Cross-points in the Dirichlet-Neumann method I: well-posedness and convergence issues

Area of Science:

• Numerical Analysis
• Computational Mathematics
• Scientific Computing

Background:

• Domain decomposition methods are crucial for solving large-scale partial differential equations.
• Cross-points, where more than two subdomains intersect, pose challenges for standard domain decomposition techniques.
• The Dirichlet-Neumann method, often involving derivatives, requires special attention at these cross-points.

Purpose of the Study:

• To investigate the convergence of the Dirichlet-Neumann method at the continuous level in the presence of cross-points.
• To provide a theoretical analysis of the behavior of the Dirichlet-Neumann method iterates at cross-points.
• To understand the impact of cross-points on the overall convergence properties of the method.

Main Methods:

• Analysis of the Dirichlet-Neumann method at the continuous level.
• Decomposition of iterates into even and odd symmetric parts.
• Mathematical analysis of convergence and singularity generation at cross-points.
• Numerical experiments to illustrate theoretical findings.

Main Results:

• The iterates of the Dirichlet-Neumann method at cross-points can be uniquely decomposed.
• An even symmetric part of the iterates converges geometrically, similar to cases without cross-points.
• An odd symmetric part generates a singularity at the cross-point and does not converge.

Conclusions:

• Cross-points introduce a non-convergent component in the Dirichlet-Neumann method iterates.
• The presence of singularities at cross-points limits the applicability or requires modifications to the standard Dirichlet-Neumann method.
• Further research may be needed to develop domain decomposition strategies that effectively handle cross-point singularities.