Divergence-free and curl-free wavelets on the square for numerical simulations

We present a construction of divergence-free and curl-free wavelets on the square, that could satisfy suitable boundary conditions. This construction is based on the existence of biorthogonal multiresolution analyses (BMRA) on [0, 1], linked by differentiation and integration. We introduce new BMRAs and wavelets for the spaces of divergence-free and curl-free vector functions on the square. The interest of such constructions is illustrated on examples including the Helmholtz-Hodge decomposition of vector flows and the Stokes problem.