Algorithms and visualization for solutions of nonlinear elliptic equations

In this paper, we compute and visualize solutions of several major types of semilinear elliptic boundary value problems with a homogeneous Dirichlet b oundary condition in 2D. We present the mountain-pass algorithm (MPA), the scaling iterative algorithm (SIA), the monotone iteration and the direct it eration algorithms (MIA and DIA). Semilinear elliptic equations are well kn own to be rich in their multiplicity of solutions. Many such physically sig nificant solutions are also known to lack stability and, thus, are elusive to capture numerically. We will compute and visualize the profiles of such multiple solutions, thereby exhibiting the geometrical effects of the domai ns on the multiplicity. Special emphasis is placed on SIA and MPA, by which multiple unstable solutions are computed. The domains include the disk, sy mmetric or nonsymmetric annuli, dumbbells, and dumbbells with cavities. The nonlinear partial differential equations include the Lane-Emden equation, Chandrasekhar's equation, Henon's equation, a singularly perturbed equation , and equations with sublinear growth. Relevant numerical data of solutions are listed as possible benchmarks for other researchers. Commentaries from the existing literature concerning solution behavior will be made, whereve r appropriate. Some further theoretical properties of the solutions obtaine d from visualization will also be presented.