Solving interface problems on three-dimensional microstructures is challenging due to the computational cost of resolving complex geometries and generating fitted meshes. Such problems arise, for example, in electrochemical systems, where microscopic transport processes occur in complex domains and are coupled across interfaces by reaction kinetics.
In this work, we derive a homogenized model using an asymptotic expansion approach. The resulting macroscopic model contains effective transport coefficients obtained from auxiliary cell problems, which capture the influence of the microscopic geometry. The cell problems are solved using a CutCell approach, while the fully resolved microscopic problem is computed with the Extended Finite Element Method (XFEM), allowing interface conditions to be imposed on linearized interfaces without requiring fitted meshes.
We demonstrate the approach for three-dimensional interface problems by comparing a fully resolved microscopic problem with the homogenized model. In particular, we investigate the convergence behavior and compare the XFEM discretization with a voxel-based approach.

Particles embedded into or attached to biological membranes
play a crucial role for a variety of cell functions. A key aspect
is the mechanical interaction of moving particles with the
membrane geometry. We consider hybrid models that combine
discrete particles with continuous membranes and investigate
gradient flows for the respective coupled problems. The
spatial discretization of incremental problems uses an
unfitted finite element method based on Bogner-Fox-Schmit
elements for a linearized Canham-Helfrich energy. We present
a time discretization for the coupled model and error estimates
for the unfitted finite element discretization.

We propose a hybrid method, the Neural Enrichment Finite Element Method (NEFEM), designed for problems involving strong oscillations or interface problems with weak discontinuities. This method is based on the stable generalized finite element method (SGFEM) framework, wherein neural networks (NNs) are introduced as enrichment functions for adaptivity, and the Ritz functional is applied for the training process. This works makes two main contributions. First, the method constructs local subspaces with superior approximation properties, significantly reducing the required number of degrees of freedom (DoFs). Second, minimal \emph{a priori} knowledge is required to define enrichment functions, as the NNs evolve heuristically during training. Furthermore, for smooth problems, we provide a residual-based error estimator and prove both its reliability and efficiency. For interface problems, a theoretical analysis on the optimal convergence of the SGFEM is studied, notably without imposing additional regularity assumptions. These analytic results guide the network architecture design and training strategies. The performance and effectiveness of the proposed method is validated through several numerical experiments.