A General Framework for Finite Element A Posteriori Error Control: Application to Linear and Nonlinear Convection-Dominated Problems (original) (raw)
arXiv: Numerical Analysis, 2018
Even though substantial progress has been made in the numerical approximation of convection-dominated problems, its major challenges remain in the scope of current research. In particular, parameter robust a posteriori error estimates for quantities of physical interest and adaptive mesh refinement strategies with proved convergence are still missing. Here, we study numerically the potential of the Dual Weighted Residual (DWR) approach applied to stabilized finite element methods to further enhance the quality of approximations. The impact of a strict application of the DWR methodology is particularly focused rather than the reduction of computational costs for solving the dual problem by interpolation or localization.
Stabilization techniques for finite element analysis of convection-diffusion problems
The accurate solution of convection type problems on practical grids has been ever a challenging issue, and invariably some sort of stabilization is needed in order to get a physical solution. This has pushed researchers to develop various stabilization algorithms used in every day practice by numerical analysts. In this chapter some methods are presented along with a new finite increment calculus approach to obtain the different algorithms using higher order conservation equations.
An optimal control approach to a posteriori error estimation in finite element methods
Acta Numerica, 2001
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 L 2 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.
Adaptavity for the Control Volume Finite Element Method in Convection-Diffusion Problems
1992
In Computational Fluid Dynamics it is usual to find the problem of increasing the accuracy of a solution without adding unnecessary degrees of freedom. It is therefore necessary to update the mesh so as to ensure that it becomes fine enough in the critical region while remaining reasonably coarse in the rest of the domain. Local a posteriori error estimators are the adequate tool for identifying automatically these critical regions. They should use only given data and the numerical solution itself. In this work the Control Volume Finite Element Method (CVFEM) for the Conveetion:Diffusion equation is considered. This is a nonconforming method in the sense that the interpolant space for the solution is not a subset of Hl. Despite of this fact, many years of numerical experiences have established the excellent behaviour of this method in non-selfadjoint problems. In the conforming case several approaches have been introduced for selfadjoint problems by using the residual equations. In ...
International Journal of Apllied Mathematics, 2015
Computing solutions of convection-diffusion equations is an important and challenging problem from the numerical point of view. We present in this work a numerical scheme to study this problem. The scheme combines a stabilized finite element method introduced in [Serghini Mounim, A stabilized finite element method for convection-diffusion problems, Mumer. Methods Partial Differential Eq 28: 1916-1943, 2012], with an adaptive mesh refinement procedure which is based on the residual a posteriori error estimators. It is worthwhile to point out that the numerical results indicate that the stabilization parameter introduced in [Serghini Mounim, A stabilized finite element method for convection-diffusion problems, Numer. Methods Partial Differential Eq. 28 (2012), 1916-1943] gives much better results than the standard Streamline upwind/Petrov-Galerkin (SUPG) one.
A Residual Based A Posteriori Error Estimators for AFC Schemes for Convection-Diffusion Equations
2020
In this work, we propose a residual-based a posteriori error estimator for algebraic flux-corrected (AFC) schemes for stationary convection-diffusion equations. A global upper bound is derived for the error in the energy norm for a general choice of the limiter which defines the nonlinear stabilization term. In the diffusion-dominated regime, the estimator has the same convergence properties as the true error. A second approach is discussed, where the upper bound is derived in a posteriori way using the SUPG estimator proposed in John & Novo, (2013). Numerical examples study the effectivity index and the adaptive grid refinement for two limiters.
Analysis of finite element schemes for convection-type problems
International Journal for Numerical Methods in Fluids, 1995
Various finite element schemes of the Bubnov–Galerkin and Taylor–Galerkin types are analysed to obtain the expressions of truncation errors. This way, dispersion errors in the transient, and diffusion errors both in the transient and in the steady state, are identified. Then, with reference to the transient advection–diffusion equation, stability limits are determined by means of a general von Neumann procedure. Finally, the operational equivalence between Taylor–Galerkin methods, utilized for pseudo-transient calculations, and Petrov–Galerkin methods, derived for the steady state forms of the advection–diffusion equation, is illustrated. Theoretical conclusions are supported by the results of numerical experiments.
A posteriori error control for finite element approximations of elliptic eigenvalue problems
2001
We develop a new approach to a posteriori error estimation for Galerkin finite element approximations of symmetric and nonsymmetric elliptic eigenvalue problems. The idea is to embed the eigenvalue approximation into the general framework of Galerkin methods for nonlinear variational equations. In this context residual-based a posteriori error representations are available with explicitly given remainder terms. The careful evaluation of these error representations for the concrete situation of an eigenvalue problem results in a posteriori error estimates for the approximations of eigenvalues as well as eigenfunctions. These suggest local error indicators that are used in the mesh refinement process.