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We consider the reaction diffusion problem and present efficient ways to discretize and precondition in the singular perturbed case when the reaction term dominates the equation. Using the concepts of optimal test norm and saddle point reformulation, we provide efficient discretization processes for uniform and non-uniform meshes. We present a preconditioning strategy that works for a large range of the perturbation parameter. Numerical examples to illustrate the efficiency of the method are included for a problem on the unit square. 

We present useful connections between the finite difference and the finite element methods for a model boundary value problem. We start from the observation that, in the finite element context, the interpolant of the solution in one dimension coincides with the finite element approximation of the solution. This result can be viewed as an extension of the Green function formula for the solution at the continuous level. We write the finite difference and the finite element systems such that the two corresponding linear systems have the same stiffness matrices and compare the right hand side load vectors for the two methods. Using evaluation of the Green function, a formula for the inverse of the stiffness matrix is extended to the case of non-uniformly distributed mesh points. We provide an error analysis based on the connection between the two methods and estimate the energy norm of the difference of the two solutions. Interesting extensions to the 2D case are provided.