Coudière, Yves; Hubert, Florence A 3D discrete duality finite volume method for nonlinear elliptic equations. (English) Zbl 1243.35061 SIAM J. Sci. Comput. 33, No. 4, 1739-1764 (2011). The authors study a finite volume approximation of solutions to the nonlinear diffusion problem \(-\operatorname{div}(\phi(z,\nabla u(z))) = f(z)\) in \(\Omega\), \(u=g\) on \(\partial\Omega\), where \(\Omega\) is a bounded polyhedral domain in \(\mathbb{R}^3\). A three-dimensional extension of the discrete duality finite volume schemes is proposed. It is based on a three-mesh finite volume formulation. The constructed discrete nonlinear system of equations is proved to be well-posed, and uniform a priori estimates on its solutions are found. Error estimates are also obtained. The efficiency of this 3D scheme is illustrated on a linear anisotropic problem. Reviewer: Marlène Frigon (Montréal) Cited in 24 Documents MSC: 35J60 Nonlinear elliptic equations 65N15 Error bounds for boundary value problems involving PDEs 74S10 Finite volume methods applied to problems in solid mechanics 35B45 A priori estimates in context of PDEs Keywords:finite volume methods; error estimates; Leray-Lions operators Software:TetGen; PELICANS PDFBibTeX XMLCite \textit{Y. Coudière} and \textit{F. Hubert}, SIAM J. Sci. Comput. 33, No. 4, 1739--1764 (2011; Zbl 1243.35061) Full Text: DOI