Numerical solution of a two-dimensional nonlinear singularly perturbed elliptic partial differential equation ∈ Δu = f(x, u), 0 < x, y < 1, with Dirichlet boundary condition is discussed here. The modified Newton method of third-order convergence is employed to linearize the nonlinear problem in place of the standard Newton method. The finite-element method is used to find the solution of the nonlinear differential equation. Numerical results are provided to demonstrate the usefulness of the method.
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Kumar, P., Kumar, M. Finite-element approximation for singularly perturbed nonlinear elliptic boundary-value problems. Comput Math Model 23, 88–95 (2012). https://doi.org/10.1007/s10598-012-9121-6
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DOI: https://doi.org/10.1007/s10598-012-9121-6