Adaptive BDDC algorithms for the system arising from plane wave discretization of Helmholtz equations

TL;DR

This paper introduces adaptive BDDC algorithms tailored for the PWLS discretization of Helmholtz equations, effectively handling high wave numbers and complex coefficients with proven condition number bounds and demonstrated numerical efficiency.

Contribution

The paper develops a novel two-level adaptive BDDC algorithm for PWLS discretization of Helmholtz equations, with a multilevel extension and theoretical condition number bounds.

Findings

Condition number bounded by user tolerance and interface complexity

Algorithm effective for high wave numbers and complex coefficients

Numerical results confirm theoretical efficiency and robustness

Abstract

Balancing domain decomposition by constraints (BDDC) algorithms with adaptive primal constraints are developed in a concise variational framework for the weighted plane wave least-squares (PWLS) discritization of Helmholtz equations with high and various wave numbers. The unknowns to be solved in this preconditioned system are defined on elements rather than vertices or edges, which are different from the well-known discritizations such as the classical finite element method. Through choosing suitable "interface" and appropriate primal constraints with complex coefficients and introducing some local techniques, we developed a two-level adaptive BDDC algorithm for the PWLS discretization, and the condition number of the preconditioned system is proved to be bounded above by a user-defined tolerance and a constant which is only dependent on the maximum number of interfaces per subdomain. A multilevel algorithm is also attempted to resolve the bottleneck in large scale coarse problem. Numerical results are carried out to confirm the theoretical results and illustrate the efficiency of the proposed algorithms.