We derive a posteriori error estimates for nonconforming discretizations of Poisson's and Stokes' equations. The estimates are residual based and make use of weight factors obtained by a duality argument. Crouzeix-Raviart elements on triangles and rotated bilinear elements are considered. The quadrilateral case involves the introduction of additional local trial functions. We show that their in...

In the talk, we discuss a posteriori error estimates for elliptic and parabolic viscous flow problems and give an overview of the results obtained in this field with the help of a new (functional) approach that was earlier applied to many problems in mathematical physics. This method provides guaranteed and computable error bounds that do not involve mesh dependent constants and are valid for a...

It is known that the energy technique for a posteriori error analysis of finite element discretizations of parabolic problems yields suboptimal rates in the norm L∞(0, T ; L2(Ω)). In this paper we combine energy techniques with an appropriate pointwise representation of the error based on an elliptic reconstruction operator which restores the optimal order (and regularity for piecewise polynomi...

We consider space-time discretizations of non-linear parabolic equations. The temporal discretizations in particular cover the implicit Euler scheme and the mid-point rule. For linear equations they correspond to the well-known A-stable θ-schemes. The spatial discretizations consist of standard conforming finite element spaces that can vary from one time-level to the other. The spatial meshes m...

An posteriori error analysis for the virtual element method (VEM) applied to general elliptic problems is presented. The resulting error estimator is of residual-type and applies on very general polygonal/polyhedral meshes. The estimator is fully computable as it relies only on quantities available from the VEM solution, namely its degrees of freedom and element-wise polynomial projection. Uppe...

Optimal a posteriori error estimates in L∞(0, T ; L(Ω)) are derived for the finite element approximation of Allen-Cahn equations. The estimates depend on the inverse of a small parameter only in a low order polynomial and are valid past topological changes of the evolving interface. The error analysis employs an elliptic reconstruction of the approximate solution and applies to a large class of...

under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Abstract In this paper, we give an a posteriori error estimates with constitutive law for some obstacle problem. The error estimator involves some parameter ε appeared in some penalized equation.

Here we survey some previously published results and announce some that have been newly obtained. We first review some of the results in [3] on estimates for the finite element error at a point. These estimates and analogous ones in [4] and [7] have been applied to problems in a posteriori estimates [2], [8], superconvergence [5] and others [9], [10]. We then discuss the extension of these esti...

Finite element exterior calculus (FEEC) has been developed over the past decade as a framework for constructing and analyzing stable and accurate numerical methods for partial differential equations by employing differential complexes. The recent work of Arnold, Falk, and Winther includes a well-developed theory of finite element methods for Hodge–Laplace problems, including a priori error esti...

Inference problems in dynamically data-driven application systems use physical measurements along with a physical model to estimate the parameters or state of a physical system. Errors in measurements and uncertainties in the model lead to inaccurate inference results. This work develops a methodology to estimate the impact of various errors on the variational solution of a DDDAS inference prob...