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Following the spirit of Hilbert space methods, a function is decomposed on a complete set of nearly orthogonal basis functions
These finite-elements (FEM) basis functions span only as far as the neighboring mesh points and need not to be homogeneously distributed. Most common are the normalized ``roof-top'' functions
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.
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.
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