Three-dimensional elliptic solvers for interface problems and applications |
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| Affiliation: | 1. Department of Mathematics, University of North Carolina at Charlotte, Charlotte, NC 28223, USA;2. Center for Research in Scientific Computation and Department of Mathematics, North Carolina State University, Raleigh, NC 27695, USA;1. Graduate School of Computational Engineering, Technische Universität Darmstadt, Dolivostrasse 15, 64293 Darmstadt, Germany;2. ETH Zurich, Institute of Electromagnetic Fields (IFH), Gloriastrasse 35, 8092 Zurich, Switzerland;3. Institut für Theorie Elektromagnetischer Felder, Technische Universität Darmstadt, Schlossgartenstrasse 8, 64289 Darmstadt, Germany;1. Department of Mathematical Sciences, University of Nevada Las Vegas, Las Vegas, Nevada 89154-4020, USA;2. Department of Applied Mathematics, Hong Kong Polytechnic University, Hung Hom, Hong Kong;1. Department of Mathematics,The Chinese University of Hong Kong (CUHK), Hong Kong Special Administrative Region;2. Department of Mathematics, Texas A&M University, College Station, TX 77843, United States;1. College of Data Science, Jiaxing University, Jiaxing, Zhejiang 314001, China;2. College of Information Engineering, Jiaxing Nanhu University, Jiaxing, Zhejiang 314001, China;1. School of Mathematical Sciences, South China Normal University, Guangzhou 510631, China;2. School of Mathematics and Information Science, Guangzhou University, Guangzhou 510006, China |
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| Abstract: | Second-order accurate elliptic solvers using Cartesian grids are presented for three-dimensional interface problems in which the coefficients, the source term, the solution and its normal flux may be discontinuous across an interface. One of our methods is designed for general interface problems with variable but discontinuous coefficient. The scheme preserves the discrete maximum principle using constrained optimization techniques. An algebraic multigrid solver is applied to solve the discrete system. The second method is designed for interface problems with piecewise constant coefficient. The method is based on the fast immersed interface method and a fast 3D Poisson solver. The second method has been modified to solve Helmholtz/Poisson equations on irregular domains. An application of our method to an inverse interface problem of shape identification is also presented. In this application, the level set method is applied to find the unknown surface iteratively. |
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