This edition contains four new sections on the following topics: the BDDCdomain decomposition preconditioner (Section 7.8), a convergent adaptivealgorithm (Section 9.5), interior penalty methods (Section 10.5) andPoincar´e-Friedrichs inequalities for piecewise W1p functions (Section 10.6).We have made improvements throughout the text, many of which weresuggested by colleagues, to whom we are grateful. New exercises have beenadded and the list of references has also been expanded and updated.Some of the new material originated from our research and we wouldlike to thank the National Science Foundation for support. The first authorwould also like to thank the Alexander von Humboldt Foundation forsupporting her visit to Germany in the Summer of 2007, during which thework on this edition was completed. The second author would also like tothank the Universit´e Pierre et Marie Curie for supporting his visits to Parisduring the past several years, during which work related to this edition wascarried out.