Subdivision and the Construction of Smooth Bases for Discrete Differential Forms

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In order to simulate physical phenomena, such as fluid flow or electromagnetism, one must map the underlying equations to algorithms that can execute efficiently and reliably on a computer. This process of mapping involves discretization, for example, mapping a smooth surface to a triangle mesh. To ensure that the resulting computations are still meaningful this discretization step must be performed so as to preserve as many of the important physical structures as possible. In mathematical terms one must find discrete analogs for the operators of differential and integral calculus which follow the same theorems and structures as their abstract mathematical counterparts. To this end the PI together with his students will bring tools from geometric modeling (subdivision schemes) to the construction of discrete basis forms (which are the foundation of suitable numerical algorithms). Such tools will enable higher order and more accurate simulation of the equations at the heart of physical phenomena of interest. Building such foundational algorithms for numerical simulation will increase their reliability and speed and extend their impact in numerous scientific and engineering disciplines. When discretizing partial differential equations for purposes of numerical simulation much emphasis has been placed on numerical accuracy, stability, and convergence. Only in recent years has it been fully appreciated that preservation of continuous structures such as symmetry groups and their associated momenta can have equally dramatic impact on the performance of numerical simulations, and in some cases appears to be essential to fully predictive simulations. Discrete Exterior Calculus has emerged as a framework in which to study such discrete realizations of the basic operators of integral and differential calculus. So far the underlying theory has essentially been one of local approximation without attendant global smoothness properties (in fact, much of it has been piecewise linear only). This research uses tools known from the construction of refinable functions to construct smooth bases of discrete differential k-forms on simplicial three manifolds (with boundary). In particular the underlying deRham complex is induced by the simplicial co-boundary operator. Piecewise linear forms of this type have been known classically. The research team recently demonstrated that the concept of refinability together with certain discrete commutative relations are sufficient to produce bases of higher smoothness in the simplicial two manifold setting. The three manifold case (using tetrahedra rather than triangles) is considerably more challenging since no regular tetrahedralizations of three space exist. Furthermore the smoothness analysis of the resulting constructions will require new tools since new classes of singular cases need to be examined and standard assumption of refinable function theory are violated. The researchers will both construct (in the sense of providing algorithms and their implementation) and analyze such constructions.