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Especially, in solving the eigenvalue problem of Biharmonic operators, the Fujino\u2013Morley FEM along with the Fujino\u2013Morley interpolation function can be utilized to find explicit lower bound for the eigenvalues; see the work of Carstensen\u2013Gallistl [3] and Liu [8, 9]. For the Fujino\u2013Morley interpolation, there are two fundamental constants that are playing important roles in bounding the eigenvalues. Rough bounds of the two constants have been given in [3] by using theoretical analysis. In this study, the optimal estimation of two interpolation error constants, denoted by C_0 and C_1, for the FujinoMorley interpolation function is considered. The problem of error constant estimation is converted to eigenvalue evaluation problem related with Biharmonic operators. To give concrete values for the constants, a new algorithm based on finite element method along with verified computation is proposed to estimate the eigenvalues corresponding to interpolation error constant. Notice that the lower eigenvalue bound provides upper bound for the interpolation error constants. Given a triangle K \u2208 R^2 with unit diameter, assume that K has vertices O(0, 0), A(1, 0) and B(x, y). Since C_0(K) and C_1(K) depend on the shape of K, we have to obtain an estimation of C0(K) and C1(K) that holds for triangle of different shapes. From the geometric structure of triangle domain and symmetry of functions, we only need to consider the case that (x, y) satisfies {x^2 + y^2 \u2264 1, x \u2265 1/2 , y \u003e 0}. The contour lines of approximate numerical evaluation of constants, denoted by C0(x, y) and C1(x, y), are displayed in Fig. 1. From Fig. 1, we have the information: (i) the maximum of C_0 is obtained when B(x, y) = (cos \u03c0/3, sin \u03c0/3 ) and (ii) the maximum of C_1 is obtained when B(x, y) \u2192 (1, 0). For a given specific triangle K, we apply the method of Liu (2015) to obtain a lower bound for the corresponding eigenvalues. For general triangle K, we propose a new theory to estimate the eigenvalue perturbation upon the variation of triangle shape. Specifically, n the perturbation analysis, we define a linear transformation on element K to introduce perturbation to vertex B(x, y). Let the linear transformation Q map K to triangle K^^~. We use the invariance of a linear composition on FujinoMorley space to show V^ \u003cFM\u003e(K) and V^~\u003cFM\u003e(K^^~) is b\u0133ective, and obtain the relationship between of constants on triangle K and K^^~. Assume the diameter of triangle element K is h, our proposed algorithm tells that the following estimation of the above two constants holds. 0.07349h^2 \u2264 sup_\u003cdiam(K)\u2264h\u003e C_0(K) \u2264 0.07353h^ 2, 0.18863h \u2264 sup_\u003cdiam(K)\u2264h\u003e C_1(K) \u2264 0.18868h. The method proposed in this research for estimating the FujinoMorley interpolation constant C_0 and C_1 can also be used to estimate the constants of other interpolation operators. For example, let \u03a0 be the Lagrange interpolation operator over triangle T, that is, for v \u2208 H^2(T), \u03a0v is a linear polynomial and v \u2212 \u03a0v = 0 at each vertex. In case that the diameter of T equals to one, the following interpolation error estimation holds. \u03a0u  u_\u003c0,T\u003e \u2264 C_\u003cL,0\u003eh^2u_\u003c2,T\u003e, \u03a0u \u2013 u_\u003c1,T\u003e \u2264 C_\u003cL,1\u003ehu_\u003c2,T\u003e. With several selection of vertex B(a, b) of triangle T, we list the estimation of upper bounds of C_\u003cL,0\u003e and C_\u003cL,1\u003e in Table 1. The underline in tables tells that the lower bound and upper bound evaluation of the constants agree with each other at the underlined digits. The computing code is shared at