Related papers: Discrete approximation of the Griffith functional …
We consider the numerical computation of a variational problem that arises from materials science. The target functional is a type of elastic energy that is influenced by obstacles and adhesion. Owing to its strong nonlinearity and…
This article considers the error analysis of finite element discretizations and adaptive mesh refinement procedures for nonlocal dynamic contact and friction, both in the domain and on the boundary. For a large class of parabolic…
A new domain decomposition method for Maxwell's equations in conductive media is presented. Using this method reconstruction algorithms are developed for determination of dielectric permittivity function using time-dependent scattered data…
We prove that E. De Giorgi's conjecture for the nonlocal approximation of free-discontinuity problems extends to the case of functionals defined in terms of the symmetric gradient of the admissible field. After introducing a suitable class…
Motivated by conforming finite element methods for elliptic problems of second order, we analyze the approximation of the gradient of a target function by continuous piecewise polynomial functions over a simplicial mesh. The main result is…
We investigate a physical characterization of the gradient flow structure of variational fracture models for brittle materials: a Griffith-type fracture model and an irreversible fracture phase field model. We derive the Griffith-type…
We shall establish the convergence of an adaptive conforming finite element method for the reconstruction of the distributed flux in a diffusion system. The adaptive method is based on a posteriori error estimators for the distributed flux,…
We study invariant sets and measures generated by iterated function systems defined on countable discrete spaces that are uniform grids of a finite dimension. The discrete spaces of this type can be considered as models of spaces in which…
We analyze optimal complexity of adaptive finite element methods (AFEMs) for general second-order linear elliptic partial differential equations (PDEs) in the Lax-Milgram setting. To this end, we formulate an adaptive algorithm which steers…
In this paper we analyze a two-dimensional discrete model of nearest-neighbour Lennard-Jones interactions under the microscopical constraint that points on a lattice triangle maintain their order. This can be understood as a microscopical…
We propose, analyze, and numerically validate a correction adaptive two-grid finite element method (CAT-GFEM) for nonselfadjoint or indefinite elliptic problems. In contrast to the adaptive two-grid finite element method (ATGFEM) of Li and…
This research rigorously investigates the convergence of adaptive finite element methods for regularized variational models of quasi-static brittle fracture in elastic solids. We specifically examine a novel Ambrosio-Tortorelli (AT1)…
We consider finite approximations of a fractal generated by an iterated function system of affine transformations on $\mathbb{R}^d$ as a discrete set of data points. Considering a signal supported on this finite approximation, we propose a…
We present a numerical approximation method for linear diffusion-reaction problems with possibly discontinuous Dirichlet boundary conditions. The solution of such problems can be represented as a linear combination of explicitly known…
In this paper, we study an adaptive finite element method for multiple eigenvalue problems of a class of second order elliptic equations. By using some eigenspace approximation technology and its crucial property which is also presented in…
In the context of Discontinuous Galerkin methods, we study approximations of nonlinear variational problems associated with convex energies. We propose element-wise nonconforming finite element methods to discretize the continuous…
We present an effective adaptive procedure for the numerical approximation of the steady-state Gross-Pitaevskii equation. Our approach is solely based on energy minimization, and consists of a combination of gradient flow iterations and…
This work is motivated by the classical discrete elastic rod model by Audoly et al. We derive a discrete version of the Kirchhoff elastic energy for rods undergoing bending and torsion and prove $\Gamma$-convergence to the continuous model.…
We develop all of the components needed to construct an adaptive finite element code that can be used to approximate fractional partial differential equations, on non-trivial domains in $d\geq 1$ dimensions. Our main approach consists of…
This paper is devoted to the complete convergence study of the finite-element approximation of Maxwell's equations in the case where the magnetic permeability is constant. Standard linear finite elements for the space discretization are…