Local convergence of the boundary element method on polyhedral domains

Area of Science:

• Numerical Analysis
• Computational Mathematics
• Boundary Element Methods

Background:

• The boundary element method (BEM) is a powerful numerical technique for solving integral equations.
• Analyzing the local behavior of BEM is crucial for understanding its accuracy and convergence properties, especially on complex geometries.
• Symm's integral equation and hyper-singular integral equations are frequently encountered in various fields, including fluid dynamics and electromagnetics.

Purpose of the Study:

• To analyze the local behavior of the lowest order boundary element method for Symm's integral equation and the stabilized hyper-singular integral equation.
• To derive local a priori estimates in L² and H¹ spaces for these equations on polygonal/polyhedral Lipschitz domains.
• To investigate the factors limiting the local rate of convergence.

Main Methods:

• The study employs the lowest order boundary element method applied to quasi-uniform meshes.
• Local a priori estimates are proven in L² for Symm's integral equation and in H¹ for the hyper-singular equation.
• The analysis considers polygonal/polyhedral Lipschitz domains.

Main Results:

• Local a priori estimates in L² and H¹ are established for the analyzed integral equations.
• The local rate of convergence is shown to be dependent on the local regularity of the solution.
• The convergence rate is further influenced by the sum of global regularity and additional regularity from the shift theorem for a dual problem.

Conclusions:

• The local behavior of the boundary element method for Symm's and hyper-singular integral equations is well-characterized.
• The derived estimates provide a theoretical foundation for understanding the method's performance on complex domains.
• The findings highlight the interplay between solution regularity and numerical method convergence.