This monograph presents a summary account of the subject of a posteriori error estimation for finite element approximations of problems in mechanics. The study primarily focuses on methods for linear elliptic boundary value problems. However, error estimation for unsymmetrical systems, nonlinear problems, including the Navier-Stokes equations, and indefinite problems, such as represented by the Stokes problem are included. The main thrust is to obtain error estimators for the error measured in the energy norm, but techniques for other norms are also discussed.

A Posteriori Error Estimation in Finite Element Analysis

This book is a tutorial written by researchers and developers behind the FEniCS Project and explores an advanced, expressive approach to the development of mathematical software. The presentation spans mathematical background, software design and the use of FEniCS in applications. Theoretical aspects are complemented with computer code which is available as free/open source software. The book begins with a special introductory tutorial for beginners. Followingare chapters in Part I addressing fundamental aspects of the approach to automating the creation of finite element solvers. Chapters in Part II address the design and implementation of the FEnicS software. Chapters in Part III present the application of FEniCS to a wide range of applications, including fluid flow, solid mechanics, electromagnetics and geophysics.

Automated Solution of Differential Equations by the Finite Element Method: The FEniCS Book

This review covers Verification, Validation, Confirmation and related subjects for computational fluid dynamics (CFD), including error taxonomies, error estimation and banding, convergence rates, surrogate estimators, nonlinear dynamics, and error estimation for grid adaptation vs Quantification of Uncertainty.

/pdf/quantification-of-uncertainty-in-computational-fluid-3imbkhwf41.pdf

Quantification of uncertainty in computational fluid dynamics

This article surveys a general approach to error control and adaptive mesh design in Galerkin finite element methods that is based on duality principles as used in optimal control. Most of the existing work on a posteriori error analysis deals with error estimation in global norms like the ‘energy norm’ or the L2 norm, involving usually unknown ‘stability constants’. However, in most applications, the error in a global norm does not provide useful bounds for the errors in the quantities of real physical interest. Further, their sensitivity to local error sources is not properly represented by global stability constants. These deficiencies are overcome by employing duality techniques, as is common in a priori error analysis of finite element methods, and replacing the global stability constants by computationally obtained local sensitivity factors. Combining this with Galerkin orthogonality, a posteriori estimates can be derived directly for the error in the target quantity. In these estimates local residuals of the computed solution are multiplied by weights which measure the dependence of the error on the local residuals. Those, in turn, can be controlled by locally refining or coarsening the computational mesh. The weights are obtained by approximately solving a linear adjoint problem. The resulting a posteriori error estimates provide the basis of a feedback process for successively constructing economical meshes and corresponding error bounds tailored to the particular goal of the computation. This approach, called the ‘dual-weighted-residual method’, is introduced initially within an abstract functional analytic setting, and is then developed in detail for several model situations featuring the characteristic properties of elliptic, parabolic and hyperbolic problems. After having discussed the basic properties of duality-based adaptivity, we demonstrate the potential of this approach by presenting a selection of results obtained for practical test cases. These include problems from viscous fluid flow, chemically reactive flow, elasto-plasticity, radiative transfer, and optimal control. Throughout the paper, open theoretical and practical problems are stated together with references to the relevant literature.

An optimal control approach to a posteriori error estimation in finite element methods

https://www.math.tamu.edu/~guermond/PUBLICATIONS/guermond_minev_shen_CMAME_2006.pdf

An overview of projection methods for incompressible flows

This is the first part of a work dealing with the rigorous error analysis of finite element solutions of the nonstationary Navier–Stokes equations. Second-order error estimates are proven for spatial discretization, using conforming or nonconforming elements. The results indicate a fluid-like behavior of the approximations, even in the case of large data, so long as the solution remains regular. The analysis is based on sharp a priori estimates for the solution, particularly reflecting its behavior as$t \to 0$ and as $t \to \infty $. It is shown that the regularity customarily assumed in the error analysis for corresponding parabolic problems cannot be realistically assumed in the case of the Navier–Stokes equations, as it depends on nonlocal compatibility conditions for the data. The results which are presented here are independent of such compatibility conditions, which cannot be verified in practice.

Finite element approximation of the nonstationary Navier-Stokes problem. I : Regularity of solutions and second-order error estimates for spatial discretization

This paper provides an error analysis for the Crank–Nicolson method of time discretization applied to spatially discrete Galerkin approximations of the nonstationary Navier–Stokes equations. Second-order error estimates are proven locally in time under realistic assumptions about the regularity of the solution. For approximations of an exponentially stable solution, these local error estimates are extended uniformly in time as ${\text{t}} \to \infty $.

Finite-element approximations of the nonstationary Navier-Stokes problem. Part IV: error estimates for second-order time discretization

An overview of benchmark computations for 2D and 3D laminar flows around a cylinder is given, which have been defined for a comparison of different solution approaches for the incompressible Navier-Stokes equations developed within the Priority Research Programme. The exact definitions of the benchmarks are recapitulated and the numerical schemes and computers employed by the various participating groups are summarized. A detailed evaluation of the results provided is given, also including a comparison with a reference experiment. The principal purpose of the benchmarks is discussed and some general conclusions which can be drawn from the results are formulated.

https://www.mathematik.tu-dortmund.de/lsiii/cms/papers/SchaeferTurek1996.pdf

Benchmark Computations of Laminar Flow Around a Cylinder

A simple nonconforming quadrilateral Stokes element based on “rotated” multi-linear shape functions is analyzed. On strongly nonuniform meshes the usual parametric version of this element suffers from a lack of consistency, while its nonparametric counterpart turns out to be convergent with optimal orders. This theoretical result is confirmed by numerical tests.

Rolf Rannacher

Papers

An optimal control approach to a posteriori error estimation in finite element methods

Finite element approximation of the nonstationary Navier-Stokes problem. I : Regularity of solutions and second-order error estimates for spatial discretization

Finite-element approximations of the nonstationary Navier-Stokes problem. Part IV: error estimates for second-order time discretization

Benchmark Computations of Laminar Flow Around a Cylinder

Simple nonconforming quadrilateral Stokes element