Orthonormal Wavelet Bases Adapted for Partial Differential Equations with Boundary Conditions

Abstract

We adapt ideas presented by Auscher to impose boundary conditions on the construction of multiresolution analyses on the interval, as introduced by Cohen, Daubechies, and Vial. We construct new orthonormal wavelet bases on the interval satisfying homogeneous boundary conditions. This construction can be extended to wavelet packets in the case of one boundary condition at each edge. We present in detail the numerical computation of the filters and the derivative operators associated with these bases. We derive quadrature formulae in order to study the approximation error at the edge of the interval. Several examples illustrate the present construction. ). Orthonormal wavelet bases on R.We briefly review wavelets and MRA of L 2 (R) (for further details, see [Daub 92, Mall 89, Meye 90]). An MRA is a set (V j ) j∈Z of closed subspaces of L 2 (R) satisfying:it is possible to construct a function Φ (called the scaling function) of V 0 such that {Φ(. − k)} k∈Z is an orthonormal basis of V 0 and +∞ −∞ Φ(x)dx = 1. Moreover, P i (m)h j−2m .

The Mortar Method in the Wavelet Context

“…Without loss of generality, we can assume that dimV j = dimṼ j = 2 j + 2 (see [2,21,32] for example). The spaces V j andṼ j are respectively spanned by biorthogonal scaling function Riesz bases (ϕ j,k ) k=0,... ,2 j +1 and (φ j,k ) k=0,... ,2 j +1 , verifying:…”

“…This can be proven in particular cases, such as for biorthogonal spline wavelets [20,25] or orthogonal wavelets. In this last example ϕ j,k =φ j,k and for instance on the left boundary the coefficientsã n k = 1 0 x nφ j,k can be written asã n k = p n (k), where p n is a polynomial of degree n [32]. [ã n k ] 0≤k,n≤Ñ −1 is then a nonsingular Vandermonde type matrix.…”

Turbulence analysis, modelling and computing using wavelets

“…The simplest and most popular (due to the development of Fourier spectral methods) are periodic boundary conditions for which periodic wavelets, in one or several dimensions, can be easily constructed [148]. For Dirichlet, or Neumann boundary conditions, compactly supported bases have recently been constructed in one dimension [42], [139], [140], and these bases are also associated to fast orthogonal wavelet transforms, like for the periodic case. They can easily be included in some of the previous algorithms, since the extension to cubic domains in several dimensions is trivial using tensor products of wavelets (in practice all two-dimensional orthogonal wavelet bases are tensor products, which raises the problem of the lack of isotropy).…”

Divergence-Free Wavelet Projection Method for Incompressible Viscous Flow on the Square

“…In practice, the scale index j must be greater than some index j min , to avoid boundary effects [29]. The biorthogonality between bases writes:…”

“…Homogeneous Dirichlet boundary conditions can be easily imposed on (V 1 j ,Ṽ 1 j ) by removing scaling functions that reproduce constant functions at edges 0 and 1, prior biorthogonalization [27,29]. Then, the spaces…”