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If one chooses a Fourier basis for E, then one gets a so-called [[Spectral method]], which from our point of view is closely related to the finite element method. Typically, in the finite element method, the set F is chosen directly, and is not usually construed to be a subset of some E.
Instead, the functions of F can be chosen to be ''[[piecewise linear]]''. A ''tesselation'' of the ___domain T is chosen, which decomposes T into triangles (say) and the functions of F are those that are linear on each component of the tesselation of T. (An illustration would be good here.) Each triangle is referred to as an "element."
To obtain better algorithms, one can attempt to vary the primitives of the tesselation; it may be more natural to use rectangular elements, and in some cases curvilinear elements are called for. Conversely, once the elements are chosen, one still has a choice of how to define the test functions on each element. Test functions are usually, but not always chosen to be piecewise polynomial.
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