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1.4.3 Finite elements
Following the spirit of Hilbert space methods, a function
is
decomposed on a complete set of nearly orthogonal basis functions

(1.4.3#eq.1) 
These finiteelements (FEM) basis functions span only as far as the
neighboring mesh points and need not to be homogeneously distributed.
Most common are the normalized ``rooftop'' functions

(1.4.3#eq.2) 
which yield a piecewise linear approximation for
and a piecewise
constant derivative
that is defined almost everywhere in
the interval
.
Boundary conditions are imposed by modifying the functional space
, e.g. by taking ``unilateral roofs'' at the boundaries.
Figure:
Approximation of
with homogeneous linear finite elements.

Generalizations with ``piecewise constant'' or higher order ``quadratic''
and ``cubic'' FEM are constructed along the same lines. The capability of
densifying the mesh where short spatial scales require a higher accuracy is
of particular interest. Figure (1.4.3#fig.1) doesn't exploit this, showing
instead what happens when the numerical resolution becomes insufficient:
around 20 linear or 2 cubic FEM are typically required per wavelength to
achieve a precision around 1%. A minimum of 2 is of course necessary
to resolve the oscillation.
SYLLABUS Previous: 1.4.2 Sampling on a
Up: 1.4 Numerical discretization
Next: 1.4.4 Splines
