Derivation of the discrete problem

A.

Number of degrees of freedom in :

.
Number of elements in :
,
effective (those that have at least one variable vertex):
.
B.

Take
,
then

integrating by parts we have:
,
,
which implies:

or

C.
Checking the assumptions:

1. Symmetry:
a(u,v) = a(v,u)
2. Continuity:

3. V-ellipticity:
Poincare–Friedrichs Inequality implies that for

,
s = 1 for our domain.
4. Continuity of L(v)=(1,v)
If we apply Schwartz inequality we have:
|L(v)|=|(1,v)|≤||1||||v||=||v||

E.

If we move to a finite dimensional space we will have to find the projection of the solution on this space

Instead of v we choose basis functions and get:

a(.,.) is a bilinear form, that allows us to take the sum out of it
,
for all j.
As both the left and the right side are integrals over the domain, we can divide them into parts, each corresponding to one element and compute the stiffness matrix and the load vector as the sum of these integrals in all finite elements. And since the functions have local support, there will be at most 6 triangles where one of the basis functions is non-zero. And each element will have at most 3 non zero basis functions.

G.

Transformation to standard triangle is
,
where
.
So, in order to use the solution from the reference triangle we should multiply gradient

and multiply the integral by Jacobian.