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The diffuse domain method (DDM) effectively solves parabolic partial differential equations on irregular domains. This study proves DDM

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Area of Science:

  • Numerical Analysis
  • Computational Mathematics
  • Partial Differential Equations

Background:

  • Solving partial differential equations on irregular domains presents significant challenges.
  • The diffuse domain method (DDM) offers a novel approach by extending problems to larger, regular domains.

Purpose of the Study:

  • To rigorously analyze the convergence properties of the diffuse domain method (DDM).
  • To establish error estimates for DDM applied to parabolic partial differential equations with Neumann boundary conditions.

Main Methods:

  • The study utilizes weighted Sobolev spaces for theoretical analysis.
  • The diffuse domain method extends the original problem to a larger rectangular domain.
  • Convergence is analyzed as the interface thickness parameter approaches zero.

Main Results:

  • Rigorous proof of DDM solution convergence to the original solution.
  • Derivation of optimal error estimates in weighted L² and H¹ norms.
  • Validation of theoretical findings through numerical experiments.

Conclusions:

  • The diffuse domain method is a convergent and accurate numerical technique for parabolic PDEs.
  • The established error estimates provide theoretical guarantees for DDM's performance.
  • Numerical results confirm the method's efficacy on irregular domains.