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.
Since Mathematica 10.3 ToElementMesh function can be used, along with FEM solver. For details see this link
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]]
Another approach is to use Imtek's package. These deal with both 2 and 3D with interfaces to Shewchuck's triangle and Tetgen respectively.
Imtek can be had from the University of Freiburg. Documentation is extensive.