Algorithms and visualization for solutions of nonlinear elliptic equations
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In this paper, we compute and visualize solutions of several major types of semilinear elliptic boundary value problems with a homogeneous Dirichlet boundary condition in 2D. We present the mountainpass algorithm (MPA), the scaling iterative algorithm (SIA), the monotone iteration and the direct iteration algorithms (MIA and DIA). Semilinear elliptic equations are well known to be rich in their multiplicity of solutions. Many such physically significant 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 domains on the multiplicity. Special emphasis is placed on SIA and MPA, by which multiple unstable solutions are computed. The domains include the disk, symmetric or nonsymmetric annuli, dumbbells, and dumbbells with cavities. The nonlinear partial differential equations include the LaneEmden 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, wherever appropriate. Some further theoretical properties of the solutions obtained from visualization will also be presented.
