Q:

How to generate 2D FEM mesh in Mathematica? 

To generate 3D mesh TetGen can be easily used.  Are there similar functions (or a way to use TetGen) to generate 2d mesh?  I know that such functionality can be easily implemented but I would like to use a Mathematica provided function, as I need to experiment with number of nodes in elements and so on.  I just want to solve PDE using FEM not really to play around with mesh generation. 

A:

Since Mathematica 10.3 ToElementMesh[] function can be used, along with FEM solver.  For details see this link

A:

You can use TetGen and just make a z axis which is set to 0 (or any other constant).   The issue with DelaunayTriangulation (if you want to generate a mesh from a list of points) is that it returns an adjacency list of the edges, which is very hard to turn into the polygons.   This thread describes the issue with it. 
Adding a z dimention of 0 is simply:
pts3d = Map[Append[#, 0]&, pts2d]

And then to turn the 3d points back into 2d:
newpts2d = pts3d[[All,1;;2]]

