Related papers: Maximum-norm a posteriori error bounds for an extr…
In this paper we study the a posteriori bounds for a conforming piecewise linear finite element approximation of the Signorini problem. We prove new rigorous a posteriori estimates of residual type in $L^{p}$, for $p \in (4,\infty)$ in two…
We use the elliptic reconstruction technique in combination with a duality approach to prove aposteriori error estimates for fully discrete back- ward Euler scheme for linear parabolic equations. As an application, we com- bine our result…
It has recently been established that the numerical solution of ordinary differential equations can be posed as a nonlinear Bayesian inference problem, which can be approximately solved via Gaussian filtering and smoothing, whenever a…
A priori and a posteriori error analysis of $hp$ finite element method for elliptic control problem with Robin boundary condition and boundary observation are presented. are presented. Through the Cl\'ement-type approach and the…
Space-time finite element discretizations of time-optimal control problems governed by linear parabolic PDEs and subject to pointwise control constraints are considered. Optimal a priori error estimates are obtained for the control variable…
Time-fractional parabolic equations with a Caputo time derivative are considered. For such equations, we explore and further develop the new methodology of the a-posteriori error estimation and adaptive time stepping proposed in [7]. We…
We consider finite element solutions to optimization problems, where the state depends on the possibly constrained control through a linear partial differential equation. Basing upon a reduced and rescaled optimality system, we derive a…
This work is concerned with the proof of \emph{a posteriori} error estimates for fully-discrete Galerkin approximations of the Allen-Cahn equation in two and three spatial dimensions. The numerical method comprises of the backward Euler…
This article provides a brief introduction to the a posteriori error analysis of parabolic partial differential equations, with an emphasis on challenges distinct from those of steady-state problems. Using the heat equation as a model…
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…
This work is concerned with the development of a space-time adaptive numerical method, based on a rigorous a posteriori error bound, for the semilinear heat equation with a general local Lipschitz reaction term whose solution may blow-up in…
Maximal regularity is a fundamental concept in the theory of partial differential equations. In this paper, we establish a fully discrete version of maximal regularity for a parabolic equation. We derive various stability results in…
This paper is concerned with the analysis and implementation of robust finite element approximation methods for mixed formulations of linear elasticity problems where the elastic solid is almost incompressible. Several novel a posteriori…
The main goal of this paper is to analyze a family of "simplest possible" initial data for which, as shown by numerical simulations, the incompressible Euler equations have multiple solutions. We take here a first step toward a rigorous…
We consider a semilinear parabolic equation with a large class of nonlinearities without any growth conditions. We discretize the problem with a discontinuous Galerkin scheme dG(0) in time (which is a variant of the implicit Euler scheme)…
We provide a posteriori error estimates for a discontinuous Galerkin scheme for the parabolic-elliptic Keller-Segel system in 2 or 3 space dimensions. The estimates are conditional, in the sense that an a posteriori computable quantity…
In this paper, we present a study of an a posteriori estimator for the discretization error of a non-standard finite difference scheme applied to boundary value problems defined on an infinite interval. In particular, we show how…
We derive efficient and reliable goal-oriented error estimations, and devise adaptive mesh procedures for the finite element method that are based on the localization of a posteriori estimates. In our previous work [SIAM J. Sci. Comput.,…
We shall develop a fully discrete space-time adaptive method for linear parabolic problems based on new reliable and efficient a posteriori analysis for higher order dG(s) finite element discretisations. The adaptive strategy is motivated…
This work is concerned with the derivation of a robust a posteriori error estimator for a discontinuous Galerkin method discretisation of linear non-stationary convection-diffusion initial/boundary value problems and with the implementation…