Basis Functions

 This chapter is devoted to outlining the polynomial bases that we use to represent a function defined on an element. A desirable basis has three properties: (i) it should be orthogonal or nearly orthogonal under a convenient norm, (ii) well behaved at high polynomial order, and (iii) it should be computationally efficient to take inner products with this basis if it is to be used in a Galerkin framework. The bases we present are designed to satisfy these conditions. The computational complexity of taking inner products has been minimized for these hierarchical bases by insisting that each basis function is a tensor product of one-dimensional shape functions.