Adaptive-Multilevel BDDC algorithm for three-dimensional plane wave Helmholtz systems

TL;DR

This paper develops multi-level adaptive BDDC algorithms for 3D Helmholtz systems using PWLS, demonstrating robustness and efficiency in large wave number problems through theoretical bounds and numerical tests.

Contribution

It introduces a novel multi-level adaptive BDDC method with carefully designed local eigenproblems for 3D Helmholtz equations, improving scalability and robustness.

Findings

Condition number bounded by user tolerance

Robustness with respect to wave number and mesh size

Efficient solution of large-scale wave problems

Abstract

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.