Conforming and non-conforming FEM Can anyone explain me what is the basic difference between a conforming finite element space and a non conforming finite element space.
Say $\Omega\subset \mathbb R^2$ is some domain and $\tau_h$ be its triangulation and say $K_1$ and $K_2$ are two neighboring triangles in the triangulation with say common edge $'e'$ and common vertices as $a_1$ and $a_2$ then what is the basic condition to say that the finite element space  is conforming and non conforming.
Note : Assume for the time being that these two triangles constitute the triangulation. Will it make any difference in the theory of the definition of these spaces.
Any type of help will be appreciated. Thanks in advance.
 A: FEM is for solving a variational problem: Find $u\in V$ such that $a(u,v)=f(v)$ for all $v\in V$ with $a$ bilinear, $f$ linear.
The space $V$ is approximated by a finite dimensional space $V_h$.
Conforming FEM means that $V_h \subset V$.
For the standard case: $V=H^1(\Omega)$, mesh of triangles, $V_h$ consisting of functions that are smooth on each triangle, conformity is equivalent to continuity of the trial functions.
A: Conforming FEM uses a mesh that is made with elements that are conforming. 
Non Conforming FEM uses a mesh that is made with elements that are nonconforming.
Conforming FEM is the standard FEM that you will learn in undergraduate courses or in general if you just type in "Finite Element Method" online.
In the standard FEM or just FEM, there is a restriction on how the mesh can be designed. Consider the case that the elements are triangular. Any two triangle elements that intersect on an edge need to do so completely. So that the edge of one triangle completely overlaps the edge of another triangle. 
If the mesh is designed with all triangle elements that do this, we call that a conforming mesh.
