A numerical method for solving three-dimensional elliptic interface problems with triple junction points
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A numerical method for solving three-dimensional elliptic interface problems with triple junction points Liqun Wang1 · Songming Hou2 · Liwei Shi3
Received: 4 February 2016 / Accepted: 3 May 2017 © Springer Science+Business Media New York 2017
Abstract Elliptic interface problems with multi-domains have wide applications in engineering and science. However, it is challenging for most existing methods to solve three-dimensional elliptic interface problems with multi-domains due to local geometric complexity, especially for problems with matrix coefficient and sharp-edged interface. There are some recent work in two dimensions for multidomains and in three dimensions for two domains. However, the extension to three dimensional multi-domain elliptic interface problems is non-trivial. In this paper, we present an efficient non-traditional finite element method with non-body-fitting grids for three-dimensional elliptic interface problems with multi-domains. Numerical experiments show that this method achieves close to second order accurate in the L∞ norm for piecewise smooth solutions.
Communicated by: John Lowengrub Liwei Shi
[email protected] Liqun Wang [email protected] Songming Hou [email protected] 1
Department of Mathematics, College of Science, China University of Petroleum-Beijing, Beijing, 102249, China
2
Department of Mathematics and Statistics, Louisiana Tech University, Ruston, LA 71272, USA
3
Department of Science and Technology Teaching, China University of Political Science and Law, Beijing, 102249, China
L. Wang et al.
Keywords Elliptic equations · Non-body-fitting mesh · Finite element method · Multi-domains · Jump condition Mathematics Subject Classification (2010) 65N30
1 Introduction Elliptic interface problems have wide applications in a variety of disciplines, such as electromagnetism, fluid dynamics, material science and so on. However, designing highly efficient methods for these problems is a difficult job, especially for threedimensional problems with multi-domains. In the past three decades, much attention has been paid to the numerical solution of elliptic equations with discontinuous coefficients and singular sources on regular Cartesian grids since the pioneering work of Peskin [1] on the first order accurate immersed boundary method. In many applications, particularly for free boundary and moving interface problems, simple Cartesian grids are preferred. In this way, the procedure of generating an unstructured grid can be bypassed, and well developed fast solvers on Cartesian grids can be utilized. Motivated by the immersed boundary method, to improve accuracy, in [9], the “immersed interface” method (IIM) was presented. This method achieves second order accuracy by incorporating the interface conditions into the finite difference stencil in a way that preserves the interface conditions in both solution and its flux, [u] = 0 and [βun ] = 0. The corresponding linear system is sparse, but may not be symmetric or positive definite if there is a jump in the coefficient. In
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