## 5 Conclusions  
This paper presents the accuracy and computational efficiency assessment results of a frequency-domain FEM based on the lossy Helmholtz equation approach, which can incorporate air absorption effects using the pure-tone sound attenuation coefficient according to the ISO 9613-1 model. In the approach, air absorption is added as volume absorption to the spaces. Two numerical problems were used for the assessment. The plane-wave propagation problem in a long duct with a length of 1 km revealed that the lossy Helmholtz equation based FEM can include a sound attenuation effect accurately up to 20 kHz with relative error below 1% from theoretical attenuation levels when using FE meshes having spatial resolution of about seven elements per wavelength. As a fundamental error characteristic, the error in the attenuation levels increases linearly over frequencies in both logarithmic axis graphs. The error level can be reduced using a mesh with higher spatial resolution. Then, the sound propagation problem in a real-scale office revealed that the lossy Helmholtz equation based FEM is more computationally efficient than the lossless one at higher frequencies when using CSQMOR iterative solver as a linear system equation solver at each frequency. The increased effect of computational efficiency is related strongly with the magnitude of the pure-tone sound attenuation coefficient. The atmospheric condition with a larger air absorption effect, having a larger pure-tone sound attenuation coefficient \begin{table}
\begin{tabular}{c c c c c} \hline \hline \multirow{2}{*}{**Frequency, Hz**} & \multicolumn{2}{c}{**FEM (Lossless)**} & \multicolumn{2}{c}{**FEM (Lossy)**} \\  & \(n_{\text{lossless}}\) & \(t_{\text{cpu}}\), s & \(n_{\text{loss}}\) (\(R_{\text{iter}}\) \%) & \(t_{\text{cpu}}\), s (\(R_{\text{cpu}}\) \%) \\ \hline
3000 & 40,561 & 200 & 29,334 (27.7) & 145 (27.5) \\
4000 & 36,883 & 183 & 24,615 (33.3) & 123 (32.7) \\
5000 & 24,477 & 121 & 16,567 (32.3) & 83 (31.1) \\
6000 & 37,811 & 187 & 23,132 (38.8) & 114 (39.2) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Iteration number and CPU time \(t_{\text{cpu}}\) for FEMs based on the lossy and lossless Helmholtz equations: reduction rate of the iteration number \(R_{\text{iter}}\) and CPU time \(R_{\text{cpu}}\) are also shown in parentheses. \begin{table}
\begin{tabular}{c c c c} \hline \hline \multicolumn{4}{c}{\(n_{\text{loss}}/\text{DOF}\times 100\,\%\)} \\
**Frequency, Hz** & **Mesh1** & **Mesh2** & \(n_{\text{loss}}^{\text{Mesh2}}/n_{\text{loss}}^{\text{Mesh1}}\) \\ \hline
3000 & 19.1 & 1.8 & 1.5 \\
4000 & 15.2 & 1.5 & 1.6 \\
5000 & 9.3 & 1.0 & 1.8 \\
6000 & 14.4 & 1.4 & 1.6 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Ratio of the iteration number over DOF and iteration number increase rate for meshes with a different spatial resolution. value, engenders higher computational efficiency with fewer iteration numbers. From these results, we can infer that the lossy Helmholtz equation-based FEM might be helpful when one simulates room acoustics at high frequencies of the computational efficiency aspect. However, our discussion is limited to 2D cases. Therefore, further examination of three-dimensional room acoustics simulation is crucially important. Although we did not examine lossy Helmholtz equation-based discretization, it might be beneficial for other numerical methods, such as a high-order FEM and BEM with iterative solvers. For example, a recent work [59] focused on solving car interior acoustics up to 20 kHz with the partition of unity isogeometric analysis. Finally, as mentioned in the present paper's introduction, the following three essential research directions currently exist in room acoustics simulation, each with a different attractive capability and applicability. 1. The use of geometrical acoustics simulation methods and presenting methods to improve accuracy, such as phase beam tracing and the novel scattering algorithm [60]. 2. The use of wave-based acoustics simulation methods and presenting methods to improve their efficiency.