Numerical simulation of singularly perturbed non-linear elliptic boundary value problems using finite element method

The present paper presents the finite element solution of two-dimensional non-linear singularly perturbed elliptic partial differential equation subject to appropriate Dirichlet boundary conditions. A new fifth order convergent Newton type iterative method has been described and used to linearize the non-linear problem. The inclusion of this Newton's method of fifth order convergence in finite element method for solving non-linear system of equations reduces the number of iterations and hence the cost of computation. To demonstrate the usefulness of the proposed scheme, a non-convex variational Ginzburg-Landau equation is considered.