Convex approximation of an inhomogeneous anisotropic functional
This paper is concerned with numerical minimization of a model functional which is of a form encountered in applications such as phase transitions and crystal growth: the functional is \[ {\mathcal F}(u)= \int_ \Omega \phi(x,\nu_ u)| Du|+ \int_{\partial\Omega} \mu u d{\mathcal H}^{n-1}- \int_ \Omega \kappa u dx \] in which \(\Omega\) is a bounded smooth domain in \(\mathbb{R}^ n\), \(\phi\) a convex, positively homogeneous and continuous function with linear growth, and where \({\mathcal H}^{n-1}\) is the \((n-1)\)-dimensional Hausdorff measure in \(\mathbb{R}^ n\). The minimum is sought on the set \(BV(\Omega;[-1,1])\). The functional \(\mathcal F\) is not strictly convex, and is therefore regularized by a family \({\mathcal F}_ \varepsilon\) of strictly convex functionals. Then in addition, a regular family of partitions of \(\Omega\) parametrized by a mesh parameter \(h\) is introduced, as is the piecewise linear finite element space, and corresponding discrete functionals \({\mathcal F}_ h\) and \({\mathcal F}_{\varepsilon,h}\) are defined on the finite-dimensional spaces. Results on the uniform convergence of the discrete functionals to the continuous functionals are proved, and the main result of the paper is the \(\Gamma\)-convergence of \({\mathcal F}_{\varepsilon,h}\) to \(\mathcal F\) when \(\varepsilon\) and \(h\) go to zero independently. A convergence result for the minimizers \(u_{\varepsilon,h}\) of the discrete regularized functional to the minimizer \(u\) of \(\mathcal F\) is also proved: \(\{u_{\varepsilon,h}\}\) contains a subsequence which converges to \(u\) and, furthermore, \({\mathcal F}_{\varepsilon,h}(u_{\varepsilon,h})\) converges to \({\mathcal F}(u)\).
scientific article; zbMATH DE number 650745