Finite element representations are accurate and convergent
in an integral sense, such as in 
norm.
This does not mean, however, that the error at any given
point will converge to zero as the mesh is refined. For this
reason, the errors in the finite element Laplacian
can be a source of noise,
unless the mesh is uniform.
This is not a problem when
one wants to invert the Laplacian, as in
eq.(19). This is a smoothing operation and the
solution is quite acceptable. On the other hand, calculating
the Laplacian, as in eq.(20) can be quite
inaccurate locally.
Forming the Laplacian by multiplying by the stiffness matrix
and the inverse mass matrix, 
, yields
excellent results on a uniform mesh.
The
Laplacian of a smooth function on this mesh is also smooth.
Unfortunately, it is very easy to encounter a non uniform mesh.
Fig.13(b)
shows
with 
on
a mesh with uniformly spaced mesh points,
but the triangle vertices have alternately
4 and 8 neighbors. A mesh of this type is used below, and shown in
Fig.13(a).
Refining the mesh is of no help, if it retains the
alternating number of neighbors. This example shows that in some
cases, the usual Laplacian is not locally convergent.
A uniform mesh in which all triangle vertices have 6 neighbors can be obtained from this mesh by a simple reconnection operation. One approach is to use vertex reconnection to try to equalize the number of neighbors of each vertex. However, with more complex meshes, it is virtually impossible to do this.
A way to improve the calculation of
the Laplacian
is to obtain the discretization of 
by
applying the gradient operator
twice.
First, the components of 
are calculated and re expanded in basis functions. Then, the discretized
derivatives of these quantities are obtained to get
which is again expanded in basis
functions. The lumped mass is used to make the solution for the
derivatives trivial.
This method improves the accuracy because
the components are first averaged from triangles
to vertices, smoothing the gradient. The divergence calculation performs
a similar smoothing. The method works because taking a single
derivative is first order accurate. Taking a second derivative
in two steps is formally first order accurate, and the method
in practice is found to be convergent, unlike the original
method.
The discretized version of the gradient is
The effect of applying eq. (36) twice in succession is equivalent to using a new stiffness matrix, S', given by
In the above eqs. (36) and (37), it is
sufficient to use the lumped mass matrix (33).
This stiffness matrix has the same symmetry properties as the
standard stiffness matrix. Fig.13(c) was obtain using
the standard lumped mass and 
Although the Laplacian has much better accuracy using this method, there is some loss of numerical stability, because the effective stencil of the Laplacian is now larger.
For adaptive problems, the best method appears to be the use of the current - vorticity formulation. In this approach, the Laplacian does not appear at all on the right hand side of the equations, only on the left hand sides of the two Poisson equations for the potentials. There appears to be no problem with smoothness of the potentials, using the original stiffness matrix, even on an irregular mesh. This because, as remarked above, the inversion of the Poisson equation is a smoothing operation. Both the currents and potentials have sufficient smoothness to obtain acceptable solutions.
