information:filtering, structural:scholar_edition
For problem P2

We need V {\displaystyle V} to be a set of functions of Ω {\displaystyle \Omega }. In the figure on the right, we have illustrated a triangulation of a 15-sided polygonal region Ω {\displaystyle \Omega } in the plane (below), and a piecewise linear function (above, in color) of this polygon which is linear on each triangle of the triangulation; the space V {\displaystyle V} would consist of functions that are linear on each triangle of the chosen triangulation.

One hopes that as the underlying triangular mesh becomes finer and finer, the solution of the discrete problem (3) will, in some sense, converge to the solution of the original boundary value problem P2. To measure this mesh fineness, the triangulation is indexed by a real-valued parameter h > 0 {\displaystyle h>0} which one takes to be very small. This parameter will be related to the largest or average triangle size in the triangulation. As we refine the triangulation, the space of piecewise linear functions V {\displaystyle V} must also change with h {\displaystyle h}. For this reason, one often reads V h {\displaystyle V_{h}} instead of V {\displaystyle V} in the literature. Since we do not perform such an analysis, we will not use this notation.