Finite Element Multiwavelets

Abstract

Finite elements with support on two intervals span the space of piecewise polynomomials with degree 2n ‒ 1 and n ‒ 1 continuous derivatives. Function values and n ‒ 1 derivatives at each meshpoint determine these “Hermite finite elements”. The n basis functions satisfy a dilation equation with n by n matrix coefficients. Orthogonal to this scaling subspace is a wavelet subspace. It is spanned by the translates of n wavelets W _i(t), each supported on three intervals. The wavelets are orthogonal to all rescalings W _i (2^j t ‒ k), but not to translates at the same level (j = 0). These new multiwavelets achieve 2n vanishing moments and high regularity with symmetry and short support.