Wavelet adaptive method for second order elliptic problems: boundary conditions and domain decomposition

Wavelet methods allow to combine high order accuracy, multilevel preconditi oning techniques and adaptive approximation, in order to solve efficiently elliptic operator equations. One of the main difficulty in this context is the efficient treatment of non-homogeneous boundary conditions. In this pap er, we propose a strategy that allows to append such conditions in the sett ing of space refinement (i.e. adaptive) discretizations of second order pro blems. Our method is based on the use of compatible multiscale decompositio ns for both the domain and its boundary, and on the possibility of characte rizing various function spaces from the numerical properties of these decom positions. In particular, this allows the construction of a lifting operato r which is stable for a certain range of smoothness classes, and preserves the compression of the solution in the wavelet basis. An explicit construct ion of the wavelet bases and the lifting is proposed on fairly general doma ins, based on CO conforming domain decomposition techniques.