Mimetic finite difference approximation of quasilinear elliptic problems
The authors consider the following quasilinear elliptic problem \[ -\mathrm{div} (\kappa (|\nabla u|^2)\nabla u)=f \quad \text{in}\, \Omega, \qquad u=0\quad \text{on}\, \partial \Omega , \tag{1} \] where \(f\) is a given function and \(\Omega\) is an open, bounded set of \(\mathbb{R}^2\) with boundary \(\partial \Omega\). For the approximation of the problem (1), the mimetic finite difference (MFD) method is used. It is shown that the MFD-solution converges to the solution of (1) in a suitable discrete energy norm. The nonlinear discrete problem is solved iteratively via linearization by applying the Kačanov method. The convergence of this method is proved in the discrete mimetic framework. In numerical examples the function \(\kappa(t)\) is chosen equal to the Carreau law: \(\kappa(t)=\eta_\infty+(\eta_0-\eta_\infty)(1+\lambda t)^{\frac{p-2}{p}}\), \(t\geq 0\), with \(\eta_0=3\), \(\eta_\infty=1\) and \(p=1.7\) and with a known solution.
scientific article; zbMATH DE number 6418685
scientific article