{"created":"2021-03-01T06:40:36.099493+00:00","id":34137,"links":{},"metadata":{"_buckets":{"deposit":"b6bcc651-f67e-4a27-8548-279891db6ee9"},"_deposit":{"id":"34137","owners":[],"pid":{"revision_id":0,"type":"depid","value":"34137"},"status":"published"},"_oai":{"id":"oai:niigata-u.repo.nii.ac.jp:00034137","sets":["453:455","468:563:564"]},"item_6_alternative_title_1":{"attribute_name":"その他のタイトル","attribute_value_mlt":[{"subitem_alternative_title":"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"}]},"item_6_biblio_info_6":{"attribute_name":"書誌情報","attribute_value_mlt":[{"bibliographicPageEnd":"46","bibliographicPageStart":"1","bibliographic_titles":[{}]}]},"item_6_date_granted_51":{"attribute_name":"学位授与年月日","attribute_value_mlt":[{"subitem_dategranted":"2020-09-23"}]},"item_6_degree_grantor_49":{"attribute_name":"学位授与機関","attribute_value_mlt":[{"subitem_degreegrantor":[{"subitem_degreegrantor_name":"新潟大学"}]}]},"item_6_degree_name_48":{"attribute_name":"学位名","attribute_value_mlt":[{"subitem_degreename":"博士(理学)"}]},"item_6_description_4":{"attribute_name":"抄録","attribute_value_mlt":[{"subitem_description":"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 Fujino-Morley 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 Fujino-Morley 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 Fujino-Morley 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^2||u||_<2,T>, ||Πu – u||_<1,T> ≤ C_<L,1>h||u||_<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.","subitem_description_type":"Abstract"}]},"item_6_description_53":{"attribute_name":"学位記番号","attribute_value_mlt":[{"subitem_description":"新大院博(理)甲第454号","subitem_description_type":"Other"}]},"item_6_dissertation_number_52":{"attribute_name":"学位授与番号","attribute_value_mlt":[{"subitem_dissertationnumber":"13101甲第4812号"}]},"item_6_select_19":{"attribute_name":"著者版フラグ","attribute_value_mlt":[{"subitem_select_item":"ETD"}]},"item_creator":{"attribute_name":"著者","attribute_type":"creator","attribute_value_mlt":[{"creatorNames":[{"creatorName":"Liao, Shihkang"}],"nameIdentifiers":[{"nameIdentifier":"178186","nameIdentifierScheme":"WEKO"}]}]},"item_files":{"attribute_name":"ファイル情報","attribute_type":"file","attribute_value_mlt":[{"accessrole":"open_date","date":[{"dateType":"Available","dateValue":"2020-11-02"}],"displaytype":"detail","filename":"r2fsk454.pdf","filesize":[{"value":"2.1 MB"}],"format":"application/pdf","licensetype":"license_note","mimetype":"application/pdf","url":{"label":"本文","url":"https://niigata-u.repo.nii.ac.jp/record/34137/files/r2fsk454.pdf"},"version_id":"d12dd8fd-a005-4964-b564-b318f03bfe2a"},{"accessrole":"open_date","date":[{"dateType":"Available","dateValue":"2020-11-02"}],"displaytype":"detail","filename":"r2fsk454_a.pdf","filesize":[{"value":"113.0 kB"}],"format":"application/pdf","licensetype":"license_note","mimetype":"application/pdf","url":{"label":"要旨","url":"https://niigata-u.repo.nii.ac.jp/record/34137/files/r2fsk454_a.pdf"},"version_id":"a270bc2a-9477-4b53-b8bd-4dd41b5a7182"}]},"item_language":{"attribute_name":"言語","attribute_value_mlt":[{"subitem_language":"eng"}]},"item_resource_type":{"attribute_name":"資源タイプ","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":"タイトル","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":["455","564"],"pubdate":{"attribute_name":"公開日","attribute_value":"2020-11-02"},"publish_date":"2020-11-02","publish_status":"0","recid":"34137","relation_version_is_last":true,"title":["Eigenvalue estimation for Biharmonic operators and its application to interpolation error constant estimation"],"weko_creator_id":"1","weko_shared_id":2},"updated":"2022-12-15T04:04:27.618008+00:00"}