SIAM Journal on Scientific Computing, Volume 42, Issue 4, Page B894-B920, January 2020.
In this paper we consider wave propagation problems in two-dimensional unbounded domains, including dissipative effects, reformulated in terms of space-time boundary integral equations. For their solution, we employ a convolution quadrature (CQ) for the temporal and a Galerkin boundary element method (BEM) for the spatial discretization. It is known that one of the main advantages of the CQ-BEMs is the use of the FFT algorithm to retrieve the discrete time integral operators with an optimal linear complexity in time, up to a logarithmic term. It is also known that a key ingredient for the success of such methods is the efficient and accurate evaluation of all the integrals that define the matrix entries associated to the full space-time discretization. This topic has been successfully addressed when standard Lagrangian basis functions are considered for the space discretization. However, it results that, for such a choice of the basis, the BEM matrices are in general fully populated, a drawback that prevents the application of CQ-BEMs to large-scale problems. In this paper, as a possible remedy to reduce the global complexity of the method, we consider approximant functions of wavelet type. In particular, we propose a numerical procedure that, by taking advantage of the fast wavelet transform, allows us on the one hand to compute the matrix entries associated to the choice of wavelet basis functions by maintaining the accuracy of those associated to the Lagrangian basis ones and, on the other hand, to generate sparse matrices without the need of storing a priori the fully populated ones. Such an approach allows in principle the use of wavelet basis of any type and order, combined with CQ based on any stable ordinary differential equations solver. Several numerical results, showing the accuracy of the solution and the gain in terms of computer memory saving, are presented and discussed.

中文翻译：

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CQ-小波边界元方法高效求解二维波传播问题：算法与应用

SIAM科学计算杂志，第42卷，第4期，第B894-B920页，2020年1月。
在本文中，我们考虑了二维无界域中的波传播问题，包括耗散效应，这些问题是根据时空边界积分方程重新制定的。对于他们的解决方案，我们对时间采用卷积正交（CQ），对空间离散采用Galerkin边界元方法（BEM）。众所周知，CQ-BEM的主要优点之一是使用FFT算法来检索离散时间积分算子，该算子具有最佳的线性时间复杂度，直至对数项。还已知的是，这种方法成功的关键因素是对定义与全时空离散化相关的矩阵项的所有积分进行有效且准确的评估。当考虑将标准拉格朗日基函数用于空间离散化时，此主题已成功解决。但是，结果是，对于这样的基础选择，通常会完全填充BEM矩阵，这是一个缺点，导致无法将CQ-BEM应用于大规模问题。在本文中，作为降低该方法整体复杂性的一种可能的补救措施，我们考虑了小波类型的近似函数。特别是，我们提出了一种数值程序，该程序利用快速小波变换，一方面允许我们通过保持与小波基函数相关的矩阵项的精度来计算与小波基函数选择相关的矩阵项。另一方面，生成稀疏矩阵而无需先验存储完全填充的矩阵。这种方法原则上允许使用任何类型和阶数的小波基，并与基于任何稳定的常微分方程求解器的CQ结合使用。提出并讨论了一些数值结果，这些结果显示了解决方案的准确性以及计算机内存节省方面的收益。