An adaptive finite element method is adopted to simulate the steady state
coupled Schrödinger equations with a small parameter. We use damped Newton iteration
to solve the nonlinear algebraic system. When the solution domain is elliptic, our
numerical results with Dirichlet or Neumann boundary conditions are consistent with
previous theoretical results. For the dumbbell and circular ring domains with Dirichlet
boundary conditions, we obtain some new results that may be compared with future
Schrödinger equations, damped Newton iteration, adaptive finite element method, spike-layer solution.