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

Abstract

A semilinear elliptic equation d∆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. 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.

Spectra of Linearized Operators for NLS Solitary Waves

“…If the components of q 0 are nonnegative, this property is preserved by each iteration q j , and hence also by the limit vector if it exists (see [15,Theorem 3.1]). The convergence of a subsequence of this iteration method to a nonzero vector is proved in [15,Theorem 2.1]. Although the convergence of the entire sequence is not proved, it is observed numerically to be very robust.…”

Energy states of vertically aligned quantum dot array with nonparabolic effective mass

“…Note that fine meshes are used in the heterojunctions and a half of the mesh length is shined in the radial coordinate• coordinate r and the natural axial coordinate z with the following two special treatments• First, in the heterojunction area, fine meshes are used to capture the rapid change in the wave functions. Secondly, half of the mesh length is shifted in the radial coordinate to avoid incorporating the pole condition [38,39]. See Figure 2 for details• Based on the grid points, equation (1) is discretized by the 3D centered seven-point finite difference method …”