An asymptotic finite element method for singularly perturbed third and fourth order ordinary differential equations with discontinuous source term

We consider singularly perturbed Boundary Value Problems (BVPs) for third and fourth order Ordinary Differential Equations (ODEs) with discontinuous source term and a small positive parameter multiplying the highest derivative. Because of the type of Boundary Conditions (BCs) imposed on these equations these problems can be transformed into weakly coupled systems. In this system, the first equation does not have the small parameter but the second contains it. In this paper a computational method named as “An asymptotic finite element method” for solving these systems is presented. In this method we first find an zero order asymptotic approximation to the solution and then the system is decoupled by replacing the first component of the solution by this approximation in the second equation. Then the second equation is independently solved by a fitted mesh Finite Element Method (FEM). Numerical experiments support our theoritical results.