Abstract
On the n-dimensional hypercube, for given \(k\in {\mathbb {N}}\), wavelet Riesz bases are constructed for the subspace of divergence-free vector fields of the Sobolev space \(H^k((0,1)^n)^n\) with general homogeneous Dirichlet boundary conditions, including slip or no-slip boundary conditions. Both primal and suitable dual wavelets can be constructed to be locally supported. The construction of the isotropic wavelet bases is restricted to the square, but that of the anisotropic wavelet bases applies for any space dimension n.
Highlights
1.1 Overview This paper concerns the construction of a Riesz basis, consisting of wavelets, for the spaceCommunicated by Wolfgang Dahmen
The aim of the current paper is to extend the approach to general boundary conditions, including no-slip boundary conditions
To further explain the difficulties and possibilities with transferring the construction of a divergence-free wavelet basis on Rn from [21] to one on a hypercube, first we describe it in some detail
Summary
Since functions in the aforementioned subspaces have components that are constants as function of some variables, the divergence-free wavelets that were obtained do not satisfy boundary conditions beyond having vanishing normals This construction was restricted to slip boundary conditions. The key to achieve this will be the replacement of the orthogonal decomposition of L2(In)n into 2n − 1 subspaces, by a biorthogonal decomposition of (L2(In)n, L2(In)n) into 2n − 1 pairs of subspaces With this approach, we will be able to construct that for given k, renormalized, will be a basis afowravH◦ekle(It Rn)ie∩szHb0a(sdiisvf0o;rIHn0)(,dwivi0th; In) the first space being the closed subspace of H k(In)n defined by imposing (very) general homogeneous Dirichlet boundary conditions up to order k. The case k = 1 is most relevant for the application in solving flow problems This construction of a divergence-free wavelet basis generalizes to Rn for n ≥ 2. For efficient implementations of such applications, it is needed that, as the primal functions, the dual functions are locally supported
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