Discontinuous Galerkin method for an elliptic problem with nonlinear Newton boundary conditions in a polygon

• ID: 2797, RIV: 10403968
• ISSN: 0272-4979, ISBN: not specified
• source: IMA Journal of Numerical Analysis
• keywords: elliptic equation; nonlinear Newton boundary condition; monotone operator method; discontinuous Galerkin method; regularity and singular behaviour of the solution; compactness in DG spaces; erro
• authors: Miloslav Feistauer, Filip Roskovec, Anna-Margarete Saendig
• authors from KNM: Roskovec Filip, Feistauer Miloslav

Abstract

This article is concerned with the analysis of the discontinuous Galerkin method (DGM) for the numerical solution of an elliptic boundary value problem with a nonlinear Newton boundary condition in a two-dimensional polygonal domain. The growth of the nonlinearity is not compatible with the differential equation, which represents an obstacle in the analysis of the problem. Using monotone operator theory, it is possible to prove the existence and uniqueness of the weak solution and the approximate DG solution. The main emphasis is on the study of error estimates. To this end, the regularity of the weak solution is investigated, and it is shown that due to the singular boundary points, the solution loses regularity in the vicinity of these points. It transpires that the error estimation depends essentially on the opening angle of the corner points and the nonlinearity in the boundary term. It also depends on the parameter defining the nonlinear behaviour of the Newton boundary condition. At the end of this article, some computational experiments are presented.