記述言語：英語   掲載種別：研究論文（学術雑誌）   出版者・発行元：ELSEVIER SCIENCE INC  

For eigenvalue problems of self-adjoint differential operators, a universal framework is proposed to give explicit lower and upper bounds for the eigenvalues. In the case of the Laplacian operator, by applying Crouzeix-Raviart finite elements, an efficient algorithm is developed to bound the eigenvalues for the Laplacian defined in 1D, 2D and 3D spaces. Moreover, for nonconvex domains, for which case there may exist singularities of eigen-functions around re-entrant corners, the proposed algorithm can easily provide eigenvalue bounds. By further adopting the interval arithmetic, the explicit eigenvalue bounds from numerical computations can be mathematically correct. (C) 2015 Elsevier Inc. All rights reserved.

DOI： 10.1016/j.amc.2015.03.048

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記述言語：英語   掲載種別：研究論文（学術雑誌）   出版者・発行元：SPRINGER JAPAN KK  

This paper proposes a verified numerical method of proving the invertibility of linear elliptic operators. This method also provides a verified norm estimation for the inverse operators. This type of estimation is important for verified computations of solutions to elliptic boundary value problems. The proposed method uses a generalized eigenvalue problem to derive the norm estimation. This method has several advantages. Namely, it can be applied to two types of boundary conditions: the Dirichlet type and the Neumann type. It also provides a way of numerically evaluating lower and upper bounds of target eigenvalues. Numerical examples are presented to show that the proposed method provides effective estimations in most cases.

DOI： 10.1007/s13160-014-0156-2

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記述言語：英語   掲載種別：研究論文（学術雑誌）   出版者・発行元：一般社団法人 電子情報通信学会  

For Poisson's equation over a polygonal domain of general shape, the solution of which may have a singularity around re-entrant corners, we provide an explicit a priori error estimate for the approximate solution obtained by finite element methods of high degree. The method used herein is a direct extension of the one developed in preceding paper of the second and third listed authors, which provided a new approach to deal with the singularity by using linear finite elements. In the present paper, we also give a detailed discussion of the dependency of the convergence order on solution singularities, mesh sizes and degrees of the finite element method used.

DOI： 10.1587/nolta.5.53

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記述言語：英語   掲載種別：研究論文（学術雑誌）   出版者・発行元：一般社団法人 電子情報通信学会  

In this paper, a numerical verification method is presented for second-order semilinear elliptic boundary value problems on arbitrary polygonal domains. Based on the Newton-Kantorovich theorem, our method can prove the existence and local uniqueness of the solution in the neighborhood of its approximation. In the treatment of polygonal domains with an arbitrary shape, which gives a singularity of the solution around the re-entrant corner, the computable error estimate of a projection into the finite-dimensional function space plays an essential role. In particular, the lack of smoothness of the solution makes classical error estimates fail on nonconvex domains. By using the Hyper-circle equation, an alternative error estimate of the projection has been proposed. Additionally, a new residual evaluation method based on the mixed finite element method works well. It yields more accurate evaluation than the existing method. The efficiency of our method is shown through illustrative numerical results on several polygonal domains.

DOI： 10.1587/nolta.4.34

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記述言語：英語   掲載種別：研究論文（学術雑誌）   出版者・発行元：KINOKUNIYA CO LTD  

We consider an explicit estimation for error constants from two basic constant interpolations on triangular finite elements. The problem of estimating the interpolation constants is related to the eigenvalue problems of the Laplacian with certain boundary conditions. By adopting the Lehmann-Goerisch theorem and finite element spaces with a variable mesh size and polynomial degree, we succeed in bounding the leading eigenvalues of the Laplacian and the error constants with high precision. An online demo for the constant estimation is also available at http://www.xfliu.org/onlinelab/.

DOI： 10.1007/s13160-013-0120-6

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