# Browsing by Subject "divergence-free"

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Item Approximation and interpolation employing divergence-free radial basis functions with applications(Texas A&M University, 2004-09-30) Lowitzsch, SvenjaShow more Approximation and interpolation employing radial basis functions has found important applications since the early 1980's in areas such as signal processing, medical imaging, as well as neural networks. Several applications demand that certain physical properties be fulfilled, such as a function being divergence free. No such class of radial basis functions that reflects these physical properties was known until 1994, when Narcowich and Ward introduced a family of matrix-valued radial basis functions that are divergence free. They also obtained error bounds and stability estimates for interpolation by means of these functions. These divergence-free functions are very smooth, and have unbounded support. In this thesis we introduce a new class of matrix-valued radial basis functions that are divergence free as well as compactly supported. This leads to the possibility of applying fast solvers for inverting interpolation matrices, as these matrices are not only symmetric and positive definite, but also sparse because of this compact support. We develop error bounds and stability estimates which hold for a broad class of functions. We conclude with applications to the numerical solution of the Navier-Stokes equation for certain incompressible fluid flows.Show more Item Refined error estimates for matrix-valued radial basis functions(Texas A&M University, 2007-09-17) Fuselier, Edward J., Jr.Show more Radial basis functions (RBFs) are probably best known for their applications to scattered data problems. Until the 1990s, RBF theory only involved functions that were scalar-valued. Matrix-valued RBFs were subsequently introduced by Narcowich and Ward in 1994, when they constructed divergence-free vector-valued functions that interpolate data at scattered points. In 2002, Lowitzsch gave the first error estimates for divergence-free interpolants. However, these estimates are only valid when the target function resides in the native space of the RBF. In this paper we develop Sobolev-type error estimates for cases where the target function is less smooth than functions in the native space. In the process of doing this, we give an alternate characterization of the native space, derive improved stability estimates for the interpolation matrix, and give divergence-free interpolation and approximation results for band-limited functions. Furthermore, we introduce a new class of matrix-valued RBFs that can be used to produce curl-free interpolants.Show more