On the norm of the hyperinterpolation operator on the unit disc and its use for the solution of the nonlinear Poisson equation

Abstract

In this article, we study the properties of the hyperinterpolation operator on the unit disc D in ℝ2. We show how hyperinterpolation can be used in connection with the Kumar–Sloan method to approximate the solution of a nonlinear Poisson equation on the unit disc (discrete Galerkin method). A bound for the norm of the hyperinterpolation operator in the space C(D) is derived. Our results prove the convergence of the discrete Galerkin method in the maximum norm if the solution of the Poisson equation is in the class C1,δ(D), δ > 0. Finally, we present numerical examples which show that the discrete Galerkin method converges faster than O(n–k), for every k ∈ ℕ if the solution of the nonlinear Poisson equation is in C∞(D).