Related papers: Interior numerical approximation of boundary value…

We consider the isoparametric finite element method (FEM) for the Poisson equation in a smooth domain with the homogeneous Dirichlet boundary condition. Because the boundary is curved, standard triangulated meshes do not exactly fit it.…

Here we consider the following fractional Hamiltonian system \begin{equation*} \begin{cases} \begin{aligned} (-\Delta)^{s} u&=H_v(u,v) \;\;&&\text{in}~\Omega,\\ (-\Delta)^{s} v&=H_u(u,v) &&\text{in}~\Omega,\\ u &= v = 0 &&\text{in} ~…

We study the regularity in weighted Sobolev spaces of Schr\"{o}dinger-type eigenvalue problems, and we analyse their approximation via a discontinuous Galerkin (dG) $hp$ finite element method. In particular, we show that, for a class of…

In this paper, we study a newly developed shearlet system on bounded domains which yields frames for $H^s(\Omega)$ for some $s\in \mathbb{N}$, $\Omega \subset \mathbb{R}^2$. We will derive approximation rates with respect to $H^s(\Omega)$…

In this paper, we propose and analyze a finite element discretization for the computation of fractional minimal graphs of order~$s \in (0,1/2)$ on a bounded domain $\Omega$. Such a Plateau problem of order $s$ can be reinterpreted as a…

Diffuse domain methods (DDMs) have gained significant attention for solving partial differential equations (PDEs) on complex geometries. These methods approximate the domain by replacing sharp boundaries with a diffuse layer of thickness…

We present a coupling of the Finite Element and the Boundary Element Method in an isogeometric framework to approximate either two-dimensional Laplace interface problems or boundary value problems consisting in two disjoint domains. We…

In this note we show that conforming Galerkin approximations for p-harmonic functions tend to infinity-harmonic functions in the limit p \to \infty and h \to 0, where h denotes the Galerkin discretisation parameter.

Strong approximation errors of both finite element semi-discretization and spatio-temporal full discretization are analyzed for the stochastic Allen-Cahn equation driven by additive noise in space dimension $d \leq 3$. The full…

We consider an initial/boundary value problem for one-dimensional fractional-order parabolic equations with a space fractional derivative of Riemann-Liouville type and order $\alpha\in (1,2)$. We study a spatial semidiscrete scheme with the…

We prove that if $\Omega\subseteq\mathbb{R}^N$ is a set with finite perimeter with $\mathscr{H}^{N-1}(\partial \Omega\setminus\partial^* \Omega)=0$, then any set of finite perimeter $E\subseteq\mathbb{R}^N$ can be approximated by a…

We present a numerical analysis of a higher order unfitted space-time Finite Element method applied to a convection-diffusion model problem posed on a moving bulk domain. The method uses isoparametric space-time mappings for the geometry…

We prove several discrete Gagliardo-Nirenberg-Sobolev and Poincar\'e-Sobolev inequalities for some approximations with arbitrary boundary values on finite volume meshes. The keypoint of our approach is to use the continuous embedding of the…

We present a finite element algorithm that computes eigenvalues and eigenfunctions of the Laplace operator for two-dimensional problems with homogeneous Neumann or Dirichlet boundary conditions or combinations of either for different parts…

Given a bounded open set $\Omega\subset \mathbb{R}^n$, we study sequences of quadratic functionals on the Sobolev space $H^1_0(\Omega)$, perturbed by sequences of bounded linear functionals. We prove that their $\Gamma$-limits, in the weak…

Elliptic partial differential equations on surfaces play an essential role in geometry, relativity theory, phase transitions, materials science, image processing, and other applications. They are typically governed by the Laplace-Beltrami…

Generalized or extended finite element methods (GFEM/XFEM) are in general badly conditioned and have numerous additional degrees of freedom (DOF) compared with the FEM because of introduction of enriched functions. In this paper, we develop…

In this paper, we study a time-fractional initial-boundary value problem of Kirchhoff type involving memory term for non-homogeneous materials. The energy argument is applied to derive the a priori bounds on the solution of the considered…

This work is devoted to the study of the boundary value problem \begin{eqnarray}\nonumber (-1)^\alpha \Delta^\alpha u = (-1)^k S_k[u] + \lambda f, \qquad x &\in& \Omega \subset \mathbb{R}^N, \\ \nonumber u = \partial_n u = \partial_n^2 u =…

We prove that any function in $GSBD^p(\Omega)$, with $\Omega$ a $n$-dimensional open bounded set with finite perimeter, is approximated by functions $u_k\in SBV(\Omega;\mathbb{R}^n)\cap L^\infty(\Omega;\mathbb{R}^n)$ whose jump is a finite…