Constructive spectral approaches for the Helmholtz decomposition of a vector field

This paper presents spectral approaches on staggered grids to extract the solenoidal (i.e. divergence free) and non-solenoidal (i.e. curl free or, gradient of a scalar field) components of a given vector field. Basically, the vector field is described using a polynomial expansion on a GLL/GL-GL/GLL mesh, and the scalar field is computed using a polynomial expansion PN−1⊗PN−1 on the GL/GL grid. These approaches use the same spectral element and consist in a projection either onto the kernel of the −grad(div) operator, or onto its range. Two variational formulations are presented and studied to approximate the divergence free and the curl free parts of the spectrum. Each one is well adapted to ensure the expected constraints. The approaches are detailed and illustrated by several numerical examples.