Augmented coupling interface method for solving eigenvalue problems with sign-changed coefficients |
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Authors: | Yu-Chen Shu Chiu-Yen Kao I-Liang Chern Chien C Chang |
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Institution: | 1. Division of Mechanics, Research Center for Applied Sciences, Academia Sinica, Taipei 115, Taiwan;2. Institute of Applied Mechanics, National Taiwan University, Taipei 106, Taiwan;3. Department of Mathematics and Taida Institute for Mathematical Sciences, National Taiwan University, Taipei 106, Taiwan;4. National Center for Theoretical Sciences, Math. Division, Taipei Office, Taipei 106, Taiwan;5. Department of Mathematics, The Ohio State University, OH 43210, USA;6. Mathematical Biosciences Institute (MBI), The Ohio State University, OH 43210, USA |
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Abstract: | In this paper, we propose an augmented coupling interface method on a Cartesian grid for solving eigenvalue problems with sign-changed coefficients. The underlying idea of the method is the correct local construction near the interface which incorporates the jump conditions. The method, which is very easy to implement, is based on finite difference discretization. The main ingredients of the proposed method comprise (i) an adaptive-order strategy of using interpolating polynomials of different orders on different sides of interfaces, which avoids the singularity of the local linear system and enables us to handle complex interfaces; (ii) when the interface condition involves the eigenvalue, the original problem is reduced to a quadratic eigenvalue problem by introducing an auxiliary variable and an interfacial operator on the interface; (iii) the auxiliary variable is discretized uniformly on the interface, the rest of variables are discretized on an underlying rectangular grid, and a proper interpolation between these two grids are designed to reduce the number of stencil points. Several examples are tested to show the robustness and accuracy of the schemes. |
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Keywords: | Coupling interface method Sign-changed coefficients Eigenvalue problems |
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