In this paper, we propose and analyze a linear time stepping finite element method for abstract second order linear evolution problems. For such methods, we derive optimal order a posteriori error estimates and sharp a posteriori nodal error estimates using the energy approach and the duality argument. Based on these estimates, we further design an adaptive time stepping strategy for the previo...

Adaptive methods for solving systems of partial diierential equations have become widespread. Much of the eeort has focused on nite element methods. In this paper modiied nite diierence approximations are obtained for grids with irregular nodes. The modiications are required to ensure consistency and stability. Asymptotically exact a posteriori error estimates of the spatial error are presented...

In this note we look at constant-free a posteriori error estimates from a different perspective. We show that they can be interpreted as an alternative way of expressing the residual of a finite element approximation and thus fit into the same framework as other a posteriori error estimates such as residual error indicators. Our approach also reveals that, when applied to singularly perturbed r...

In this paper we prove a priori and a posteriori error estimates for a multiscale numerical method for computing equilibria of multilattices under an external force. The error estimates are derived in a W 1,∞ norm in one space dimension. One of the features of our analysis is that we establish an equivalent way of formulating the coarse-grained problem which greatly simplifies derivation of the...

These lecture notes comprise the talks of the author on “A posteriori error estimates for modelling errors” and on “A posteriori error estimates for highly indefinite problems” given at the Zürich Summerschool 2012. 1 Lecture 1: Combined A Posteriori Modeling Discretization Error Estimate for Elliptic Problems with Complicated Interfaces Remark. This part of the lecture notes is a slightly exte...

In this paper we prove a priori and a posteriori error estimates for a multiscale numerical method for computing equilibria of multilattices under an external force. The error estimates are derived in a W 1,∞ norm in one space dimension. One of the features of our analysis is that we establish an equivalent way of formulating the coarse-grained problem which greatly simplifies derivation of the...

Here a posteriori error estimate for the numerical solution of nonlinear Voltena- Hammerstein equations is given. We present an error upper bound for nonlinear Voltena-Hammastein integral equations, in which the form of nonlinearity is algebraic and develop a posteriori error estimate for the recently proposed method of Brunner for these problems (the implicitly linear collocation method)...

In this thesis we develop and evaluate the performance of adaptive finite element methods for multiphysics problems. In particular, we propose a methodology for deriving computable error estimates when solving unidirectionally coupled multiphysics problems using segregated finite element solvers. The error estimates are of a posteriori type and are derived using the standard framework of dual w...