2001 Fiscal Year Final Research Report Summary
Error Analysis of Ritz Finite Element Methods
| Project/Area Number |
12640128
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| Research Category |
Grant-in-Aid for Scientific Research (C)
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| Allocation Type | Single-year Grants |
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| Section | 一般 |
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| Research Field |
General mathematics (including Probability theory/Statistical mathematics)
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| Research Institution | Ehime University |
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Principal Investigator |
TSUCHIYA Takuya Faculty of Science, Ehime University, Associate Professor, 理学部, 助教授 (00163832)
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| Co-Investigator(Kenkyū-buntansha) |
FANG Qing Faculty of Science, Ehime University, Instructor, 理学部, 助手 (10243544)
YAMAMOTO Tetsuro Faculty of Science, Ehime University, Professor, 理学部, 教授 (80034560)
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| Project Period (FY) |
2000 – 2001
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| Keywords | finite element method / error analysis / conformal mappings / minimal surfaces / mean curvature / variational principle |
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| Research Abstract |
In this research project, we have tried to develop an error analysis of Ritz finite element methods, and have obtained the following results.
We have considered finite element approximations of conformal mappings from the unit disk to Jordan domains defined in two-dimensional Euclidean space. The set of admissible mappings is defined in piecewise linear finite element space. We then define the finite element (FE) conformal mappings as the minimizer of the Dirichlet integral in the set of admissible mappings. Under certain mild assumptions we have shown FE conformal mappings converge to a exact conformal mapping as triangulation of the unit disk is getting refined. We also considered Jordan domains with angles. Introducing "the smoothing method", developed in optimization theory, we may also compute FE conformal mappings to those domains. Many interesting examples are given.
Surfaces in three-dimensional Euclidean space on which the mean curvature is constant at every point are called H-surfaces. We consider finite element approximation of H-surfaces. Finite element H-surfaces are defined as stationary point of an energy functional in a certain set of admissible mappings in the piecewise linear finite element space. It is proven that finite element H-surfaces converge to an exact H-surface. Many numerical examples are given.
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Research Products
(12 results)
