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Register a new userRobust BPX Preconditioner for Fractional Laplacians on Bounded Lipschitz Domains
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Shuonan Wu
Publication date
2021
fields
Informatics Engineering
and research's language is
English
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We propose and analyze a robust BPX preconditioner for the integral fractional Laplacian on bounded Lipschitz domains. For either quasi-uniform grids or graded bisection grids, we show that the condition numbers of the resulting systems remain uniformly bounded with respect to both the number of levels and the fractional power. The results apply also to the spectral and censored fractional Laplacians.
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This article is an account of the NABUCO project achieved during the summer camp CEMRACS 2019 devoted to geophysical fluids and gravity flows. The goal is to construct finite difference approximations of the transport equation with nonzero incoming boundary data that achieve the best possible convergence rate in the maximum norm. We construct, implement and analyze the so-called inverse Lax-Wendroff procedure at the incoming boundary. Optimal convergence rates are obtained by combining sharp stability estimates for extrapolation boundary conditions with numerical boundary layer expansions. We illustrate the results with the Lax-Wendroff and O3 schemes.
Pavarino proved that the additive Schwarz method with vertex patches and a low-order coarse space gives a $p$-robust solver for symmetric and coercive problems. However, for very high polynomial degree it is not feasible to assemble or factorize the matrices for each patch. In this work we introduce a direct solver for separable patch problems that scales to very high polynomial degree on tensor product cells. The solver constructs a tensor product basis that diagonalizes the blocks in the stiffness matrix for the internal degrees of freedom of each individual cell. As a result, the non-zero structure of the cell matrices is that of the graph connecting internal degrees of freedom to their projection onto the facets. In the new basis, the patch problem is as sparse as a low-order finite difference discretization, while having a sparser Cholesky factorization. We can thus afford to assemble and factorize the matrices for the vertex-patch problems, even for very high polynomial degree. In the non-separable case, the method can be applied as a preconditioner by approximating the problem with a separable surrogate.
Fourier extension is an approximation method that alleviates the periodicity requirements of Fourier series and avoids the Gibbs phenomenon when approximating functions. We describe a similar extension approach using regular wavelet bases on a hypercube to approximate functions on subsets of that cube. These subsets may have a general shape. This construction is inherently associated with redundancy which leads to severe ill-conditioning, but recent theory shows that nevertheless high accuracy and numerical stability can be achieved using regularization and oversampling. Regularized least squares solvers, such as the truncated singular value decomposition, that are suited to solve the resulting ill-conditioned and skinny linear system generally have cubic computational cost. We compare several algorithms that improve on this complexity. The improvements benefit from the sparsity in and the structure of the discrete wavelet transform. We present a method that requires $mathcal O(N)$ operations in 1-D and $mathcal O(N^{3(d-1)/d})$ in $d$-D, $d>1$. We experimentally show that direct sparse QR solvers appear to be more time-efficient, but yield larger expansion coefficients.
It is well known that, with a particular choice of norm, the classical double-layer potential operator $D$ has essential norm $<1/2$ as an operator on the natural trace space $H^{1/2}(Gamma)$ whenever $Gamma$ is the boundary of a bounded Lipschitz domain. This implies, for the standard second-kind boundary integral equations for the interior and exterior Dirichlet and Neumann problems in potential theory, convergence of the Galerkin method in $H^{1/2}(Gamma)$ for any sequence of finite-dimensional subspaces $(mathcal{H}_N)_{N=1}^infty$ that is asymptotically dense in $H^{1/2}(Gamma)$. Long-standing open questions are whether the essential norm is also $<1/2$ for $D$ as an operator on $L^2(Gamma)$ for all Lipschitz $Gamma$ in 2-d; or whether, for all Lipschitz $Gamma$ in 2-d and 3-d, or at least for the smaller class of Lipschitz polyhedra in 3-d, the weaker condition holds that the operators $pm frac{1}{2}I+D$ are compact perturbations of coercive operators -- this a necessary and sufficient condition for the convergence of the Galerkin method for every sequence of subspaces $(mathcal{H}_N)_{N=1}^infty$ that is asymptotically dense in $L^2(Gamma)$. We settle these open questions negatively. We give examples of 2-d and 3-d Lipschitz domains with Lipschitz constant equal to one for which the essential norm of $D$ is $geq 1/2$, and examples with Lipschitz constant two for which the operators $pm frac{1}{2}I +D$ are not coercive plus compact. We also give, for every $C>0$, examples of Lipschitz polyhedra for which the essential norm is $geq C$ and for which $lambda I+D$ is not a compact perturbation of a coercive operator for any real or complex $lambda$ with $|lambda|leq C$. Finally, we resolve negatively a related open question in the convergence theory for collocation methods.
In this article, an advanced differential quadrature (DQ) approach is proposed for the high-dimensional multi-term time-space-fractional partial differential equations (TSFPDEs) on convex domains. Firstly, a family of high-order difference schemes is introduced to discretize the time-fractional derivative and a semi-discrete scheme for the considered problems is presented. We strictly prove its unconditional stability and error estimate. Further, we derive a class of DQ formulas to evaluate the fractional derivatives, which employs radial basis functions (RBFs) as test functions. Using these DQ formulas in spatial discretization, a fully discrete DQ scheme is then proposed. Our approach provides a flexible and high accurate alternative to solve the high-dimensional multi-term TSFPDEs on convex domains and its actual performance is illustrated by contrast to the other methods available in the open literature. The numerical results confirm the theoretical analysis and the capability of our proposed method finally.
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