https://ganjin.online/shihkang/BiharmonicEig.", "subitem_description_type": "Abstract"}]}, "item_6_description_53": {"attribute_name": "\u5b66\u4f4d\u8a18\u756a\u53f7", "attribute_value_mlt": [{"subitem_description": "\u65b0\u5927\u9662\u535a(\u7406)\u7532\u7b2c454\u53f7", "subitem_description_type": "Other"}]}, "item_6_dissertation_number_52": {"attribute_name": "\u5b66\u4f4d\u6388\u4e0e\u756a\u53f7", "attribute_value_mlt": [{"subitem_dissertationnumber": "13101\u7532\u7b2c4812\u53f7"}]}, "item_6_select_19": {"attribute_name": "\u8457\u8005\u7248\u30d5\u30e9\u30b0", "attribute_value_mlt": [{"subitem_select_item": "ETD"}]}, "item_creator": {"attribute_name": "\u8457\u8005", "attribute_type": "creator", "attribute_value_mlt": [{"creatorNames": [{"creatorName": "Liao, Shihkang"}], "nameIdentifiers": [{"nameIdentifier": "178186", "nameIdentifierScheme": "WEKO"}]}]}, "item_files": {"attribute_name": "\u30d5\u30a1\u30a4\u30eb\u60c5\u5831", "attribute_type": "file", "attribute_value_mlt": [{"accessrole": "open_date", "date": [{"dateType": "Available", "dateValue": "20201102"}], "displaytype": "detail", "download_preview_message": "", "file_order": 0, "filename": "r2fsk454.pdf", "filesize": [{"value": "2.1 MB"}], "format": "application/pdf", "future_date_message": "", "is_thumbnail": false, "licensetype": "license_free", "mimetype": "application/pdf", "size": 2100000.0, "url": {"label": "\u672c\u6587", "url": "https://niigatau.repo.nii.ac.jp/record/34137/files/r2fsk454.pdf"}, "version_id": "d12dd8fda0054964b564b318f03bfe2a"}, {"accessrole": "open_date", "date": [{"dateType": "Available", "dateValue": "20201102"}], "displaytype": "detail", "download_preview_message": "", "file_order": 1, "filename": "r2fsk454_a.pdf", "filesize": [{"value": "113.0 kB"}], "format": "application/pdf", "future_date_message": "", "is_thumbnail": false, "licensetype": "license_free", "mimetype": "application/pdf", "size": 113000.0, "url": {"label": "\u8981\u65e8", "url": "https://niigatau.repo.nii.ac.jp/record/34137/files/r2fsk454_a.pdf"}, "version_id": "a270bc2a94774b53b8bd4dd41b5a7182"}]}, "item_language": {"attribute_name": "\u8a00\u8a9e", "attribute_value_mlt": [{"subitem_language": "eng"}]}, "item_resource_type": {"attribute_name": "\u8cc7\u6e90\u30bf\u30a4\u30d7", "attribute_value_mlt": [{"resourcetype": "thesis", "resourceuri": "http://purl.org/coar/resource_type/c_46ec"}]}, "item_title": "Eigenvalue estimation for Biharmonic operators and its application to interpolation error constant estimation", "item_titles": {"attribute_name": "\u30bf\u30a4\u30c8\u30eb", "attribute_value_mlt": [{"subitem_title": "Eigenvalue estimation for Biharmonic operators and its application to interpolation error constant estimation"}]}, "item_type_id": "6", "owner": "1", "path": ["453/455", "468/563/564"], "permalink_uri": "http://hdl.handle.net/10191/00051935", "pubdate": {"attribute_name": "\u516c\u958b\u65e5", "attribute_value": "20201102"}, "publish_date": "20201102", "publish_status": "0", "recid": "34137", "relation": {}, "relation_version_is_last": true, "title": ["Eigenvalue estimation for Biharmonic operators and its application to interpolation error constant estimation"], "weko_shared_id": 2}
Eigenvalue estimation for Biharmonic operators and its application to interpolation error constant estimation
http://hdl.handle.net/10191/00051935
550756e36f294e2bbade07c5752384ae
名前 / ファイル  ライセンス  アクション  

本文 (2.1 MB)



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Item type  学位論文 / Thesis or Dissertation(1)  

公開日  20201102  
タイトル  
タイトル  Eigenvalue estimation for Biharmonic operators and its application to interpolation error constant estimation  
言語  
言語  eng  
資源タイプ  
資源  http://purl.org/coar/resource_type/c_46ec  
タイプ  thesis  
その他のタイトル  
その他のタイトル  重調和微分作用素の固有値評価と補間関数の誤差定数評価への応用  
著者 
Liao, Shihkang
× Liao, Shihkang 

抄録  
内容記述タイプ  Abstract  
