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Interior Penalty Discontinuous Galerkin Based Isogeometric Analysis for Allen-Cahn Equations on Surfaces

Part of: Parabolic equations and systems Numerical approximation and computational geometry Partial differential equations, initial value and time-dependent initial-boundary value problems

Published online by Cambridge University Press:  23 November 2015

Futao Zhang ,
Yan Xu ,
Falai Chen  and
Ruihan Guo
Futao Zhang*
Affiliation:
School of Mathematical Sciences, University of Science and Technology of China, Hefei, Anhui 230026, P.R. China.
Yan Xu
Affiliation:
School of Mathematical Sciences, University of Science and Technology of China, Hefei, Anhui 230026, P.R. China.
Falai Chen
Affiliation:
School of Mathematical Sciences, University of Science and Technology of China, Hefei, Anhui 230026, P.R. China.
Ruihan Guo
Affiliation:
School of Mathematical Sciences, University of Science and Technology of China, Hefei, Anhui 230026, P.R. China.
*
*Corresponding author. Email addresses:zhft00@mail.ustc.edu.cn (F. Zhang), yxu@ustc.edu.cn (Y. Xu), chenf l@ustc.edu.cn (F. Chen), guoguo88@mail.ustc.edu.cn (R. Guo)
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Abstract

We propose a method that combines Isogeometric Analysis (IGA) with the interior penalty discontinuous Galerkin (IPDG) method for solving the Allen-Cahn equation, arising from phase transition in materials science, on three-dimensional (3D) surfaces consisting of multiple patches. DG ideology is adopted at patch level, i.e., we employ the standard IGA within each patch, and employ the IPDG method across the patch interfaces. IGA is very suitable for solving Partial Differential Equations (PDEs) on (3D) surfaces and the IPDG method is used to glue the multiple patches together to get the right solution. Our method takes advantage of both IGA and the IPDG method, which allows us to design a superior semi-discrete (in time) IPDG scheme. First and most importantly, the time-consuming mesh generation process in traditional Finite Element Analysis (FEA) is no longer necessary and refinements, including h-refinement and p-refinement which both maintain the original geometry, can be easily performed at any level. Moreover, the flexibility of the IPDG method makes our method very easy to handle cases with non-conforming patches and different degrees across the patch interfaces. Additionally, the geometrical error is eliminated (for all conic sections) or significantly reduced at the beginning due to the geometric flexibility of IGA basis functions, especially the use of multiple patches. Finally, this method can be easily formulated and implemented. We present our semi-discrete IPDG scheme after generally describe the problem, and then briefly introduce the time marching method employed in this paper. Theoretical analysis is carried out to show that our method satisfies a discrete energy law, and achieves the optimal convergence rate with respect to the L2 norm. Furthermore, we propose an elliptic projection operator on (3D) surfaces and prove an approximation error estimate which are vital for us to obtain the error estimate in the L2 norm. Numerical tests are given to validate the theory and gauge the good performance of our method.

MSC classification

Secondary: 65M60: Finite elements, Rayleigh-Ritz and Galerkin methods, finite methods 65D17: Computer aided design (modeling of curves and surfaces) 68U07: Computer-aided design 35K93: Quasilinear parabolic equations with mean curvature operator
Type
Research Article
Information
Communications in Computational Physics , Volume 18 , Issue 5 , November 2015 , pp. 1380 - 1416
Copyright
Copyright © Global-Science Press 2015 

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