This paper reviews standard oversampling strategies as performed in the multiscale finite
element method (MsFEM). Common to those approaches is that the oversampling is …

The FEniCS Project set out in 2003 with an idea to automate the solution of mathematical
models based on differential equations. Initially, the FEniCS Project consisted of two …

Numerical homogenization, ie, the finite-dimensional approximation of solution spaces of
PDEs with arbitrary rough coefficients, requires the identification of accurate basis elements …

In order to make the field development decision making and planning process tractable, the
decision-makers usually need a few representative models (for example, P10, P50, P90 …

This article presents new immersed finite element (IFE) methods for solving the popular
second order elliptic interface problems on structured Cartesian meshes even if the involved …

In this paper we study multiscale finite element methods (MsFEMs) using spectral multiscale
basis functions that are designed for high-contrast problems. Multiscale basis functions are …

A simple and efficient interface-fitted mesh generation algorithm which can produce a semi-
structured interface-fitted mesh in two and three dimensions quickly is developed in this …

In this paper, robust preconditioners for multiscale flow problems are investigated. We
consider elliptic equations with highly varying coefficients. We design and analyze two-level …

In this paper, we propose a rigorous and accurate non-local (in the oversampled region)
upscaling framework based on some recently developed multiscale methods [10]. Our …

In this paper, we propose a mesh-free method to solve interface problems using the deep
learning approach. Two types of PDEs are considered. The first one is an elliptic PDE with a …