Conforming Finite Elements - Finite Elements: Theory, Fast Solvers, and Applications in Elasticity Theory - page 100

Complexity of Setting up the Matrix

In setting up the system matrix, we need to perform Ms 2 matrix element calcula-
tions, where M is the number of elements, and s is the number of local degrees of
freedom. Thus, clearly one tries to avoid calculating with finite elements that have
a large number of local degrees, if possible. Only recently computations with poly-
nomials of high degree have impact on the design of finite element programs. They
are so designed that their good approximation properties more than compensate for
the increase of the computational effort; see Schwab [1998].
It is for this reason that in practice C 1 elements are not used for systems of
partial differential equations. For planar C 1 elements, it is well known that we need
at least 12 degrees of freedom per function. Thus, for elliptic systems with three
variables, we would have to set up a
36 × 36 matrix

for each element.

Effect on the Choice of a Grid

Once we have selected an element type, the work required to set up the stiffness
matrix is approximately proportional to the number of unknowns. However, the
work required for the solution of the corresponding system of linear equations
using classical methods increases faster than linearly. For large systems, this can
quickly lead to memory problems.
These considerations suggest individually tailoring the grid to the problem in
order to reduce the number of variables as much as possible.
With the development of newer methods for solving systems of equations,
such as the ones in Chapters IV and V, this problem has become less critical, and
once again assembling the matrix constitutes the main part of the work. Thus, it
makes more sense to save computation time there if possible. One way to do this
is to build the grid so that the elements are all translations of a few basic ones. If
the coefficients of the differential equation are piecewise constant functions, the
computational effort can be reduced. Dividing each triangle into four congruent
parts means here that the matrix elements for the subtriangles can be obtained from
those of the original triangles with just a few calculations.

Local Mesh Refinement

A triangle can easily be decomposed into four congruent subtriangles. Thus, using
bisection we can easily perform a global grid refinement to halve the mesh size.
This process leaves the regularity parameter κ (the maximum ratio of circumcircle
radius to the radius of an inscribed circle) unchanged.