## 4 Computational Efficiency Assessment with a 2D Office Problem
### Results and Discussion  
First, as an example to demonstrate how a room's frequency response changes with the consideration of the air absorption effect, Figure 7 presents a comparison of frequency responses computed using FEMs based on the lossy and lossless Helmholtz equation approaches for the case with the atmospheric condition (a) and with GW24K. Condition (a) is the highest air absorption case, as shown in Figure 2. It can be found that SPL reduction because of the air absorption is especially pronounced at higher frequencies, as expected. Additionally, the air absorption does not influence SPL at frequencies lower than 2 kHz. The magnitude of the pure-tone sound attenuation coefficient is 0.044-0.1 [dB/m] at 2 kHz-6 kHz. Although the GW24K sound absorber has \(\alpha_{0}>0.6\) at frequencies higher than 2 kHz, consideration of the air absorption still leads to SPL reduction in this case. To measure the degree of the SPL reduction by considering the air absorption, we define the following mean SPL difference \(D\) in a 1/3 octave band level over 15 sound receivers, calculated using the FEMs based on the lossy and lossless Helmholtz equations as  
\[D=\frac{1}{n_{r}}\sum_{i=1}^{n_{r}}[\overline{L}_{\text{lossless}}(\mathbf{r}_{i},\omega_{c})-\overline{L}_{\text{loss}}(\mathbf{r}_{i},\omega_{c})]\ [\text{dB}], \tag{12}\]  
where \(\overline{L}_{\text{lossless}}(\mathbf{r}_{i},\omega_{c})\) and \(\overline{L}_{\text{loss}}(\mathbf{r}_{i},\omega_{c})\) represent the 1/3 octave band SPL at receiver \(\mathbf{r}_{i}\) and at the center frequency \(\omega_{c}\) computed, respectively, by the FEMs based on the lossless and lossy Helmholtz equations. The \(n_{r}(=15)\) is the number of sound receivers. The positive value in \(D\) expresses the magnitude of SPL reduction by the air absorption. Figure 8 presents the mean SPL difference \(D\) at the three atmospheric conditions (a)-(c) for cases with GW24K, GW32K, and GW96K sound absorbers. Results showed that the order of SPL reduction is consistent with the order of air absorption magnitude, as shown in Figure 2. For the atmospheric conditions (b) and (c), the air absorption effect shows a small impact on the 1/3 octave band SPL with \(D<0.5\) dB for all frequency bands up to 5 kHz. For the atmospheric condition (a), the air absorption effect is also small, with \(D<0.6\) dB at frequencies below 2 kHz. We can find an SPL reduction with \(D\) values of around 1.0 dB at 4 and 5 kHz. This order of the mean SPL difference may lead to uncertainty in room acoustics prediction, considering the standard deviation of 0.26\(\sim\)0.36 dB. Figure 6: Sound propagation problem in a 2D office. A total of 15 sound receivers represented by gray circles are placed inside the room. The rigid-backed porous sound absorber is modeled with surface impedance \(Z_{\text{surf}}\) as a local-reaction material in which a plane wave with incidence angle \(\theta_{i}\) propagates inside materials as the transmission angle \(\theta_{i}=0\). The computational efficiency of the FEMs based on the lossy and lossless Helmholtz equations is evaluated by comparing the iteration numbers of both approaches. Results demonstrate that including the air absorption effect has an additional benefit on the computational efficiency at high frequencies. In this evaluation, a smaller iteration number represents higher computational speed. As a quantitative measure, we define the reduction rate \(R_{\text{iter}}\) of the iteration numbers as  
\[R_{\text{iter}}(\omega)=\frac{n_{\text{lossless}}(\omega)-n_{\text{loss}}( \omega)}{n_{\text{lossless}}(\omega)}\times 100[\%], \tag{13}\]  
where \(n_{\text{lossless}}(\omega)\) and \(n_{\text{loss}}(\omega)\), respectively, denote iteration numbers of the CSQMOR method at frequency \(\omega\). It is noteworthy that we used the iteration number required for convergence as a measure to evaluate the computational efficiency because the iteration process of iterative solvers is the most time-consuming part of FEM in the frequency domain.