Domain decomposition methods for CAD.

*(English)*Zbl 0937.68140Summary: Constructive solid geometry in CAD leads to domain decompositions which are based on primitive shapes, as briefly explained in the introduction below. A special role is played by “holes”, which can be viewed in several different ways. We combine this remark with the method of virtual controls. In a previous note [the authors, ibid. 327, No. 11, 947-952 (1998; Zbl 0918.65071)] where this method was introduced, the distributions resulting from the decomposition and Green’s formula where thought of as virtual controls – here the virtual controls are introduced a priori with support in the holes, or outside the domain, or in the intersections of the domains. They are then chosen so as to (approximately) satisfy the boundary conditions, which is possible by virtue of approximate controlability results (which have to be proven!). These ideas are presented here on an example (Section 1) which is certainly not the most general one but which is sufficient to show how everything extends to very many other situations. Algorithms, following the approximate controllability lemma, are given in Section 2 and a numerical example is presented in Section 4.