Maher Khrais: Adaptive Iterative Numerical Homogenization for Quasilinear Nonmonotone Elliptic PDE скачать в хорошем качестве

Maher Khrais: Adaptive Iterative Numerical Homogenization for Quasilinear Nonmonotone Elliptic PDE

In our talk, we present an adaptive iterative numerical homogenization method in the framework of the localized orthogonal decomposition (LOD). The algorithm is based on a Kačanov iteration, where a new multiscale space construction is performed in each iteration. In our method, we do not update all the correction operators that might be unnecessary if the multiscale solutions slightly change in certain parts of the domain. However, the algorithm adaptively (using local error indicators) recomputes the local corrector problems where it only improves the accuracy of the solution. The motivation is to enhance the efficiency of method when using LOD to solve quasilinear PDE. We also present the well-posedness of the method and the proof of convergence under suitable assumptions on the initial data of the problem. In addition, we illustrate the theoretical findings with numerical experiments, in which we also study a similar algorithm based on Newton iteration, which is only presented experimentally, since its theoretical analysis relies on assumptions that are not guaranteed to hold in our setting.