# Potential of Room Acoustic Solver with Plane-Wave Enriched Finite Element Method
## 3 Numerical Experiments
### Sound Propagation in a Single Room
#### 3.2.2 Results and Discussion  
As an example of comparison of frequency responses, Figure 4a,b respectively show comparisons of frequency response calculated using PUFEM and standard FEM at R1 with reference solution (Ref): (a) Ref versus PUFEM (\(C=13\), Mesh2), and (b) Ref versus standard FEM (\(0.01\) m mesh). This figure visually presents the effectiveness of PUFEM. The standard FEM using \(0.01\) m mesh with \(400\),\(720\) DOF shows a marked difference from the reference solution at higher frequencies than \(1\) kHz because of the inherent large dispersion error, i.e., sound speed increases at higher frequencies, and the error magnitude becomes higher for larger domains. The PUFEM result shows much better agreement with the reference solution at entire frequencies despite the use of coarse mesh with \(267\) elements. Regarding the PUFEM analysis, Figure 5a,b show changes in the DOF for the entire frequency range when using Mesh 2 with \(C=13\). The DOF in PUFEM is defined as the product of plane wave numbers \(q\) for enrichment and nodes \(N_{\text{node}}\) i.e., \(\text{DOF}=q\times N_{\text{node}}\). The DOF of PUFEM analysis changes from \(2149\) to \(16\),\(271\) at frequencies of \(20\) Hz to \(2.5\) kHz, with a change in the plane wave numbers of \(7\)-\(53\). The PUFEM analysis has only \(16\),\(271\) DOF at \(2.5\) kHz, which is \(1/25\) smaller than the DOF of standard FEM. Figure 6a,b show RMS errors of both the standard FEM (\(0.01\) m mesh) and the PUFEM with different values of \(C\) for Mesh 1 and Mesh 2. We only present results with \(C=5\), \(7\), \(9\), \(11\), and \(13\) for ease of illustration of the error behavior. In the PUFEM results, the RMS errors are reduced overall with larger \(C\). The larger value of \(C\) is necessary to reduce the error at higher frequencies. This is true for Mesh 1 and Mesh 2. In comparison with standard FEM results, the PUFEM with \(C\geq 6\) offers more accurate results at frequencies higher than \(315\) Hz in both Mesh 1 and Mesh 2. It is a very interesting capability of plane wave enrichment because the standard FEM with non-enriched elements uses very fine mesh at \(315\) Hz, where the spatial resolution is \(108\) elements per wavelength. Regarding the RMS error at frequencies below \(315\) Hz, PUFEM shows error of less than \(1\) dB when using \(C\geq 7\) for Mesh1 and \(C\geq 6\) for Mesh 2. It is an acceptable error magnitude for practical applications. Furthermore, standard FEM shows RMS error of \(8.1\) dB at \(2\) kHz although the used mesh has \(400\),\(720\) DOF. By contrast, PUFEM shows a significantly low error value of \(0.76\) dB in both Mesh 1 and Mesh 2 when using \(C=13\). Additionally, we performed the standard FEM analysis with  
Figure 3: PUFEM mesh with two spatial resolutions: (a) Mesh 1 and (b) Mesh 2. The range of \(n_{w}\) is also shown for reference. 0.005 m mesh having 1,599,840 DOF, which has spatial resolution of 34 elements per wavelength at 2 kHz. However, the result still shows an RMS error of 7.4 dB at 2 kHz because, for large scale sound field analysis at high frequencies, the standard FEM results include large dispersion error as described previously. That result also demonstrates that the standard FEM requires a finer mesh to obtain the equivalent level of accuracy as the PUFEM results at 2 kHz. Based on those results, the PUFEM with Mesh 2 can probably perform more accurate analysis with at least \(1/100\) fewer DOF than the standard FEM does. Furthermore, we present the convergence behavior of a solution against DOF and examine whether a coarse mesh or a fine mesh is effective. Figure 7a,d show relations between DOF and RMS error in PUFEM analysis using Mesh 1 and Mesh 2 at four higher frequency bands of 1 kHz, 1.25 kHz, 1.6 kHz, and 2 kHz. They show that the coarse mesh, Mesh 2, achieves a practically acceptable error  
Figure 4: Comparison of frequency responses at R1: (**a**) reference solution (Ref) vs. PUFEM (\(C=13\), Mesh 2), and (**b**) reference solution vs. standard FEM (0.01 m mesh).