Boundary-aware hodge decompositions for piecewise constant vector fields

• Discrete Hodge decompositions on simplicial surfaces with boundary are proposed.
• The discretization scheme is structurally consistent with the smooth theory.
• Refined decompositions distinguish between boundary homology and inner homology.
• The intersection of discrete Neumann and Dirichlet fields depends on the mesh.
• Algorithms for computing various harmonic fields and decompositions are presented.

We provide a theoretical framework for discrete Hodge-type decomposition theorems of piecewise constant vector fields on simplicial surfaces with boundary that is structurally consistent with decomposition results for differential forms on smooth manifolds with boundary. In particular, we obtain a discrete Hodge–Morrey–Friedrichs decomposition with subspaces of discrete harmonic Neumann fields Hh,NHh,N and Dirichlet fields Hh,DHh,D, which are representatives of absolute and relative cohomology and therefore directly linked to the underlying topology of the surface. In addition, we discretize a recent result that provides a further refinement of the spaces Hh,NHh,N and Hh,DHh,D, and answer the question in which case one can hope for a complete orthogonal decomposition involving both spaces at the same time.As applications, we present a simple strategy based on iterated L2L2-projections to compute refined Hodge-type decompositions of vector fields on surfaces according to our results, which give a more detailed insight than previous decompositions. As a proof of concept, we explicitly compute harmonic basis fields for the various significant subspaces and provide exemplary decompositions for two synthetic vector fields.