## 1 Introduction  
Computational acoustics simulation methods [1] are indispensable tools for designing room acoustics in various architectural spaces and acoustic virtual reality applications [2]. Two simulation methods, called geometrical acoustic methods and wave-based numerical methods, are available, and they have unique strengths and shortcomings. The former is already a general-purpose method, and the latter is gradually gaining interest thanks to recent developments in computer technology. Geometrical acoustic methods [3], such as acoustic ray tracing [4], beam tracing, and the image source method [5], are widely used practical acoustic simulation tools with high computational efficiency. The geometrical acoustic methods model, sound propagation as ray propagation, and wave phenomena such as diffraction and interference are not addressed accurately. Therefore, efforts to improve the accuracy have been undertaken. For example, the phase beam tracing method [6; 7; 8; 9] can include phase information essential for dealing with the interference effect and phase shift on reflection at boundaries. On the other hand, room acoustic simulations with wave-based numerical methods, such as the finite difference time-domain (FDTD) method [10; 11; 12; 13; 14], finite element method (FEM) [15; 16; 17; 18; 19; 20], and boundary element method (BEM) [21; 22; 23; 24; 25], have become applicable for the accurate prediction of indoor sound environments. Compared to practical geometrical room acoustic simulations using ray tracing methods [3], wave-based room acoustic simulations have attractive benefits, providing a reliable prediction by solving the partial differential equations of wave phenomena, such as the wave equation in the time-domain analysis and the Helmholtz equation in the frequency-domain analysis. However, because wave-based room acoustics simulation requires vast computational times for practical-sized rooms, many researchers have been striving to develop computationally efficient wave-based methods. Through outcomes established from their studies, some recent works [18; 19; 20] have demonstrated that recently developed wave-based room acoustic solvers with FEM in the time domain can predict indoor acoustics up to 6 kHz in small room spaces, such as classrooms and meeting rooms, within practical computational times. Moreover, they have shown a fundamental effectiveness as a virtual indoor sound environment creation tool for virtual reality applications. It is also worth mentioning that a hybrid technique [26; 27] that tries to combine the strength of geometrical acoustic methods and wave-based numerical methods is an attractive direction for room acoustics simulation. It can enhance accuracy at low frequencies while keeping computational efficiency. Among the above three directions on room acoustics simulation, i.e., the use of geometrical acoustic methods, wave-based numerical methods, and their hybrid techniques, the present paper specifically focuses on wave-based numerical methods using FEM. Apart from developing highly efficient methods on wave-based numerical methods, an essential key to reliable room acoustics prediction is the use of accurate sound attenuation models for acoustic materials installed in spaces and sound wave propagation in the air. Acoustic materials such as porous and resonant absorbers [28; 29], which are common sound-absorbing materials for room acoustics control, have frequency-dependent and incident-angle-dependent sound absorption characteristics. Impedance boundary conditions, which use theoretically computed or measured complex-valued specific acoustic impedance of materials, are used to describe the frequency dependence of acoustic properties of local-reaction materials. Impedance tube measurements described in ISO 10534-2 [30] and in-situ measurements [31] are available as measurement-based approaches for this purpose. The transfer matrix method [32] is a theoretical method of computing the specific acoustic impedance of materials.