In this paper, we are concerned with the weighted plane wave least-squares (PWLS) method for three-dimensional Helmholtz equations, and develop the multi-level adaptive BDDC algorithms for solving the resulting discrete system. In order to form the adaptive coarse components, the local generalized eigenvalue problems for each common face and each common edge are carefully designed. The condition number of the two-level adaptive BDDC preconditioned system is proved to be bounded above by a user-defined tolerance and a constant which is dependent on the maximum number of faces and edges per subdomain and the number of subdomains sharing a common edge. The efficiency of these algorithms is illustrated on a benchmark problem. The numerical results show the robustness of our two-level adaptive BDDC algorithms with respect to the wave number, the number of subdomains and the mesh size, and illustrate that our multi-level adaptive BDDC algorithm can reduce the scale of the coarse problem and can be used to solve large wave number problems efficiently.

在本文中，我们关注三维Helmholtz方程的加权平面波最小二乘（PWLS）方法，并开发了多级自适应BDDC算法来求解所得离散系统。为了形成自适应粗糙分量，精心设计了每个公共面和每个公共边的局部广义特征值问题。事实证明，两级自适应BDDC预处理系统的条件数在上面受用户定义的公差和常数的限制，该常数取决于每个子域的最大面和边缘数以及共享公共边缘的子域数。在一个基准问题上说明了这些算法的效率。数值结果显示了我们的两级自适应BDDC算法相对于波数的鲁棒性，