Moritz Hauck

Moritz Hauck, Axel Målqvist, Malin Mosquera

We propose a multiscale method for mixed-dimensional elliptic problems with highly heterogeneous coefficients arising, for example, in the modeling of fractured porous media. The method is based on the Localized Orthogonal Decomposition (LOD) framework and constructs locally supported, problem-adapted basis functions on a coarse mesh that does not need to resolve the coefficient oscillations. These basis functions are obtained in parallel by solving localized fine-scale problems. Our a priori error analysis shows that the method achieves optimal convergence with respect to the coarse mesh size, independent of the coefficient regularity, with an exponentially decaying localization error. Numerical experiments validate these theoretical findings and demonstrate the computational viability of the method.

Moritz Hauck, Roland Maier, Timo Sprekeler

We study the finite element approximation of linear second-order elliptic partial differential equations in nondivergence form with highly heterogeneous diffusion and drift coefficients. A generalized Cordes condition is imposed to guarantee that a suitably renormalized version of the nondivergence-form differential operator is near the Laplacian. Based on a stabilized symmetric formulation for the gradient that enables the use of $H^1$-conforming approximation spaces, we construct a multiscale method following the methodology of the localized orthogonal decomposition with coarse basis functions tailored to the heterogeneous coefficients. We employ a novel post-processing strategy to obtain higher-order convergence rates, overcoming previous limitations imposed by the low regularity of the load functional. Numerical experiments demonstrate the performance of the method.