内容記述  The Fujino–Morley finite element method (FEM) [5, 12, 13] provides a robust way to solve partial differential problems evolving Biharmonic operators. Especially, in solving the eigenvalue problem of Biharmonic operators, the Fujino–Morley FEM along with the Fujino–Morley interpolation function can be utilized to find explicit lower bound for the eigenvalues; see the work of Carstensen–Gallistl [3] and Liu [8, 9]. For the Fujino–Morley interpolation, there are two fundamental constants that are playing important roles in bounding the eigenvalues. Rough bounds of the two constants have been given in [3] by using theoretical analysis. In this study, the optimal estimation of two interpolation error constants, denoted by C_0 and C_1, for the FujinoMorley interpolation function is considered. The problem of error constant estimation is converted to eigenvalue evaluation problem related with Biharmonic operators. To give concrete values for the constants, a new algorithm based on finite element method along with verified computation is proposed to estimate the eigenvalues corresponding to interpolation error constant. Notice that the lower eigenvalue bound provides upper bound for the interpolation error constants. Given a triangle K ∈ R^2 with unit diameter, assume that K has vertices O(0, 0), A(1, 0) and B(x, y). Since C_0(K) and C_1(K) depend on the shape of K, we have to obtain an estimation of C0(K) and C1(K) that holds for triangle of different shapes. From the geometric structure of triangle domain and symmetry of functions, we only need to consider the case that (x, y) satisfies {x^2 + y^2 ≤ 1, x ≥ 1/2 , y > 0}. The contour lines of approximate numerical evaluation of constants, denoted by C0(x, y) and C1(x, y), are displayed in Fig. 1. From Fig. 1, we have the information: (i) the maximum of C_0 is obtained when B(x, y) = (cos π/3, sin π/3 ) and (ii) the maximum of C_1 is obtained when B(x, y) → (1, 0). For a given specific triangle K, we apply the method of Liu (2015) to obtain a lower bound for the corresponding eigenvalues. For general triangle K, we propose a new theory to estimate the eigenvalue perturbation upon the variation of triangle shape. Specifically, n the perturbation analysis, we define a linear transformation on element K to introduce perturbation to vertex B(x, y). Let the linear transformation Q map K to triangle K^^~. We use the invariance of a linear composition on FujinoMorley space to show V^ <FM>(K) and V^~<FM>(K^^~) is bĳective, and obtain the relationship between of constants on triangle K and K^^~. Assume the diameter of triangle element K is h, our proposed algorithm tells that the following estimation of the above two constants holds. 0.07349h^2 ≤ sup_<diam(K)≤h> C_0(K) ≤ 0.07353h^ 2, 0.18863h ≤ sup_<diam(K)≤h> C_1(K) ≤ 0.18868h. The method proposed in this research for estimating the FujinoMorley interpolation constant C_0 and C_1 can also be used to estimate the constants of other interpolation operators. For example, let Π be the Lagrange interpolation operator over triangle T, that is, for v ∈ H^2(T), Πv is a linear polynomial and v − Πv = 0 at each vertex. In case that the diameter of T equals to one, the following interpolation error estimation holds. Πu  u_<0,T> ≤ C_<L,0>h^2u_<2,T>, Πu – u_<1,T> ≤ C_<L,1>hu_<2,T>. With several selection of vertex B(a, b) of triangle T, we list the estimation of upper bounds of C_<L,0> and C_<L,1> in Table 1. The underline in tables tells that the lower bound and upper bound evaluation of the constants agree with each other at the underlined digits. The computing code is shared at https://ganjin.online/shihkang/BiharmonicEig.  
書誌情報  p. 146  
著者版フラグ  
値  ETD  
学位名  
学位名  博士(理学)  
学位授与機関  
学位授与機関名  
学位授与機関名  新潟大学  
学位授与年月日  
学位授与年月日  20200923  
学位授与番号  
学位授与番号  13101甲第4812号  
学位記番号  
内容記述タイプ  Other  
内容記述  新大院博(理)甲第454号 