This paper is concerned with linear algebra based methods for solving exactly polynomial systems through so-called Gröbner bases, which allow one to compute modulo the polynomial ideal generated by the input equations. This is a topical issue in nonlinear algebra and more broadly in computational mathematics because of its numerous applications in engineering and computing sciences. Such applications often require geometric computing features such as representing the closure of the set difference of two solution sets to given polynomial systems. Algebraically, this boils down to computing Gröbner bases of colon and/or saturation polynomial ideals. In this paper, we describe and analyze new Gröbner bases algorithms for this task and present implementations which are more efficient by several orders of magnitude than the state-of-the-art software.