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6 can also be interpreted as follows. 55) of order d for S( 1>j) imply the cancellation properties where W! c Loo :s 2-j(n(2+dlllvllwi,(suPP1/Jj,kl' is the Sobolev space of order d. 55} means that all polynomials up to order d are contained in S(1)j). One also says that S(1)j) is exact of order d. 32}, this implies that the wavelets ~j,k are orthogonal to polynomials up to order d or have dth order vanishing moments. 64}. 53). 52) is also equivalent to Ivl. 61) read IlvllHs rv Illvlll s rv Ivl s .
G. [Br, BF]. 32) PA - 'YBAYA alternately, starting with an initial approximation for PA and YA. 32) does not depend on A. 4. g. [BWY, BP, BPV1]. 9) below that yield uniformly bounded condition numbers has been constructed in [K1J, see also [K3]. 2 Elliptic Boundary Value Problems as Saddle Point Problems Now we are ready to formulate elliptic boundary value problems as saddle point problems, placing particular emphasis on a flexible treatment of essential, non-homogeneous boundary conditions. 1 with Lipschitz continuous boundary an and let r <:: an be a smooth subset.
1) to pose a well-defined problem. 13). 13) possesses a unique solution. ) is the same for functions in Hl(O) and Hl(Sl). Different fictitious domain approaches have been employed for quite a long time, both on the continuous as well as on the matrix level where this is called matrix capacitance method, see for instance [Ag, As, FK, GPP, HHK, KP, MKM, Nep, PW, Ril]. Lately they have become even more interesting since, for 3D problems, mesh generation of Sl can become quite expensive. Of course, the above mentioned advantages will only prevail if there is no need to adapt the discretization on 0 to the particular current domain geometry Sl.