Efficient approximation of the solution of certain nonlinear reaction-diffusion equations with small absorption

DRATMAN, EZEQUIEL

JOURNAL OF COMPLEXITY

ACADEMIC PRESS INC ELSEVIER SCIENCE

Lugar: Amsterdam; Año: 2013 vol. 29 p. 263 - 282

0885-064X

We study the positive stationary solutions of a standard finite-difference discretization of the semilinear heat equation with nonlinear Neumann boundary conditions. We prove that, if the absorption is small enough, compared with the flux in the boundary, there exists a unique solution of such a discretization, which approximates the unique positive stationary solution of the ?continuous? equation. Furthermore, we exhibit an algorithm computing an ε-approximation of such a solution by means of a homotopy continuation method. The cost of our algorithm is linear in the number of nodes involved in the discretization and the logarithm of the number of digits of approximation required.