Adaptive Methods and a Posteriori Error Estimators for Partial Differential Equations

Analysis for Discretizations which do not Posses the Galerkin Orthogonality

Two-level methods for the Navier-Stokes equations
V. John, "Residual A posteriori error estimates for two-level finite element methods for the Navier-Stokes equations" , Applied Numerical Mathematics 37, 503 - 518, 2001
Residual based a posteriori error estimates for conforming finite element solutions of the incompressible Navier--Stokes equations which are computed with four recently proposed two--level methods are derived. The a posteriori error estimates contain additional terms in comparison to the estimates for the solution obtained by the standard one--level method. The importance of these additional terms in the error estimates is investigated by studying their asymptotic behaviour. For optimally scaled meshes, these bounds are not of higher order than the order of convergence of the discrete solution.
Non-conforming discretization of the Stokes equations
V. John, "A Posteriori L²-Error Estimates for the Nonconforming P1/P0-Finite Element Discretization of the Stokes Equations", J. Comp. Appl. Math. 96 (2): 99 - 116, 1998
This paper focusses on an residual based a posteriori error estimator for the $L^2$--error of the velocity for the nonconforming $P_1/P_0$--finite element discretization of the Stokes equations. We derive an a posteriori error estimator which yields a local lower as well as a global upper bound on the error. Numerical tests demonstrate the efficiency of the global error estimator and give a comparison with respect to the adaptive grid refinement to an a posteriori error estimator in the discrete energy norm proposed by Dari et al. (1995).

Computational Results and Estimates for Convection-Diffusion Equations

Javier de Frutos, Bosco Garcia-Archilla, Volker John, Julia Novo An adaptive SUPG method for evolutionary convection-diffusion equations, Comput. Methods Appl. Mech. Engrg. 273, 219 - 237, 2014
An adaptive algorithm for the numerical simulation of time-dependent convection-diffusion-reaction equations will be proposed and studied. The algorithm allows the use of the natural extension of any error estimator for the steady-state problem for controlling local refinement and coarsening. The main idea consists in considering the SUPG solution of the evolutionary problem as the SUPG solution of a particular steady-state convection-diffusion problem with data depending on the computed solution. The application of the error estimator is based on a heuristic argument by considering a certain term to be of higher order. This argument is supported in the one-dimensional case by numerical analysis. In the numerical studies, particularly the residual-based error estimator from \cite{JN13} will be applied, which has proved to be robust in the SUPG norm. The effectivity of this error estimator will be studied and the numerical results (accuracy of the solution, fineness of the meshes) will be compared with results obtained by utilizing the adaptive algorithm proposed in \cite{JBJ_adap}.
Volker John, Julia Novo A robust SUPG norm a posteriori error estimator for stationary convection-diffusion equations , Comput. Methods Appl. Mech. Engrg. 255, 289 - 305, 2013
A robust residual-based a posteriori estimator is proposed for the SUPG finite element method applied to stationary convection-diffusion-reaction equations. The error in the natural SUPG norm is estimated. The main concern of this paper is the consideration of the convection-dominated regime. A global upper bound and a local lower bound for the error are derived, where the global upper estimate relies on some hypotheses. Numerical studies demonstrate the robustness of the estimator and the fulfillment of the hypotheses. A comparison to other residual-based estimators with respect to the adaptive grid refinement is also provided.
V. John, "A Numerical Study of A Posteriori Error Estimators for Convection--Diffusion Equations", Comput. Methods Appl. Mech. Engrg., 190, 757 - 781, 2000
This paper presents a numerical study of a posteriori error estimators for convection--diffusion equations. The study involves the gradient indicator, an a posteriori error estimator which is based on gradient recovery, three residual based error estimators for different norms, and two error estimators which are defined by solutions of local Neumann problems. They are compared with respect to the reliable estimation of the global error and with respect to the accuracy of the computed solutions on adaptively refined grids. The numerical study shows for both criteria of comparison that no one of the considered error estimators works satisfactory in all tests.

A Posteriori Error Estimates for the Large Eddies

Equilibrium Navier-Stokes equations
A. Dunca, V. John, W.J. Layton, "Approximating Local Averages of Fluid Velocities: the equilibrium Navier--Stokes equations" , Appl. Numer. Math., accepted for publication, 2003
In the approximation of higher Reynolds number flow problems, the usual approach is to seek to approximate suitable velocity averages rather than the pointwise fluid velocity itself. We consider an approach to this question wherein the averages are local, spatial averages computed with the Gaussian filter (as in large eddy simulation) and the averages are approximated without using either turbulent closure models or wall laws. The approach we consider is a (underresolved) direct numerical simulation followed by postprocessing to extract accurate flow averages. A priori and a posteriori estimates are given for $\| g_\delta\ast(\bu-\bu^h)\|_0$ which can give guidance for the coupling between the averaging radius $\delta$ and the mesh width $h$. Numerical experiments support the error estimates and illustrate the adaptive grid refinement procedure.
Stokes equations
V. John, W.L. Layton, "Approximating Local Averages of Fluid Velocities: The Stokes Problem" , Computing 66, 269 - 287, 2001
As a first step to developing mathematical support for finite element approximation to the large eddies in fluid motion we consider herein the Stokes problem. We show that the local average of the usual approximate flow field $\1^h$ over radius $\delta$ provides a very accurate approximation to the flow structures of $O(\delta)$ or greater. The extra accuracy appears for quadratic or higher velocity elements and degrades to the usual finite element accuracy as the averaging radius $\delta \rightarrow h$ (the local meshwidth). We give both a priori and a posteriori error estimates incorporating this effect.