We consider an adaptive refinement method for numerical solution of
elliptic and parabolic partial differential equations. The resulting
linear system contains discretizations of the equation at different
grids. In order to find a good approximation of its solution, we propose
a reduction procedure that is inverse to the refinement one. The
reduction step results a less complicated system at a coarser grid. It
is defined by using discretizations at two levels - the finest one and
one level coarser.