Table of Content

20 June 2019, Volume 1 Issue 2

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A Two-Level Additive Schwarz Preconditioner for Local C⁰ Discontinuous Galerkin Methods of Kirchhof Plates

Jianguo Huang, Xuehai Huang

2019, 1(2):  167-185.  doi:10.1007/s42967-019-0003-1

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A two-level additive Schwarz preconditioner based on the overlapping domain decomposition approach is proposed for the local C⁰ discontinuous Galerkin (LCDG) method of Kirchhof plates. Then with the help of an intergrid transfer operator and its error estimates, it is proved that the condition number is bounded by O(1 + (H⁴/δ⁴)), where H is the diameter of the subdomains and δ measures the overlap among subdomains. And for some special cases of small overlap, the estimate can be improved as O(1 + (H³/δ³)). At last, some numerical results are reported to demonstrate the high efciency of the two-level additive Schwarz preconditioner.

Solving Interface Problems of the Helmholtz Equation by Immersed Finite Element Methods

Tao Lin, Yanping Lin, Qiao Zhuang

2019, 1(2):  187-206.  doi:10.1007/s42967-019-0002-2

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This article reports our explorations for solving interface problems of the Helmholtz equation by immersed fnite elements (IFE) on interface independent meshes. Two IFE methods are investigated:the partially penalized IFE (PPIFE) and discontinuous Galerkin IFE (DGIFE) methods. Optimal convergence rates are observed for these IFE methods once the mesh size is smaller than the optimal mesh size which is mainly dictated by the wave number. Numerical experiments also suggest that higher degree IFE methods are advantageous because of their larger optimal mesh size and higher convergence rates.

A New Spectral Method Using Nonstandard Singular Basis Functions for Time-Fractional Diferential Equations

Wenjie Liu, Li-Lian Wang, Shuhuang Xiang

2019, 1(2):  207-230.  doi:10.1007/s42967-019-00012-1

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In this paper, we introduce new non-polynomial basis functions for spectral approximation of time-fractional partial diferential equations (PDEs). Diferent from many other approaches, the nonstandard singular basis functions are defned from some generalised Birkhof interpolation problems through explicit inversion of some prototypical fractional initial value problem (FIVP) with a smooth source term. As such, the singularity of the new basis can be tailored to that of the singular solutions to a class of time-fractional PDEs, leading to spectrally accurate approximation. It also provides the acceptable solution to more general singular problems.

Multi-Domain Decomposition Pseudospectral Method for Nonlinear Fokker-Planck Equations

Tao Sun, Tian-jun Wang

2019, 1(2):  231-252.  doi:10.1007/s42967-019-00013-0

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Results on the composite generalized Laguerre-Legendre interpolation in unbounded domains are established. As an application, a composite Laguerre-Legendre pseudospectral scheme is presented for nonlinear Fokker-Planck equations on the whole line. The convergence and the stability of the proposed scheme are proved. Numerical results show the efciency of the scheme and conform well to theoretical analysis.

Public “Cloud” Provisioning for Venus Express VMC Image Processing

J. L. Vázquez-Poletti, M. P. Velasco, S. Jiménez, D. Usero, I. M. Llorente, L. Vázquez, O. Korablev, D. Belyaev, M. V. Patsaeva, I. V. Khatuntsev

2019, 1(2):  253-261.  doi:10.1007/s42967-019-00014-z

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In this paper, we consider the implementation of the "cloud" computing strategy to study data sets associated to the atmospheric exploration of the planet Venus. More concretely, the Venus Monitoring Camera (VMC) onboard Venus Express orbiter provided the largest and the longest so far set of ultraviolet (UV), visible and near-IR images for investigation of the atmospheric circulation. To our best knowledge, this is the frst time where the analysis of data from missions to Venus is integrated in the context of the "cloud" computing. The followed path and protocols can be extended to more general cases of space data analysis, and to the general framework of the big data analysis.

On Minimization of Upper Bound for the Convergence Rate of the QHSS Iteration Method

Wen-Ting Wu

2019, 1(2):  263-282.  doi:10.1007/s42967-019-00015-y

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For an upper bound of the spectral radius of the QHSS (quasi Hermitian and skew-Hermitian splitting) iteration matrix which can also bound the contraction factor of the QHSS iteration method, we give its minimum point under the conditions which guarantee that the upper bound is strictly less than one. This provides a good choice of the involved iteration parameters, so that the convergence rate of the QHSS iteration method can be signifcantly improved.

Jacobi-Sobolev Orthogonal Polynomials and Spectral Methods for Elliptic Boundary Value Problems

Xuhong Yu, Zhongqing Wang, Huiyuan Li

2019, 1(2):  283-308.  doi:10.1007/s42967-019-00016-x

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Generalized Jacobi polynomials with indexes α, β ∈ R are introduced and some basic properties are established. As examples of applications, the second- and fourth-order elliptic boundary value problems with Dirichlet or Robin boundary conditions are considered, and the generalized Jacobi spectral schemes are proposed. For the diagonalization of discrete systems, the Jacobi-Sobolev orthogonal basis functions are constructed, which allow the exact solutions and the approximate solutions to be represented in the forms of infnite and truncated Jacobi series. Error estimates are obtained and numerical results are provided to illustrate the efectiveness and the spectral accuracy.