Block diagonal Calderón preconditioning for scattering at multi-screens.

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.