Decomposition of a planar vector field into irrotational and rotational components

A formulation of the boundary value problem in a finite domain for the scalar potential and the stream function is given: the basic decomposition equation is assumed as boundary condition. The problem is singular: the existence of solutions, which are determined up to conjugate harmonic functions, is proved. The basic properties of the spectrum of the homogeneous operator associated to the boundary value problem for the potentials are derived. The discrete equations are obtained by means of the finite volume method. It is verified that the main properties of the continuous problem are maintained in the discrete equations. We address the computation of minimum norm solutions, which are obtained by means of the SVD algorithm. Numerical experiments have been performed in different situations of the assigned vector field (presence of zero points, size of the finite domain, degree of stochasticity of the field) to estimate the effects on the decomposition-reconstruction operations.