We prove a posteriori error estimates of optimal order for linear Schrödinger-type equations in the L∞(L2)- and the L∞(H1)-norm. We discretize only in time by the Crank-Nicolson method. The direct use of the reconstruction technique, as it has been proposed by Akrivis et al. in [Math. Comput.75 (2006) 511–531], leads to a posteriori upper bounds that are of optimal order in the L∞(L2)-norm, but of suboptimal order in the L∞(H1)-norm. The optimality in the case of L∞(H1)-norm is recovered by using...

We analyze Euler-Galerkin approximations (conforming finite elements in space and implicit Euler in time) to coupled PDE systems in which one dependent variable, say $u$, is governed by an elliptic equation and the other, say $p$, by a parabolic-like equation. The underlying application is the poroelasticity system within the quasi-static assumption. Different polynomial orders are used for the $u$- and $p$-components to obtain optimally convergent a priori bounds for all the terms in the error energy norm....

We analyze Euler-Galerkin approximations (conforming finite elements in space and implicit Euler in time) to coupled PDE systems in which one dependent variable, say u, is governed by an elliptic equation and the other, say p, by a parabolic-like equation. The underlying application is the poroelasticity system within the quasi-static assumption. Different polynomial orders are used for the u- and p-components to obtain optimally convergent a priori bounds for all the terms in the error energy...

The time-dependent Stokes equations in two- or three-dimensional bounded domains are discretized by the backward Euler scheme in time and finite elements in space. The error of this discretization is bounded globally from above and locally from below by the sum of two types of computable error indicators, the first one being linked to the time discretization and the second one to the space discretization.

The time-dependent Stokes equations in two- or three-dimensional bounded domains are discretized by the backward Euler scheme in time and finite elements in space. The error of this discretization is bounded globally from above and locally from below by the sum of two types of computable error indicators, the first one being linked to the time discretization and the second one to the space discretization.

In this paper, we extend the reduced-basis methods and associated a posteriori error estimators developed earlier for elliptic partial differential equations to parabolic problems with affine parameter dependence. The essential new ingredient is the presence of time in the formulation and solution of the problem – we shall “simply” treat time as an additional, albeit special, parameter. First, we introduce the reduced-basis recipe – Galerkin projection onto a space $W_N$ spanned by solutions of the...

In this paper, we extend the reduced-basis methods and associated a posteriori error estimators developed earlier for elliptic partial differential equations to parabolic problems with affine parameter dependence. The essential new ingredient is the presence of time in the formulation and solution of the problem – we shall “simply” treat time as an additional, albeit special, parameter. First, we introduce the reduced-basis recipe – Galerkin projection onto a space WN spanned by solutions...

Phase-field models, the simplest of which is Allen–Cahn’s problem, are characterized by a small parameter $\epsilon$ that dictates the interface thickness. These models naturally call for mesh adaptation techniques, which rely on a posteriori error control. However, their error analysis usually deals with the underlying non-monotone nonlinearity via a Gronwall argument which leads to an exponential dependence on ${\epsilon }^{-2}$. Using an energy argument combined with a topological continuation argument and a spectral...

Phase-field models, the simplest of which is Allen–Cahn's problem, are characterized by a small parameter ε that dictates the interface thickness. These models naturally call for mesh adaptation techniques, which rely on a posteriori error control. However, their error analysis usually deals with the underlying non-monotone nonlinearity via a Gronwall argument which leads to an exponential dependence on ε-2. Using an energy argument combined with a topological continuation argument and...

The paper presents an a posteriori error estimator for a (piecewise linear) nonconforming finite element approximation of the heat equation in ${ℝ}^d$, $d=2$ or 3, using backward Euler’s scheme. For this discretization, we derive a residual indicator, which use a spatial residual indicator based on the jumps of normal and tangential derivatives of the nonconforming approximation and a time residual indicator based on the jump of broken gradients at each time step. Lower and upper bounds form the main results...

The paper presents an a posteriori error estimator for a (piecewise linear) nonconforming finite element approximation of the heat equation in ${ℝ}^d$, d=2 or 3, using backward Euler's scheme. For this discretization, we derive a residual indicator, which use a spatial residual indicator based on the jumps of normal and tangential derivatives of the nonconforming approximation and a time residual indicator based on the jump of broken gradients at each time step. Lower and upper bounds form the main...

Systems of parabolic differential equations are studied in the paper. Two a posteriori error estimates for the approximate solution obtained by the finite element method of lines are presented. A statement on the rate of convergence of the approximation of error by estimator to the error is proved.

This paper is devoted to the study of a posteriori error estimates for the scalar nonlinear convection-diffusion-reaction equation $c_t+\nabla ·\left(𝐮f\left(c\right)\right)-\nabla ·\left(D\nabla c\right)+\lambda c=0$. The estimates for the error between the exact solution and an upwind finite volume approximation to the solution are derived in the $L^1$-norm, independent of the diffusion parameter $D$. The resulting a posteriori error estimate is used to define an grid adaptive solution algorithm for the finite volume scheme. Finally numerical experiments underline the applicability...

This paper is devoted to the study of a posteriori error estimates for the scalar nonlinear convection-diffusion-reaction equation $c_t+\nabla ·\left(𝐮f\left(c\right)\right)-\nabla ·\left(D\nabla c\right)+\lambda c=0$. The estimates for the error between the exact solution and an upwind finite volume approximation to the solution are derived in the L1-norm, independent of the diffusion parameter D. The resulting a posteriori error estimate is used to define an grid adaptive solution algorithm for the finite volume scheme. Finally numerical experiments underline the applicability...

We deal with a posteriori error estimates of the discontinuous Galerkin method applied to the nonstationary heat conduction equation. The problem is discretized in time by the backward Euler scheme and a posteriori error analysis is based on the Helmholtz decomposition.

We will investigate the possibility to use superconvergence results for the mixed finite element discretizations of some time-dependent partial differential equations in the construction of a posteriori error estimators. Since essentially the same approach can be followed in two space dimensions, we will, for simplicity, consider a model problem in one space dimension.

We consider the efficient and reliable solution of linear-quadratic optimal control problems governed by parametrized parabolic partial differential equations. To this end, we employ the reduced basis method as a low-dimensional surrogate model to solve the optimal control problem and develop a posteriori error estimation procedures that provide rigorous bounds for the error in the optimal control and the associated cost functional. We show that our approach can be applied to problems involving...