Finite Element Approximations Applied to the Nonlinear Boundary Value Problem $\Delta u=b{u}^2$

Abstract

In this paper, we consider finite element approximations for the nonlinear boundary value problem $\Delta u=b{u}^2$, based on piecewise linear polynomials, and discuss iterative methods associated with the finite element schemes. Error estimates are obtained, which imply that the approximate solutions converge uniformly to the exact solution. Finally, we give some numerical examples indicating the effectiveness of our results.