Can anyone help me with good references (books or papers) where I can learn about dual-weighted residual (DWR) method for goal-oriented adaptive mesh refinement (AMR)?


The canonical "first" reference for the method is a paper by Becker and Rannacher that was ultimately published as an article in the ENUMATH 97 proceedings, but is often cited as the following preprint:

R. Becker, R. Rannacher: "A feed-back approach to error control in finite element methods: basic analysis and examples". IWR preprint, University of Heidelberg, 1996.

Because of the difficulty of actually locating this preprint, Becker and Rannacher later wrote a longer article on the subject in Acta Numerica:

R. Becker, R. Rannacher, "An optimal control approach to a posteriori error estimation in finite element methods," Acta Numerica, vol. 10, pp. 1–102, May 2001.

Then Rannacher thought "Why stop at 102 pages if I could write a whole book about the subject", and this led to the following book (disclaimer, if it's not obvious: I'm one of the co-authors):

W. Bangerth, R. Rannacher: "Adaptive finite element methods for differential equations". Birkhäuser, 2003.

I believe that these are the three most frequently cited resources on the DWR method for error estimation and mesh adaptation.


You may want to take a look at the arXiv preprint of

In this article, the authors motivate and present a new duality theory for FEM. Some overview of dual-weighted residual is also given (Section 5).

It also links to the very widely cited:

specifically for dual-weighted residual method.


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