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Linear barycentric rational collocation method for solving generalized Poisson equations.

Jin Li1, Yongling Cheng1, Zongcheng Li1

  • 1School of Science, Shandong Jianzhu University, No. 1000 Fengming Road, Lingang Development Zone, Jinan 250101, Shandong, China.

Mathematical Biosciences and Engineering : MBE
|March 10, 2023
PubMed
Summary

This study introduces a linear barycentric rational collocation method for solving the Poisson equation, demonstrating its effectiveness and convergence rate for numerical applications.

Keywords:
barycentric rational functioncollocation methoderror functionallinear barycentric rational functionpoisson equation

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Area of Science:

  • Numerical Analysis
  • Computational Mathematics
  • Partial Differential Equations

Background:

  • The Poisson equation is a fundamental partial differential equation with wide applications in physics and engineering.
  • Existing numerical methods may face challenges with accuracy and efficiency for certain boundary conditions or complex geometries.
  • Collocation methods offer an alternative approach to discretizing and solving differential equations.

Purpose of the Study:

  • To develop and analyze a novel linear barycentric rational collocation method for the Poisson equation.
  • To investigate the convergence properties of this new method.
  • To explore the application of domain decomposition with this method for enhanced scalability.

Main Methods:

  • The Poisson equation is transformed into a matrix form using the linear barycentric rational collocation method.
  • Convergence rates are theoretically derived for the chosen basis functions.
  • A domain decomposition strategy is integrated with the barycentric rational collocation method (BRCM).

Main Results:

  • The study presents the discrete matrix form of the Poisson equation.
  • Theoretical convergence rates for the linear barycentric rational collocation method are established.
  • Numerical examples validate the accuracy and efficiency of the proposed algorithm, including its domain decomposition variant.

Conclusions:

  • The linear barycentric rational collocation method provides an effective approach for solving the Poisson equation.
  • The method exhibits favorable convergence properties.
  • The integration of domain decomposition enhances the applicability of the barycentric rational collocation method for complex problems.