Volume-preserving parametric finite element methods for axisymmetric geometric evolution equations
We propose and analyze volume-preserving parametric finite element methods for surface diffusion, conserved mean curvature flow and an intermediate evolution law in an axisymmetric setting. The weak formulations are presented in terms of the generating curves of the axisymmetric surfaces. The propos...
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| Published in | Journal of computational physics Vol. 460; p. 111180 |
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| Main Authors | , , , |
| Format | Journal Article |
| Language | English |
| Published |
Cambridge
Elsevier Inc
01.07.2022
Elsevier Science Ltd |
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| Online Access | Get full text |
| ISSN | 0021-9991 1090-2716 |
| DOI | 10.1016/j.jcp.2022.111180 |
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| Abstract | We propose and analyze volume-preserving parametric finite element methods for surface diffusion, conserved mean curvature flow and an intermediate evolution law in an axisymmetric setting. The weak formulations are presented in terms of the generating curves of the axisymmetric surfaces. The proposed numerical methods are based on piecewise linear parametric finite elements. The constructed fully practical schemes satisfy the conservation of the enclosed volume. In addition, we prove the unconditional stability and consider the distribution of vertices for the discretized schemes. The introduced methods are implicit and the resulting nonlinear systems of equations can be solved very efficiently and accurately via the Newton's iterative method. Numerical results are presented to show the accuracy and efficiency of the introduced schemes for computing the considered axisymmetric geometric flows.
•We propose volume-preserving finite element methods for axisymmetric geometric evolution equations.•The unconditional stability and vertices distributions of the introduced schemes are analyzed.•Numerical examples are presented to show the accuracy and efficiency of the schemes. |
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| AbstractList | We propose and analyze volume-preserving parametric finite element methods for surface diffusion, conserved mean curvature flow and an intermediate evolution law in an axisymmetric setting. The weak formulations are presented in terms of the generating curves of the axisymmetric surfaces. The proposed numerical methods are based on piecewise linear parametric finite elements. The constructed fully practical schemes satisfy the conservation of the enclosed volume. In addition, we prove the unconditional stability and consider the distribution of vertices for the discretized schemes. The introduced methods are implicit and the resulting nonlinear systems of equations can be solved very efficiently and accurately via the Newton's iterative method. Numerical results are presented to show the accuracy and efficiency of the introduced schemes for computing the considered axisymmetric geometric flows. We propose and analyze volume-preserving parametric finite element methods for surface diffusion, conserved mean curvature flow and an intermediate evolution law in an axisymmetric setting. The weak formulations are presented in terms of the generating curves of the axisymmetric surfaces. The proposed numerical methods are based on piecewise linear parametric finite elements. The constructed fully practical schemes satisfy the conservation of the enclosed volume. In addition, we prove the unconditional stability and consider the distribution of vertices for the discretized schemes. The introduced methods are implicit and the resulting nonlinear systems of equations can be solved very efficiently and accurately via the Newton's iterative method. Numerical results are presented to show the accuracy and efficiency of the introduced schemes for computing the considered axisymmetric geometric flows. •We propose volume-preserving finite element methods for axisymmetric geometric evolution equations.•The unconditional stability and vertices distributions of the introduced schemes are analyzed.•Numerical examples are presented to show the accuracy and efficiency of the schemes. |
| ArticleNumber | 111180 |
| Author | Garcke, Harald Nürnberg, Robert Zhao, Quan Bao, Weizhu |
| Author_xml | – sequence: 1 givenname: Weizhu surname: Bao fullname: Bao, Weizhu email: matbaowz@nus.edu.sg organization: Department of Mathematics, National University of Singapore, 119076, Singapore – sequence: 2 givenname: Harald surname: Garcke fullname: Garcke, Harald email: harald.garcke@ur.de organization: Fakultät für Mathematik, Universität Regensburg, 93040 Regensburg, Germany – sequence: 3 givenname: Robert orcidid: 0000-0002-2489-2416 surname: Nürnberg fullname: Nürnberg, Robert email: robert.nurnberg@unitn.it organization: Dipartimento di Mathematica, Università di Trento, 38123 Trento, Italy – sequence: 4 givenname: Quan orcidid: 0000-0002-3131-6863 surname: Zhao fullname: Zhao, Quan email: quan.zhao@ur.de organization: Fakultät für Mathematik, Universität Regensburg, 93040 Regensburg, Germany |
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| Keywords | Unconditional stability Surface diffusion flow Conserved mean curvature flow Volume conservation Parametric finite element method Axisymmetry |
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| SubjectTerms | Apexes Axisymmetric flow Axisymmetry Computational physics Conserved mean curvature flow Evolution Finite element method Iterative methods Mathematical analysis Nonlinear systems Numerical methods Parametric finite element method Surface diffusion Surface diffusion flow Unconditional stability Volume conservation |
| Title | Volume-preserving parametric finite element methods for axisymmetric geometric evolution equations |
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