In this paper, the second order boundary value problem −∇ · (A(x, y) ∇u) = f is discretized by the Finite Element Method using piecewise polynomial functions of degree p on a triangular mesh. On the reference element, we define integrated Jacobi polynomials as interior ansatz functions. If A is a constant function on each triangle and each triangle has straight edges, we prove that the element stiffness matrix has not more than 25 2 p 2 nonzero matrix entries. An application for preconditioning is given. Numerical examples show the advantages of the proposed basis.

We establish multiresolution norm equivalences in weighted spaces L 2 w ((0, 1)) with possibly singular weight functions w(x) ≥ 0 in (0, 1). Our analysis exploits the locality of the biorthogonal wavelet basis and its dual basis functions. The discrete norms are sums of wavelet coefficients which are weighted with respect to the collocated weight function w(x) within each scale. Since norm equivalences for Sobolev norms are by now well-known, our result can also be applied to weighted Sobolev norms. We apply our theory to the problem of preconditioning p-Version FEM and wavelet discretizations of degenerate elliptic and parabolic problems from finance.

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