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

Tsung Min Hwang, Weichung Wang

Research output: Contribution to journalArticle

4 Citations (Scopus)

Abstract

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
Volume18
Issue number3
DOIs
Publication statusPublished - 2002 May 1

Fingerprint

Semilinear Elliptic Problem
Neumann Boundary Conditions
Positive Solution
Boundary conditions
Axial Symmetry
Polar coordinates
Discretization Scheme
Semilinear Elliptic Equations
Pattern Formation
Unit ball
Monotonicity
Aggregation
Efficient Algorithms
Numerical Experiment
Converge
Experiment
Agglomeration
Experiments
Model
Knowledge

Keywords

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

ASJC Scopus subject areas

  • Analysis
  • Numerical Analysis
  • Computational Mathematics
  • Applied Mathematics

Cite this

Analyzing and visualizing a discretized semilinear elliptic problem with Neumann boundary conditions. / Hwang, Tsung Min; Wang, Weichung.

In: Numerical Methods for Partial Differential Equations, Vol. 18, No. 3, 01.05.2002, p. 261-279.

Research output: Contribution to journalArticle

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