5) shows

Then we differentiate (A.6) thrice (resp. four times) to obtain

An integration by parts twice (resp. thrice) to the last term I t ⁢ t ⁢ t ⁢ ( t ) (resp. I t ⁢ t ⁢ t ⁢ t ⁢ ( t ) ) leads to

The fact that ∥ u t ⁢ t ⁢ t ∥ 2 = ∑ n = 1 ∞ | d n ⁢ t ⁢ t ⁢ t ⁢ ( t ) | 2 (resp. ∥ u t ⁢ t ⁢ t ⁢ t ∥ 2 = ∑ n = 1 ∞ | d n ⁢ t ⁢ t ⁢ t ⁢ t ⁢ ( t ) | 2 ) from (1.5) and an approach similar to (A.8)-(A.10) leads to

Now we aim to control ∥ u t ⁢ t ∥ D ⁢ ( A ) . A differentiation of (A.6) twice and substitution of d n ⁢ t ⁢ t in (A.5) shows

We can then argue as before by using (A.7c) to conclude that, for any t ∈ [ 0 , T ] ,

As f, f t , f t ⁢ t , and f t ⁢ t ⁢ t ∈ L 2 ⁢ ( L 2 ⁢ ( Ω ) ) , from the Sobolev embedding we can infer that f, f t , and f t ⁢ t ∈ L ∞ ⁢ ( L 2 ⁢ ( Ω ) ) . Consequently, (1.5) and the fact that D ⁢ ( A ) ⊂ H 2 + γ 0 ⁢ ( Ω ) (i.e., ∥ u t ⁢ t ∥ H 2 + γ 0 ⁢ ( Ω ) 2 ≲ ∥ u t ⁢ t ∥ D ⁢ ( A ) 2 ) show that ∥ u t ⁢ t ∥ H 2 + γ 0 ⁢ ( Ω ) 2 is bounded (up to a multiplicative constant) by

Consider (A.6) with integration by parts applied to the last term I ⁢ ( t ) once to obtain

A differentiation of (A.6) once followed by integration by parts of the term I t ⁢ ( t ) appearing in the derivative twice, yields

The last two equations and an analogous to the one used to derive (A.12) from (A.11), shows that

This concludes the proof. ∎