Orthogonal Helmholtz decomposition in arbitrary dimension using divergence-free and curl-free wavelets

We present tensor-product divergence-free and curl-free wavelets, and define associated projectors. These projectors permit the definition of an iterative algorithm to compute the Helmholtz decomposition in wavelet domain of any vector field. This decomposition is localized in space, in contrast to the Helmholtz decomposition calculated by Fourier transform. Then we prove the convergence of the algorithm in any dimension for the particular case of Shannon wavelets. We also present a modification of the algorithm which allows to apply it in an adaptive context. Finally, numerical tests show the validity of this approach for any choice of wavelets. Introduction In many physical problems, like the simulation of incompressible fluids (Stokes problem, Navier-Stokes equations [2, 17]), or in electromagnetism (Maxwell’s equations [16]), the solution has to fulfill a divergence-free condition. The implementation of relevant numerical schemes often requires orthogonal projection on the set of divergence-free vector valued functions. The Helmholtz decomposition [10, 2] consists in decomposing a vector field u ∈ (L2(Rn))n, into the sum of its divergence-free component udiv and its curl-free component ucurl. More precisely, there exist a stream-function ψ and a potential-function p such that: u = udiv + ucurl (0.1) with udiv = curl ψ , (div udiv = 0) and ucurl = ∇p , (curl ucurl = 0) . Moreover, the functions curl ψ and ∇p are orthogonal in (L2(Rn))n. The stream-function ψ and the potential-function p are unique, up to an additive constant. This decomposition arises from the orthogonal direct sum of the two spaces Hdiv 0(R n), the space of divergence-free vector functions, and Hcurl 0(R n), the space of curl-free vector functions. In short: (L(R)) = Hdiv 0(R )⊕ Hcurl 0(R ) . ∗Institute of Mathematics, Polish Academy of Sciences. ul. Sniadeckich 8, 00-956 Warszawa, Poland, to whom correspondance should be addressed ([email protected]) †Laboratoire Jean Kuntzmann, INPG, BP 53, 38 041 Grenoble cedex 9, France ([email protected])