Conforming Finite Elements - Finite Elements: Theory, Fast Solvers, and Applications in Elasticity Theory - page 58

Fig. 11. Nodal basis function

Finally, we find that

a(ψ C ,ψ NW ) = [ ∂ 1 ψ C ∂ 1 ψ NW + ∂ 2 ψ C ∂ 2 ψ NW ] dxdy = 0 .

III + IV

In evaluating a(ψ C ,ψ SE ) , note that all products in the integrals vanish. Thus we
get a system of linear equations with exactly the same matrix as in the finite
difference method based on the standard five-point stencil

−

1

− 14 − 1 . ( 4 . 9 )

− 1

∗

We should emphasize that this connection with difference methods does not
hold in general. The finite element method provides the user with a great deal of
freedom, and for most other finite element approximations and other equations,
there is no equivalent finite difference star. In general, the finite element approxi-
mation does not even satisfy the discrete maximum principle. - The same holds,
by the way, for the method of finite volumes. Once again, we get the same matrix
only in the above simple case [Hackbusch 1989].
The stiffness matrix for the model problem was determined here in a node-
oriented way. We note that the matrices are assembled in a different way in real-
life computations, i.e. element-oriented . First, the contribution of each triangle
(element) to the stiffness matrix is determined by doing the computation only for
a master triangle (reference element). Finally the contributions of all triangles are
added.