Implementation of an exact finite reduction scheme for steady-state reaction-diffusion equations

In this paper we propose the numerical solution of a steady-state reaction-diffusion problem by means of application of a non-local Lyapunov-Schmidt type reduction originally devised for field theory. A numerical algorithm is developed on the basis of the discretization of the differential operator by means of simple finite differences. The eigendecomposition of the resulting matrix is used to implement a discrete version of the reduction process. By the new algorithm the problem is decomposed into two coupled subproblems of different dimensions. A large subproblem is solved by means of a fixed point iteration completely controlled by the features of the original equation, and a second problem, with dimensions that can be made much smaller than the former, which inherits most of the non-linear difficulties of the original system. The advantage of this approach is that sophisticated linearization strategies can be used to solve this small non-linear system, at the expense of a partial eigendecomposition of the discretized linear differential operator. The proposed scheme is used for the solution of a simple nonlinear one-dimensional problem. The applicability of the procedure is tested and experimental convergence estimates are consolidated. Numerical results are used to show the performance of the new algorithm.