A Nitsche-Based Element-Free Galerkin Method for Semilinear Elliptic Problems (Advances in Applied Mathematics and Mechanics, 2023)
A Nitsche-based element-free Galerkin (EFG) method for solving semilinearelliptic problems is developed and analyzed in this paper. The existence and uniqueness of the weak solution for semilinear elliptic problems are proved based on a condition that the nonlinear term is an increasing Lipschitz continuous function of theunknown function. A simple iterative scheme is used to deal with the nonlinear integral term. We proved the existence, uniqueness and convergence of the weak solution sequence for continuous level of the simple iterative scheme. A commonly usedassumption for approximate space, sometimes called inverse assumption, is proved.Optimal order error estimates in $L^2$ and $H^1$ norms are proved for the linear and semilinear elliptic problems. In the actual numerical calculation, the characteristic distance $h$ does not appear explicitly in the parameter $β$ introduced by the Nitsche method. Thetheoretical results are confirmed numerically.