A posteriori error estimation using Bank-Weiser type estimator

A posteriori error estimation using a Bank-Weiser type estimator

The main aim of a posteriori error estimation is to provide computable bounds for the error commited solving PDEs using finite elements methods. A crucial property for an a posteriori estimator is to be local, in other words to be able to bound the error on each cell of the mesh. Using this property we are able to determine on which part of the domain the error is mainly distributed. This information allows us to refine the mesh in this special zone and try to reduce the global error while maintaining a reasonable computational cost. Among all a posteriori error estimators we can cite the residual estimator described in [1] or the ZZ estimator introduced by O.C. Zienkiewicz and J.Z. Zhu in [2]. Here we will focus on an estimator introduced by R.E. Bank and A. Weiser in [3] and applied on elliptic PDEs.

 

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Comparisons between BW estimator and the true error for FE of order 1, in dimension 1, where the solution is a Gaussian.

 

Definition of the estimator

Let us call true error the difference between the solution of our PDE and its Galerkin approximation. We build the Bank-Weiser estimator (BW estimator) from a specific discontinuous polynomial function called residual solution. This function is defined locally on each cell of the mesh as the elliptic projection (in the sense of Céa’s lemma, see [4]) of the (local) true error on a specific discontinuous finite elements (FE) space.

 

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Comparison between BW estimator and the true error for FE of order 3, in dimension 1, where the solution is a Gaussian.

This FE space is chosen in order to ensure that this residual solution is well-defined. Indeed, the computation of this residual solution implies to solve local Neumann problems on each cell and specific assumptions must be made to ensure their well-posedness.

 

Since this solution is defined as a projection, it is the best approximation of the true error we can find on this subspace. Finally, we define the BW estimator as the sum over the cells of the norm of the energy of the residual solution.

 

Cons and pros of the Bank-Weiser estimator

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Residual solution in dimenson 2 for a Gaussian solution.

The first advantage of the BW estimator is its accuracy for problems admitting a rough solution (H1) as well as when the solution is more regular (H2). Indeed, numerical results has been done in the case H1 in [3] when the solution admits a singularity. Similarily, the estimator has been generalized to the Stokes problem in [5] and the numerical results suggest a good accuracy in this case as well.

 

Another advantage of the BW estimator is the (relative) rapidity of computation, due to the choice of the FE space containing the residual solution.

However, the main disadvantage of the BW estimator lays in the lack of theoretical guaranties. Indeed, the proof of the equivalence between this estimator and the true error has been proved (in [3]) but only under a constraining

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BW estimator in dimension 2 for a Gaussian solution.

assumption called saturation assumption. This assumption can be related to the regularity of the data (and then the regularity of the solution). It is possible to find very simple cases where the saturation assumption fails, see [6]. Also, the numerical examples shown in [3, 5] do not satisfy this assumption but the BW estimator remains accurate.

For the time being the theory can not fully explain this accuracy except in the case of linear FE with a proof from R. H. Nochetto in [7]. Even if the  numerical results seem to suggest it, the extension of this proof to higher order FE is still an open question.

 

 

Screenshot 2018-04-23 19.48.17Future work

Our aim is to implement an efficient algorithm computing the BW estimator using the FEniCS software. By definition the computation of the BW estimator is fully parallelisable which is very interesting in order to save computational time. Then we will be able to test it on mechanics problems and compare it to other estimators like the residual estimator or ZZ estimator and corroborate the results of the papers cited above.

 

[1] – Mark Ainsworth and J. Tinsley Oden. A posteriori error estimation in finite element analysis. Pure and Applied Mathematics (New York). Wiley-Interscience [John Wiley & Sons], New York, 2000.

[2] – O.C. Zienkiewicz and J.Z. Zhu. A simple error estimator and adaptive procedure for practical engineering analysis. Internat. J. Numer. Methods Engrg., 24:337-357, 1987.

[3] – R. E. Bank and A. Weiser. Some a posteriori error estimators for elliptic partial differential equations. Math. Comp., 44(170):283–301, 1985.

[4] – A. Ern and J.L. Guermond. Theory and Practice of Finite Elements. Applied Mathematical Sciences. Springer New York, 2004.

[5] – R. E. Bank and B. D. Welfert. A posteriori error estimates for the Stokes problem. SIAM J. Numer. Anal.591–623, 1991.

[6] – W. Dörfler and R. H. Nochetto. Small data oscillation implies the saturation assumption. Numer. Math. 1–12, 2002.

[7] – R. H. Nochetto. Removing the saturation assumption in a posteriori error analysis. Istit. Lombardo Accad. Sci. Lett. Rend. A. 67–82, 1993.