A Linear System Solver for Adaptive FEM

In this paper it is presented an approximate direct solver for systems of linear equations arising from finite element discretizations. The approximation is implemented using estimates arising from functional analysis or matrix analysis of the differential problem discretized using finite elements, or of its discrete counter-part. It results an advantage in terms of less matrix updates during the factorization, and possibly less {\it fill-in} and less equations to be included in the factorization (with a considerable computational saving, in this case). This last advantage arise frequently during successive updates of the solution after refinement/un-refinement steps in an adaptive analysis. The basic principle underlying the method here proposed is related to the knowledge of the Green's function associated to the differential problem discretized by finite elements. Early experimentation shows the method to be promising.