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Adaptive Nonoverlapping Preconditioners for the Helmholtz Equation

Yi Yu, Marcus Sarkis, Guanglian Li, Zhiwen Zhang

TL;DR

This work tackles solving the Helmholtz equation at high frequencies by introducing a purely algebraic, nonoverlapping substructuring framework (NOSAS) that stabilizes local Dirichlet problems through a global coarse interface. It builds two classes of preconditioners for the real part $ ext{Re}oldsymbol{B}_0$ and then jointly for $ ext{Re}oldsymbol{B}_0$ and $ ext{Im}oldsymbol{B}_0$, using generalized eigenvalue problems to extract near-singular modes and form a compact coarse space. Theoretical convergence results hold for near-zero thresholds, and extensive numerical experiments across FEM and HDG discretizations demonstrate robust convergence, scalability, and favorable performance for large wavenumbers. The algebraic character and parallel-friendly structure of the approach make it adaptable to heterogeneous coefficients and various discretizations, with potential extensions to multilevel schemes and broader wave propagation problems.

Abstract

The Helmholtz equation poses significant computational challenges due to its oscillatory solutions, particularly for large wavenumbers. Inspired by the Schur complement system for elliptic problems, this paper presents a novel substructuring approach to mitigate the potential ill-posedness of local Dirichlet problems for the Helmholtz equation. We propose two types of preconditioners within the framework of nonoverlapping spectral additive Schwarz (NOSAS) methods. The first type of preconditioner focuses on the real part of the Helmholtz problem, while the second type addresses both the real and imaginary components, providing a comprehensive strategy to enhance scalability and reduce computational cost. Our approach is purely algebraic, which allows for adaptability to various discretizations and heterogeneous Helmholtz coefficients while maintaining theoretical convergence for thresholds close to zero. Numerical experiments confirm the effectiveness of the proposed preconditioners, demonstrating robust convergence rates and scalability, even for large wavenumbers.

Adaptive Nonoverlapping Preconditioners for the Helmholtz Equation

TL;DR

This work tackles solving the Helmholtz equation at high frequencies by introducing a purely algebraic, nonoverlapping substructuring framework (NOSAS) that stabilizes local Dirichlet problems through a global coarse interface. It builds two classes of preconditioners for the real part and then jointly for and , using generalized eigenvalue problems to extract near-singular modes and form a compact coarse space. Theoretical convergence results hold for near-zero thresholds, and extensive numerical experiments across FEM and HDG discretizations demonstrate robust convergence, scalability, and favorable performance for large wavenumbers. The algebraic character and parallel-friendly structure of the approach make it adaptable to heterogeneous coefficients and various discretizations, with potential extensions to multilevel schemes and broader wave propagation problems.

Abstract

The Helmholtz equation poses significant computational challenges due to its oscillatory solutions, particularly for large wavenumbers. Inspired by the Schur complement system for elliptic problems, this paper presents a novel substructuring approach to mitigate the potential ill-posedness of local Dirichlet problems for the Helmholtz equation. We propose two types of preconditioners within the framework of nonoverlapping spectral additive Schwarz (NOSAS) methods. The first type of preconditioner focuses on the real part of the Helmholtz problem, while the second type addresses both the real and imaginary components, providing a comprehensive strategy to enhance scalability and reduce computational cost. Our approach is purely algebraic, which allows for adaptability to various discretizations and heterogeneous Helmholtz coefficients while maintaining theoretical convergence for thresholds close to zero. Numerical experiments confirm the effectiveness of the proposed preconditioners, demonstrating robust convergence rates and scalability, even for large wavenumbers.
Paper Structure (12 sections, 7 theorems, 85 equations, 7 tables, 1 algorithm)

This paper contains 12 sections, 7 theorems, 85 equations, 7 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Discretization and Domain Decomposition Notation
  3. A New Iterative Substructuring Method
  4. NOSAS Preconditioners for $\Re B_0$
  5. Preconditioner $P_1^{-1}$
  6. Preconditioner $P_2^{-1}$
  7. Scalability of the preconditioner
  8. Well-posedness of the Coarse Problem When $\eta$ Is Close to zero
  9. NOSAS Preconditioners for $\Re B_0$ and $\Im B_0$
  10. Numerical Experiments
  11. Conclusion
  12. Appendix

Key Result

Theorem 3.2

\newlabelinfsup_b_00 Any $u_h\in V_h$ admits the unique decomposition in the form: with $u_0\in V_0$ and $u_i\in V_i$ for $1\leq i\leq N$. Furthermore, for any $v_0\in V_0$, we have and

Theorems & Definitions (24)

  • Remark 2.2
  • Definition 3.1
  • Theorem 3.2
  • Proof 1
  • Remark 4.1
  • Remark 4.2
  • Remark 4.3
  • Remark 4.4
  • Theorem 4.5
  • Proof 2
  • ...and 14 more