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Boundary-aware Hodge decompositions for piecewise constant vector fields

Hold Date 2016-10-21 15:00~2016-10-21 16:00

Place IMI Conference Room(W1-D-414), West Zone 1, Ito campus, Kyushu University

Object person  

Speaker Konrad Polthier (Freie Universitaet Berlin)

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 $\calH_{h,N}$ and Dirichlet fields $\calH_{h,D}$, 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 $\calH_{h,N}$ and $\calH_{h,D}$, 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 $L^2$-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.

The paper, joint with Konstantin Poelke, on “Boundary-aware Hodge decompositions for piecewise constant vector fields”, published in Computer-Aided Design, Volume 78, 2017, won the 1-prize best paper award at SPM2016 at the International Geometry Summit ( ).