دانلود رایگان مقاله سرحد تجزیه هاج برای زمینه های بردار ثابت تکه ای

Abstract

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,N and Dirichlet fields Hh,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,N and Hh,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 L2-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. Keywords

5. Conclusion and outlook

We presented a complete description for the structure of the space of piecewise constant vector fields on simplicial surfaces with boundary in terms of several Hodge-type decomposition theorems that are structurally consistent with the smooth theory. We used these results to compute refined decompositions for vector fields and bases for topologically significant subspaces of Xh. Distinguishing between the inner and boundary cohomology representatives helps to identify intrinsic flows and flows that arise due to the presence of nontrivial boundary cycles individually. In addition, we found an obstruction not present in the smooth counterpart which needs to be considered for discretization strategies to obtain structurally consistent results, and we gave counterexamples to illustrate the problem. We intend to build upon this work in various directions: since the validity of Eq. (12) is not only of theoretical interest, but crucial to obtain a consistent discretization, it is important to find a criterion that is easy to check computationally for a given simplicial surface Mh. On the computational side, we want to evaluate different solving strategies (e.g. a direct method based on a sparse QR decomposition) for the involved systems in terms of efficiency and robustness.