We denote the gradient and divergence in the usual way, i. The fractional cases follow from standard interpolation arguments. For fields on bounded domains we will focus on the two fundamental decompositions given in the following propositions. The next decomposition splits a vector field into a divergence-free field and a gradient field normal to the boundary. This leads to potential functions for our decompositions that satisfy the following regularity result.
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In the case of Proposition 2. The Proposition 2. The fractional cases can be handled using standard interpolation arguments. We remark that the existence of these potentials is used only for theoretical purposes.
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The choice of cuts and the conditions 2. However, potential functions for each term in the kernel decomposition will be readily available. These same arguments can be repeated to establish a continuous extension satisfying 2. The reproducing kernel Hilbert space structure of the native space makes it possible to interpolate using a wide variety of continuous linear functionals. A concise treatment of this is given for scalar-valued RBFs in Wendland , Chapter 16 , and generalizes in a straightforward way to the matrix-valued case.
We summarize the main results we need below. For example, 3. In this section we show how to construct a kernel-based approximation to the decompositions discussed earlier. We will also show how one easily obtains potential functions from the kernel approximation. Note that the interpolation matrix in 4. In Section 5.
Also one can use the form of the kernels 3. We will review the specific results we require below and extend them slightly to suit our purposes. Next, we derive the error estimates in Sections 5. The following is from Narcowich et al. We begin with a basic interpolation estimate. Next, apply Proposition 2. An application of Lemma 5. Note that while Lemma 5.
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Now we focus on the decomposition in Proposition 2. First, we have a lemma, whose proof we omit since the arguments are similar to those of Lemma 5. With these results, one can now construct an argument very similar to the proof of Theorem 5. Theorems 5. Lemma 5. Convergence results for each numerical experiment. Nevertheless, we estimated the rate of convergence by using each approximation on the finest node set as proxies for the true solution. There is room for improvement in both the error estimates and the computational cost of this method.
First, the global basis functions used here lead to full systems. However, estimates for continuous target functions too rough for the native space have been given in other kernel approximation problems see, for example, Narcowich et al. Given that the potential functions are usually solutions to some elliptic differential equation, this assumption requires smoothness of the domain, even for smooth target fields. On nonsmooth domains we expect the convergence rates to be dictated by the regularity of the potentials, which are governed by the elliptic regularity of the domain.
The method presented in this article distinguishes itself from many existing approaches in several ways. The decomposition is approximated by analytically divergence-free and curl-free functions, can handle data from scattered sites and only discrete samples from the target field are used to construct the approximation, for example, one does not need to compute the curl or divergence of the samples in order to reconstruct one of the potentials. One important feature is that boundary conditions are enforced on the divergence-free or curl-free terms directly , with no boundary conditions required on the scalar or vector potential functions.
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This is in constrast to standard projection methods, for example, which incorporate decompositions obtained by solving a Poisson problem for the pressure. Choosing proper boundary conditions for the pressure is sometimes a difficult task; even boundary conditions consistent with the model often cause slow time convergence in unsteady flow simulations Liu et al.
The decomposition presented here, which completely avoids boundary conditions on the pressure, has been used as a projection step on test problems solving the unsteady Stokes equation, and high-order approximation in time up to order 4 was observed Fuselier et al. Lastly, the method seems to extend to other boundary conditions quite easily.enter site
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For other boundary conditions, if the functionals associated with the interpolation and boundary conditions are linearly independent and the Riesz representers are chosen as basis functions, then the kernel decomposition can be constructed. In this way, one could impose a whole host of boundary conditions in vector decomposition problems, and do so in a natural way. Oxford University Press is a department of the University of Oxford. It furthers the University's objective of excellence in research, scholarship, and education by publishing worldwide.
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