A robust domain decomposition algorithm for singularly perturbed semilinear systems

We consider a system of singularly perturbed semilinear reaction-diffusion equations. To solve this system numerically we develop an overlapping Schwarz domain decomposition algorithm, where we use the asymptotic behaviour of the exact solution for domain partitioning as well as to construct the iterative algorithm. The algorithm is analysed by defining some auxiliary problems, that allows to prove the uniform convergence of the method in two steps, splitting the discretization error and the iteration error. It is shown that the algorithm gives almost fourth uniform numerical approximations for the exact solution. More importantly, it is shown that for small values of the perturbation parameter just one iteration is required to achieve the almost fourth-order accuracy. Numerical results support our theoretical findings.