Orthogonal Helmholtz decomposition in arbitrary dimension using divergence-free and curl-free wavelets
There are many physical problems like Stokes problem, Navier-Stokes equation, Maxwell's equation, where the solution has to fulfill a divergence-free condition. From numerical point of view, appropriate numerical schemes very often requires the orthogonal projection of the solution, onto the set of divergence-free vector valued functions. So called Helmholtz decomposition consists in splitting a vector field into its divergence-free component and its curl-free component. The main goal of this paper is to propose an efficient way to compute the orthogonal Helmholtz decomposition of any vector field in wavelet domain. Since wavelet bases are localized both in physical and Fourier spaces the advantages of such decomposition in contrast with the one based on the Fourier transform, are its localization in physical space and its availability on both, the whole domain \(\mathbb{R}^n\) and bounded domains as well. Since divergence-free and curl-free wavelets are biorthogonal bases the associated projectors do not provide directly the Helmholtz decomposition of a vector field. Therefore an iterative algorithm for proving the convergence was developed in authors' previous work in dimensions 2 and 3. The extension for the arbitrary dimension is presented in this paper. For this purpose a new formulation for the divergence-free and curl-free wavelets in dimension larger than 4 is used. This formulation is based on the expression of the Leray projector in wavelet domain analogous to its expression in Fourier domain. The convergence proof is generalized for all kind of wavelets, under some hypothesis on the Fourier transform of the wavelets. A modification of the algorithm by using quasi-isotropic divergence-free and curl-free wavelets is included. Numerical test that confirm the effectivity of this algorithm for a large class of wavelets conclude the paper.
scientific article; zbMATH DE number 5532857
scientific article