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Block diagonal Calderón preconditioning for scattering at multi-screens.

Kristof Cools1, Carolina Urzúa-Torres2

  • 1Tech Lane 126, 9052 Ghent, Belgium.

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Summary

A new preconditioner for Laplace exterior boundary value problems on multi-screens is introduced. This method significantly reduces simulation costs by achieving logarithmic growth in the spectral condition number, even for complex geometries.

Keywords:
Complex screensGalerkin boundary element methodPreconditioningQuotient-space boundary element method

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Area of Science:

  • Computational Mathematics
  • Numerical Analysis
  • Boundary Value Problems

Background:

  • Laplace exterior boundary value problems present computational challenges, particularly on complex geometries like multi-screens.
  • Efficient numerical methods are crucial for solving these problems in various scientific and engineering applications.
  • Existing preconditioners may not scale effectively with decreasing mesh sizes for multi-screen configurations.

Purpose of the Study:

  • To develop and analyze a novel preconditioner for Laplace exterior boundary value problems on multi-screens.
  • To achieve a spectral condition number that grows only logarithmically with decreasing mesh size.
  • To reduce the computational cost associated with simulating these problems.

Main Methods:

  • Combination of the quotient-space boundary element method and operator preconditioning.
  • Development of block diagonal Calderón preconditioners for a general subclass of multi-screens.
  • Strategies for removing redundancy in the resulting computational scheme without compromising effectiveness.

Main Results:

  • The proposed preconditioner achieves a spectral condition number with logarithmic growth relative to mesh size, similar to simple screens.
  • Redundancy reduction strategies are presented, maintaining the preconditioner's effectiveness.
  • Numerical results validate the method's performance and suggest broad applicability to practical multi-screen geometries.

Conclusions:

  • The developed preconditioner offers an efficient and scalable solution for Laplace exterior boundary value problems on multi-screens.
  • The method significantly reduces simulation costs for a wide range of practical geometries.
  • This approach advances the numerical treatment of boundary value problems in complex domains.