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This study introduces an efficient algorithm for solving constrained elliptic problems. A simple scaling technique significantly speeds up convergence for Stokes problems using preconditioned MINRES methods.

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Area of Science:

  • Numerical analysis
  • Computational mathematics
  • Scientific computing

Background:

  • Elliptic problems with constraints are common in various scientific and engineering fields.
  • Efficient solution algorithms are crucial for tackling complex computational challenges.
  • Block diagonal preconditioners are established methods for solving Stokes problems.

Purpose of the Study:

  • To implement efficient solution algorithms for elliptic problems with constraints.
  • To enhance the convergence rate of preconditioned MINRES methods for Stokes problems.
  • To provide accessible codes for numerical studies.

Main Methods:

  • Development and implementation of efficient solution algorithms.
  • Theoretical analysis of preconditioned iterative methods.
  • Application of a simple scaling technique within block diagonal preconditioners.
  • Numerical studies using the preconditioned MINRES method.

Main Results:

  • The proposed scaling technique within block diagonal preconditioners leads to significantly faster convergence.
  • The efficiency of the enhanced preconditioners is demonstrated for Stokes problems.
  • The implemented codes are publicly available for reproducibility and further research.

Conclusions:

  • The integration of a simple scaling into block diagonal preconditioners is an effective strategy for accelerating the solution of constrained elliptic problems.
  • The findings offer practical improvements for computational fluid dynamics and other related fields.
  • Open-source code availability promotes transparency and facilitates broader adoption of the method.