A Nitsche-Based Element-Free Galerkin Method for Semilinear Elliptic Problems

A Nitsche-based element-free Galerkin (EFG) method for solving semilinear elliptic problems is developed and analyzed in this paper. The existence and unique-ness of the weak solution for semilinear elliptic problems are proved based on a con-dition that the nonlinear term is an increasing Lipschitz continuous function of the unknown function. A simple iterative scheme is used to deal with the nonlinear in-tegral term. We proved the existence, uniqueness and convergence of the weak solu-tion sequence for continuous level of the simple iterative scheme. A commonly used assumption for approximate space, sometimes called inverse assumption, is proved. Optimal order error estimates in L2 and H1 norms are proved for the linear and semi-linear elliptic problems. In the actual numerical calculation, the characteristic distance h does not appear explicitly in the parameter & beta; introduced by the Nitsche method. The theoretical results are confirmed numerically.