A posteriori error estimators are an essential ingredient for the numerical solution of boundary value problems with adaptive finite element methods. Starting with the seminal work of Babuska around 1980, a relatively complete theory for the derivation of a posteriori estimators has been developed; see e.g. the monographs by Ainsworth/Oden, Babuska/Strouboulis, and Verfürth.
An issue less clarified, which has recently attracted interest, is the robustness of estimators. An error estimator is robust with respect to a certain class of problems and discretizations if the constants in the corresponding a posteriori estimates are uniform in that class.
This special day intends to assess a posteriori error estimators and their robustness, in particular with respect to the following aspects: strong advection and reaction, jumps and anisotropy of the diffusivity, and properties of the mesh.
09:00-09:15 Opening
09:15-10:15 Rüdiger Verfürth
(Ruhr Universität Bochum)
A review of robust a posteriori error estimates
(hand out)
10:15-10:45 Coffee break in Aula C
10:45-11:30 Giancarlo Sangalli (Università di Pavia)
Robust a posteriori estimators for
advection-dominated problems
11:30-12:15 Stefano Berrone (Politecnico di Torino)
Robust a posteriori error estimates for
finite element discretizations of the heat equation with discontinuous
coefficients
12:15-14:15 Lunch in Aula C
14:15-15:00 Simona Perotto (MOX, Politecnico di Milano)
Robustness of a posteriori error estimator
on anisotropic grids
15:00-15:45 Marco Verani (MOX, Politecnico di Milano)
A safeguarded dual weighted residual
method
15:45-16:15 Coffee break in Aula C
16:15-17:00 Ilaria Perugia (Università di Pavia)
A posteriori error indicators for
discontinuous Galerkin discretizations of eddy current
models
Robust a posteriori error estimates for
finite element discretizations of the heat equation with discontinuous
coefficients
Stefano Berrone
(Politecnico di Torino)
We derive a posteriori error estimates based on equations residuals
for the heat equation with discontinuous diffusivity coefficients.
The estimates are based on a fully discrete scheme based
on conforming finite elements in each time slab and on the A-stable
theta-scheme.
Assuming a quasi-monotonicity condition, we obtain upper and lower
bounds whose ratio is independent of any mesh-size, time-step, problem
parameter and its jumps.
Numerical results allow to identify a time-discretization error-estimator
and a space-discretization error-estimator.
In this work we introduce a similar splitting for the
data-approximation error in time and in space.
A simple adaptive algorithm based on these a posteriori error
estimates is proposed and some numerical experiments are presented.
Robustness of a posteriori error estimator
on anisotropic grids
Simona Perotto (MOX,
Politecnico di Milano)
In this communication we review the basics of the error estimator proposed in [2]. This estimator merges the good properties of computationally cheapness and capability of detecting the directional features of the solution to the problem at hand. This goal is attained by properly combining the Zienkiewicz-Zhu gradient recovery procedure [4] with the anisotropic framework in [1]. The popularity of the Zienkiewicz-Zhu approach can be attributed to several factors: the method is rather independent of the considered problem and of most details of the finite element formulation (except for the finite element space); it is cheap to compute and easy to implement and, first and foremost, the method works very well in practice. However, the theoretical properties of this procedure are not very well understood yet, though many theoretical investigations have been carried out in the literature. On the other hand, it is well-known that, in the presence of physical problems exhibiting directional features (internal and/or boundary layers, shocks, etc.), the effectiveness of finite element procedures can be improved by the employment of meshes suitably oriented. In more detail, the approach we pursue is the following: we start from a standard residual-based analysis, as in [3]; then the error in the interpolation terms is bounded via suitable anisotropic error estimates; finally, the derivatives of the exact solution entering these anisotropic terms are replaced by recovered quantities, in the spirit of the Zienkiewicz-Zhu procedure. We focus on the reliability and the efficiency of the proposed estimator in the case of the Poisson problem provided with mixed boundary conditions.
A posteriori error indicators for
discontinuous Galerkin discretizations of eddy current
models
Ilaria Perugia
(Università degli Studi di Pavia)
In this talk, a residual-based a posteriori error indicator for
discontinuous Galerkin (DG) discretizations of Hcurl-elliptic
boundary value problems arising in eddy current models will be
presented. In particular, this error indicator is shown to be both
reliable and efficient with respect to the approximation error
measured in terms of a natural energy norm.
The proof of the upper bound of the proposed error indicator is
based on rewriting the method in a non-consistent manner, and on
estimates of the error between the analytical solution and a
conforming approximant of it. This approach allows for overcoming
the drawback of previous analyzes, where Helmholtz decompositions of
the error were applied, leading to error estimators which depend on
certain Sobolev regularity parameters.
The performance of the presented indicator within an adaptive mesh
refinement procedure will be validated, showing its asymptotic
exactness for a range of test problems.
Robust a posteriori estimators for
advection-dominated problems
Giancarlo
Sangalli (Università degli Studi di Pavia)
Unlike the a-posteriori error analysis for elliptic problems, which is by now very mature, the theory for advection-dominated problems is still under development. Even though in recent years some progresses have been achieved, in fact the developed analysis is not fully satisfactory even for the simplest one-dimensional problem. This is the case I first consider for the theoretical study in the present talk. In spite of its simplicity, the robust a posteriori analysis of it is harder than it might seem at the first sight. In the estimate proposed, the numerical error is measured with respect to a norm which plays the role that the energy norm has with respect to the reaction-diffusion operator (i.e., the symmetric case). I also discuss the extension of the theory to the higher dimensional case, which presents additional difficulties. Eventually, I will show numerical tests validating the theoretical results.
A safeguarded dual weighted residual
method
Marco Verani (MOX, Politecnico di
Milano)
The dual weighted residual (DWR) method yields reliable a
posteriori error bounds for linear output functionals provided that
the error incurred by the numerical approximation of the dual
solution is negligible. In that case its performance is generally
superior than that of global `energy norm' error estimators which
are `unconditionally' reliable. We present a simple numerical
example for which neglecting the approximation error leads to severe
underestimation of the functional error, thus showing that the DWR
method may be unreliable. We propose a remedy that preserves the
original performance, namely a DWR method safeguarded by additional
asymptotically higher order a posteriori terms. In particular, the
enhanced estimator is unconditionally reliable and asymptotically
coincides with the original DWR method. These properties are
illustrated via the aforementioned example.
This is a joint work with R.H. Nochetto and A. Veeser.
A review of robust a posteriori error
estimates
Rüdiger Verfürth (Ruhr
Universität Bochum)
In this lecture we give an overview of robust a posteriori error
estimates for singularly perturbed elliptic and parabolic pdes. We
start with a simple diffusion dominated model problem to elaborate
the basic ingredients. Then we add a dominant reaction term and show
that the weighting factors in the error estimate must appropriately
be adapted to maintain robustness. For proving the robustness
standard quasi-interpolation error estimates and inverse
inequalities for local cut-off functions must be refined. In a next
step we pass to the case of dominant convection. Now one has to
consider an appropriate problem-adapted norm for obtaining
robustness. We also discuss the relation of this norm to standard
mesh-dependent norms used for the analysis of SUPG-schemes. Finally
we consider the non-stationary case and show how spatial and
temporal discretizations and error estimations interplay in an
optimal way.
(hand out)