# Potential of Room Acoustic Solver with Plane-Wave Enriched Finite Element Method
## 3 Numerical Experiments
### Sound Propagation Problem in a Coupled Room
#### 3.3.2 Results and Discussion  
Figure 10 presents a comparison of RMS errors among results with different values of \(C\). The figure also includes standard FEM results obtained using 0.01 m mesh having 612,000 DOF. The RMS error of PUFEM is reduced at higher frequencies with larger values of \(C\). The PUFEM with \(C\geq 6\) were more accurate than the standard FEM results at above 500 Hz band. At the highest 2 kHz band, PUFEM using \(C=13\) shows a smaller RMS value of 1.4 dB than the 7.5 dB in the standard FEM. The magnitude is 1/5.4 of the standard FEM. Here, the change in DOF of PUFEM is the same as that shown in Figure 5; in addition, the DOFs change from 3451 to 26,129 at 20 Hz to 2.5 kHz. The results demonstrate clearly that the PUFEM can also perform well for more complex coupled fields with a small amount of DOF.  
Figure 10: Comparison of RMS errors among results with different values of \(C\): The standard FEM results obtained using 0.01 m mesh are also shown.  
Figure 9: PUFEM mesh for coupled room analysis: The \(n_{w}\) range is the same as Mesh 2 in Figure 3.  
Furthermore, Figure 11 depicts SPL distributions at 2 kHz and 2.5 kHz among Ref, PUFEM, and standard FEM, where Ref and the standard FEM used 0.005 m mesh having 2,448,000 DOF. The PUFEM used \(C\) = 12 at 2 kHz and \(C\) = 13 at 2.5 kHz. The DOFs are, respectively, 21,692 and 26,129. In addition, PUFEM results were shown at the same 2,448,000 nodal points as in Ref and standard FEM. RMS error \(L_{\text{rms}}\) was calculated with all nodal points. The \(L_{\text{rms}}\) values for PUFEM and standard FEM are also included in Figure 11. The SPL distributions at 2 kHz and 2.5 kHz of PUFEM agree very well with those of Ref, exhibiting smaller values of \(L_{\text{rms}}\) than the standard FEM results. For PUFEM results, the value is 1.3 dB at 2 kHz and 1.5 dB at 2.5 kHz. The standard FEM shows large errors of 5.6 dB at 2 kHz and 7.1 dB at 2.5 kHz. We can observe in the standard FEM results by which SPL distributions differ from those of Ref in both frequencies. For example, at 2.5 kHz, SPL values in the four rooms show lower values than Ref. Clearer dips in SPL distributions are apparent in the three soundproof rooms. These results suggest that the DOF reduction in PUFEM reaches at least 1/100 of the standard FEM even for large problems including practical boundary conditions.  
However, we must state an important aspect in the use of PUFEM at this stage. In Figure 11, we present results of PUFEM with \(C\) = 12 instead of \(C\) = 13 at 2 kHz because the result with \(C\) = 13 showed noisy SPL distributions, as shown in Figure 12. In the present figure, it is apparent that the solution did not converge to the reference solution in the three soundproof rooms because of the ill-conditioning of the linear system, as described previously. From this result, it can be inferred that there exists a proper number of plane waves for enrichment to obtain reliable results. Finally, in Figure 13, we present comparisons of SPL distributions between Ref and PUFEM with \(C\) = 13 at three lower frequencies of 125 Hz, 250 Hz, and 500 Hz, where modal behavior is enhanced because of low diffuseness. The figure also includes RMS error values calculated at 615,020 nodal points. The agreement of SPL distributions between PUFEM and Ref is excellent with RMS errors less than  
Figure 11: SPL distributions at 2 kHz (**upper**) and 2.5 kHz (**lower**) among Ref (**left**), PUFEM (**center**), and standard FEM (**right**).  
0.25 dB at all frequencies. The PUFEM results used \(C=13\), but lower values of \(C\) can be used at the low frequencies shown in Figure 10.