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H${\mathscr{H}}$ -matrix approximability of inverses of discretizations of the fractional Laplacian
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The integral version of the fractional Laplacian on a bounded domain is discretized by a Galerkin approximation based on piecewise linear functions on a quasi-uniform mesh by showing that the inverse of the associated stiffness matrix can be approximated by blockwise low-rank matrices at an exponential rate in the block rank.Abstract:
The integral version of the fractional Laplacian on a bounded domain is discretized by a Galerkin approximation based on piecewise linear functions on a quasiuniform mesh. We show that the inverse of the associated stiffness matrix can be approximated by blockwise low-rank matrices at an exponential rate in the block rank.read more
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A Reduced Basis Method For Fractional Diffusion Operators II
Tobias Danczul,Joachim Schöberl +1 more
TL;DR: A reduced basis strategy on top of a finite element method to approximate its integrand is proposed, and the choice of snapshots for the reduced basis procedure analytically is deduce in a spectral setting to evaluate the surrogate directly.
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Quasi-optimal convergence rate for an adaptive method for the integral fractional Laplacian
TL;DR: A reliable weighted residual a posteriori error estimator for the discretization of the integral fractional Laplacian based on piecewise linear functions is presented and it is proved optimal convergence rates for an $h$-adaptive algorithm driven by this error estimators.
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Local convergence of the FEM for the integral fractional Laplacian.
TL;DR: This work provides for first order discretizations of the integral fractional Laplacian sharp local error estimates on proper subdomains in both the local $H^1$-norm and the localized energy norm.
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Exponential Convergence of $hp$ FEM for Spectral Fractional Diffusion in Polygons
TL;DR: It is proved exponential convergence of two classes of $hp$ discretizations under the assumption of analytic data, without any boundary compatibility, in the natural fractional Sobolev norm of $\mathbb{H}^s(\Omega)$.
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Tensor product method for fast solution of optimal control problems with fractional multidimensional Laplacian in constraints
TL;DR: In this article, a tensor numerical method for solution of the 2D and 3D optimal control problems with fractional Laplacian type operators in constraints discretized on large $n^{\otimes d}$ tensor-product Cartesian grids is introduced.
References
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Journal ArticleDOI
An Extension Problem Related to the Fractional Laplacian
TL;DR: In this article, the square root of the Laplacian (−△) 1/2 operator was obtained from the harmonic extension problem to the upper half space as the operator that maps the Dirichlet boundary condition to the Neumann condition.
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Finite element interpolation of nonsmooth functions satisfying boundary conditions
L. Ridgway Scott,Shangyou Zhang +1 more
TL;DR: In this article, a modified Lagrange type interpolation operator is proposed to approximate functions in Sobolev spaces by continuous piecewise polynomials, and the combination of averaging and interpolation is shown to be a projection, and optimal error estimates are proved for the projection error.
Journal ArticleDOI
A sparse matrix arithmetic based on H -matrices. Part I: introduction to H -matrices
TL;DR: This paper is the first of a series and is devoted to the first introduction of the $\Cal H$-matrix concept, which allows the exact inversion of tridiagonal matrices.
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Approximation of boundary element matrices
TL;DR: From results an iterative algorithm for the low-rank approximation of blocks of large unstructured matrices generated by asymptotically smooth functions is developed.