Higher-order convergence analysis for interior and boundary layers in a semi-linear reaction-diffusion system networked by a $k$-star graph with non-smooth source terms

We investigated a nonlinear singularly perturbed elliptic reaction-diffusion coupled system having non-smooth data networked by a $k$-star graph. We considered all possible boundary conditions at the free boundary located at the tail of the edge and imposed the continuity condition with Kirchhoff's junction law at the junction point of the $k$-star graph to obtain a continuous solution for this coupled system. First, we showed the existence and uniqueness of the solution using the variational formulation approach. Then, we reformulated it into a minimization problem over a function space to conclude the uniqueness of the solution. For the approximation of the continuous problem, note that the upwind scheme for the flux condition at the free boundary leads to a parameter uniform first-order approximation. To obtain a higher-order uniform accuracy, we utilized a three-point scheme for first-order derivatives and a five-point approximation at the point of discontinuity. These approximations typically did not yield an M-matrix or strict diagonally dominant structure of the stiffness matrix. Hence, we provided a suitable transformation that could lead to a sufficient condition for preserving the strict diagonally dominant structure of the stiffness matrix. We performed a comprehensive convergence analysis to demonstrate the almost second-order uniform accuracy on each edge of the $k$-star graph. Numerical experiments highly validate the theory on the $k$-star graph.