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

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Domain decomposition methods, particularly the Dirichlet-Neumann method, face challenges at cross-points where subdomains meet. This study analyzes convergence at continuous levels, revealing a non-convergent component due to cross-point singularities.

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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.