Numerical analysis of the Allen-Cahn equation and approximation for mean curvature flows

By Xiaobing H. Feng, Andreas Prohl. Numerische Mathematik, 2003

381 citations recorded by the source corpus.

Subfields: Computational Theory and Mathematics, Materials Chemistry, Numerical Analysis

View Paper: https://www.semanticscholar.org/paper/2a22d9c57c5febb5aade91894bff9452a07bc5d0

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Computational Theory and Mathematics

1. Xiaofeng Yang
2. Jie Shen
3. Jiang Yang
4. Tao Tang
5. Charles M. Elliott
6. Gerhard Dziuk
7. Yukun Li
8. Jie Xu

In Materials Chemistry

1. Xiaofeng Yang
2. Jie Shen
3. Jiang Yang
4. Tao Tang
5. Xiaobing H. Feng
6. Jie Xu
7. Yukun Li
8. N. Mozaffari

In Numerical Analysis

1. Jie Shen
2. Xiaofeng Yang
3. Xiaobing H. Feng
4. Jiang Yang
5. Tao Tang
6. Jie Xu
7. Yukun Li
8. Charles M. Elliott

Citations in the corpus

• A posteriori error control for the Allen–Cahn problem: circumventing Gronwall's inequality (2004, result): “Our work was inspired by recent results of Caginalp and Chen [3] and Feng and Prohl [9].”
• On the maximum principle preserving schemes for the generalized Allen–Cahn equation (2016, background): “, mean curvature flow [8,13], image segmentation [2], and many problems in materials science.”
• Convergence and Error Analysis for the Scalar Auxiliary Variable (SAV) Schemes to Gradient Flows (2018, background): “Although some convergence and error analysis are available for fully implicit (such as backward Euler) [7, 10, 23, 11, 5] or nonlinearly implicit (such as convex splitting) [9, 2] schemes without restrictive assumptions on the free energy, most of the convergence and error analysis for linearly impl”
• Computation of geometric partial differential equations and mean curvature flow (2005, background): “Feng and Prohl (2003) have improved the finite element error analysis of the Allen–Cahn equation using the special structure of the solution.”
• The phase field method for geometric moving interfaces and their numerical approximations (2019, neutral): “A key idea of the second technique of the fine error analysis, which was developed in [203, 204, 207, 195, 197], is not to separate that term, instead, to estimate it together with the proceeding term.”
• Analysis of interior penalty discontinuous Galerkin methods for the Allen-Cahn equation and the mean curvature flow (2013, positive): “(1) There exists a nonnegative constant σ 1 such that (2.2) (2) There exists a nonnegative constant σ 2 such that (3) There exists nonnegative constant σ 3 such that The following solution estimates can be found in [12].”
• Nonlocal damage model using the phase field method: Theory and applications (2013, background): “(2001); Gránásy et al. (2004); Karma (2001); Ohno and Matsuura (2010)), inhomogeneous elasticity (Boussinot et al.”
• Numerical approximations of Allen-Cahn and Cahn-Hilliard equations (2010, background): “While it is possible to improve this bound to polynomial order in terms of ε−1 as is done in [12, 13, 15] and [24], this process is very technical so we decided not to carry it out as our main purpose in this paper is to study the stability properties of various schemes and to demonstrate that optim”
• Error analysis of stabilized semi-implicit method of Allen-Cahnequation (2009, positive): “In [7], the spectrum argument [2, 10, 11] was applied to derive the rigorous error bounds for the implicit scheme (for f(φ)).”