Analyzing and visualizing a discretized semilinear elliptic problem with Neumann boundary conditions

Tsung Min Hwang*, Weichung Wang

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review


A semilinear elliptic equation dΔu - u + up = 0 over the unit ball in ℝ2 with positive solution and the homogeneous Neumann boundary condition is considered. This equation models applications like chemotactic aggregation and biological pattern formation. Focusing on solving the discretized version of the equation, this work proposes an efficient algorithm that combines a newly developed discretization scheme on polar coordinates with a fast Fourier solver. An analysis of the induced matrix structures proves the algorithm converges to positive solutions; the analysis also establishes the q-axial symmetry and monotonicity behavior of the solutions. Numerical experiments were conducted to visualize various solution forms that are new to the best of our knowledge. The experiments also illustrated sensitivity behavior of the solutions.

Original languageEnglish
Pages (from-to)261-279
Number of pages19
JournalNumerical Methods for Partial Differential Equations
Issue number3
Publication statusPublished - 2002 May


  • Discretization on polar coordinates
  • Monotonicity
  • Numerical methods
  • q-axial symmetry
  • Semilinear Neumann problem
  • Visualization

ASJC Scopus subject areas

  • Analysis
  • Numerical Analysis
  • Computational Mathematics
  • Applied Mathematics


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