Related papers: Pointwise A Posteriori Error Estimators for Ellipt…
We propose and analyze novel adaptive algorithms for the numerical solution of elliptic partial differential equations with parametric uncertainty. Four different marking strategies are employed for refinement of stochastic Galerkin finite…
Based on the auxiliary subspace techniques, a hierarchical basis a posteriori error estimator is proposed for the Stokes problem in two and three dimensions. For the error estimator, we need to solve only two global diagonal linear systems…
We introduce quantitative and robust tools to control the numerical accuracy in simulations performed using the Multiscale Finite Element Method (MsFEM). First, we propose a guaranteed and fully computable a posteriori error estimate for…
In this paper, based on the combination of tensor neural network and a posteriori error estimator, a novel type of machine learning method is proposed to solve high-dimensional boundary value problems with homogeneous and non-homogeneous…
The recent work [Kurz et al., Numer. Math., 147 (2021)] proposed functional a posteriori error estimates for boundary element methods (BEMs) together with a related adaptive mesh-refinement strategy. Unlike most a posteriori BEM error…
We consider the a posteriori error estimation for an atomistic-to-continuum cou- pling scheme for a generic one-dimensional many-body next-nearest-neighbour interaction model in 1D. We derive and rigorously prove the efficiency of the…
While many methods exist to discretize nonlinear time-dependent partial differential equations (PDEs), the rigorous estimation and adaptive control of their discretization errors remains challenging. In this paper, we present a methodology…
In this work, we derive a $\gamma$-robust a posteriori error estimator for finite element approximations of the Allen-Cahn equation with variable non-degenerate mobility. The estimator utilizes spectral estimates for the linearized steady…
This paper aims to develop an efficient adaptive finite element method for the second-order elliptic problem. Although the theory for adaptive finite element methods based on residual-type a posteriori error estimator and bisection…
We present an a posteriori error estimate based on equilibrated stress reconstructions for the finite element approximation of a unilateral contact problem with weak enforcement of the contact conditions. We start by proving a guaranteed…
We present a novel \textit{a posteriori} error estimator for N\'ed\'elec elements for magnetostatic problems that is constant-free, i.e. it provides an upper bound on the error that does not involve a generic constant. The estimator is…
Kohn-Sham density functional theory is one of the most widely used electronic structure theories. The recently developed adaptive local basis functions form an accurate and systematically improvable basis set for solving Kohn-Sham density…
In this work, we investigate the numerical reconstruction of inclusions in a semilinear elliptic equation arising in the mathematical modeling of cardiac ischemia. We propose an adaptive finite element method for the resulting constrained…
We establish rigorous \emph{a posteriori} error bounds for a space-time finite element method of arbitrary order discretising linear wave problems in second order formulation. The method combines standard finite elements in space and…
This paper is concerned with adaptive mesh refinement strategies for the spatial discretization of parabolic problems with dynamic boundary conditions. This includes the characterization of inf-sup stable discretization schemes for a…
We consider the Galerkin boundary element method (BEM) for weakly-singular integral equations of the first-kind in 2D. We analyze some residual-type a posteriori error estimator which provides a lower as well as an upper bound for the…
In this paper, we study two residual-based a posteriori error estimators for the $C^0$ interior penalty method in solving the biharmonic equation in a polygonal domain under a concentrated load. The first estimator is derived directly from…
This paper develops a posteriori error estimates of residual type for conforming and mixed finite element approximations of the fourth order Cahn-Hilliard equation $u_t+\De\bigl(\eps \De u-\eps^{-1} f(u)\bigr)=0$. It is shown that the {\it…
In this paper, we discuss adaptive approximations of an elliptic eigenvalue optimization problem in a phase-field setting by a conforming finite element method. An adaptive algorithm is proposed and implemented in several two dimensional…
We consider the space-time boundary element method (BEM) for the heat equation with prescribed initial and Dirichlet data. We propose a residual-type a posteriori error estimator that is a lower bound and, up to weighted $L_2$-norms of the…