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A 3D discrete duality finite volume method for nonlinear elliptic equations. (English) Zbl 1243.35061

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.

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

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