SAMS Subject Classification Number: 23

The Lane-Emden equation is an example of a non-linear Poisson equation. In this study we consider a boundary value problem for the Lane-Emden equation which describes the behaviour of the density of a gas sphere in hydrostatic equilibrium. The non-linearity of the equation motivates the use of a numerical solver. Our choice of the solver is a Galerkin finite element method coupled with linearisation because finite element methods handle complex geometries well and they have well developed techniques for their mathematical analyses among other desirable properties. The numerical solver is implemented on the computer using MATLAB, a computer environment for performing numerical computations and visualisation. Numerical experiments are performed to show that the solver is computationally effective.

[1] Zhao-xiang Li and Zhong-qing Wang, Pseudospectral methods for computing the multiple solutions of the Lane-Emden equation, Journal of Computational Physics 255 (2013) 407-421.

[2] S. C. Brenner and L. R. Scott, The mathematical theory of finite element methods, Springer, 2008.

[3] S. Linge and H. P. Langtangen, Programming for Computations - MATLAB/Octave, Springer Open, 2016.