Analyzing and visualizing a discretized semilinear elliptic problem with Neumann boundary conditions
Tsung‐Min Hwang, Weichung Wang · Numerical Methods for Partial Differential Equations · 2002
Abstract A semilinear elliptic equation d Δ u − u + u p =0 over the unit ball in ℝ 2 with positive solution and the homogeneous Neumann boundary condition is considered. This equation models applications like chemotactic aggregation and biological pattern formation. Recent theoretical analyses on the equation suggest little continuous solution properties. 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. Based on the q‐axial symmetry property, 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. © 2002 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 18: 261–279, 2002; Published online in Wiley InterScience (www.interscience.wiley.com). DOI 10.1002/num.10006