Feistauer, Miloslav; Roskovec, Filip; Sändig, Anna-Margarete Discontinuous Galerkin method for an elliptic problem with nonlinear Newton boundary conditions in a polygon. (English) Zbl 1465.65134 IMA J. Numer. Anal. 39, No. 1, 423-453 (2019). Summary: 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. Cited in 3 Documents MSC: 65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs 65N15 Error bounds for boundary value problems involving PDEs 35B65 Smoothness and regularity of solutions to PDEs 35D30 Weak solutions to PDEs 35A01 Existence problems for PDEs: global existence, local existence, non-existence 35A02 Uniqueness problems for PDEs: global uniqueness, local uniqueness, non-uniqueness Keywords:elliptic equation; nonlinear Newton boundary condition; monotone operator method; discontinuous Galerkin method; regularity and singular behaviour of the solution; compactness in DG spaces; error estimation PDFBibTeX XMLCite \textit{M. Feistauer} et al., IMA J. Numer. Anal. 39, No. 1, 423--453 (2019; Zbl 1465.65134) Full Text: DOI Link