# Applied Numerical Mathematics 61 (February 2011) by Robert Beauwens, Martin Berzins By Robert Beauwens, Martin Berzins

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1) by the ﬁnite element method of lines. Let Th and Fh denote the partitions of Ω and Ωu into the disjoint open regular n-simplicity K , respectively. Every element has at most one face on ∂Ω or ∂Ω u . Any two elements have at most either one common vertex, or a whole edge, or a whole face. Every element K in triangulation (or rectangularity) partition Th is aﬃne equivalent to one of several reference elements. g. ). , Γh = K ∈Th S ∈∂ K S . , P r ( K ) = span x|α | , 0 |α | r , P m ( K u ) = span x|α | , 0 |α | m .

V. on behalf of IMACS. All rights reserved. 1. Introduction The optimal control or design and the ﬁnite element method are crucial to many engineering applications. It is essential to apply eﬃcient numerical methods in solving the optimal control problem. Finite element approximation of the optimal control problems has been an important and hot topic in engineering design work, and has been extensively studied in literature [13,17,14,21,18]. g. [5,20,7,19,22,15]. Recently a number of techniques for the a posteriori error estimates have been developed.

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