Related papers: Elliptic Equations in High-Contrast Media and Appl…
In this article we introduce an asymptotic preserving scheme designed to compute the solution of a two dimensional elliptic equation presenting large anisotropies. We focus on an anisotropy aligned with one direction, the dominant part of…
Many problems of theoretical and practical interest involve finding a convex or concave function. For instance, optimization problems such as finding the projection on the convex functions in $H^k(\Omega)$, or some problems in economics. In…
This article deals with the numerical approximation of effective coefficients in stochastic homogenization of discrete linear elliptic equations. The originality of this work is the use of a well-known abstract spectral representation…
We consider nonconforming methods for symmetric elliptic problems and characterize their quasi-optimality in terms of suitable notions of stability and consistency. The quasi-optimality constant is determined and the possible impact of…
Numerical computation of harmonic forms (typically called harmonic fields in three space dimensions) arises in various areas, including computer graphics and computational electromagnetics. The finite element exterior calculus framework…
We derive the optimal energy error estimate for multiscale finite element method with oversampling technique applying to elliptic system with rapidly oscillating periodic coefficients under the assumption that the coefficients are bounded…
We overview a series of recent works addressing numerical simulations of partial differential equations in the presence of some elements of randomness. The specific equations manipulated are linear elliptic, and arise in the context of…
Based on the plenty method, this paper describes a numerical method for 2D non-smooth contact problems with Coulomb friction and bilateral constraints and its application to the simulation of statics and dynamics for a frictional…
In this paper we provide a unified treatment of some convex minimization problems, which allows for a better understanding and, in some cases, improvement of results in this direction proved recently in spaces of curvature bounded above.…
This work introduces a novel, fully robust and highly-scalable, $h$-adaptive aggregated unfitted finite element method for large-scale interface elliptic problems. The new method is based on a recent distributed-memory implementation of the…
Elliptic problems along smooth surfaces embedded in three dimensions occur in thin-membrane mechanics, electromagnetics (harmonic vector fields), and computational geometry. In this work, we present a parametrix-based integral equation…
In this paper we investigate the approximation of a diffusion model problem with contrasted diffusivity and the error analysis of various nonconforming approximation methods. The essential difficulty is that the Sobolev smoothness index of…
This paper analyzes the random fluctuations obtained by a heterogeneous multi-scale first-order finite element method applied to solve elliptic equations with a random potential. We show that the random fluctuations of such solutions are…
In this paper we propose a nonconforming finite element method for the solution of the ill-posed elliptic Cauchy problem. We prove error estimates using continuous dependence estimates in the $L^2$-norm. The effect of perturbations in data…
This work is devoted to the homogenization of elliptic equations in high-contrast media in the so-called 'double-porosity' resonant regime, for which we solve two open problems of the literature. First, we prove qualitative stochastic…
We establish the existence of weak solutions of a nonlinear radiation-type boundary value problem for elliptic equation on divergence form with discontinuous leading coefficient. Quantitative estimates play a crucial role on the real…
In this paper, a direct finite element method is proposed for solving interface problems on unfitted meshes. This new method treats the two interface conditions as an $H^{\frac12}(\Gamma)\times H^{-\frac12}(\Gamma)$ pair for the mutual…
We introduce an alternative to the method of matched asymptotic expansions. In the "traditional" implementation, approximate solutions, valid in different (but overlapping) regions are matched by using "intermediate" variables. Here we…
Motivated by the problem of finding resistances among vertices in a hypercube, we derive exact expressions, generating functions, and asymptotic expansions for these resistances, then study the combinatorial interpretations of the…
The goal of this work is to present a fast and viable approach for the numerical solution of the high-contrast state problems arising in topology optimization. The optimization process is iterative, and the gradients are obtained by an…