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Under some quite general settings, we show not only the pointwise convergence but also prove that the norm of approximate operator converges to the corresponding norm for the inverse of a linear elliptic operator. Since the approximate norm estimates of linearized inverse operator play an essential role in the numerical verification method of solutions for non-linear elliptic problems, our result is also important in terms of guaranteeing its validity. Furthermore, the present method can also be applied to more general elliptic problems, e.g., biharmonic problems and so on.<\/jats:p>","DOI":"10.14232\/actacyb.294906","type":"journal-article","created":{"date-parts":[[2022,9,5]],"date-time":"2022-09-05T11:41:42Z","timestamp":1662378102000},"page":"71-82","source":"Crossref","is-referenced-by-count":0,"title":["On Some Convergence Properties for Finite Element Approximations to the Inverse of Linear Elliptic Operators"],"prefix":"10.14232","volume":"26","author":[{"ORCID":"http:\/\/orcid.org\/0000-0001-9756-4571","authenticated-orcid":false,"given":"Takehiko","family":"Kinoshita","sequence":"first","affiliation":[{"id":[{"id":"https:\/\/ror.org\/04f4wg107","id-type":"ROR","asserted-by":"publisher"}],"name":"Saga University","department":["Department of Mathematical Science"]}]},{"ORCID":"http:\/\/orcid.org\/0000-0001-6520-3552","authenticated-orcid":false,"given":"Yoshitaka","family":"Watanabe","sequence":"additional","affiliation":[{"id":[{"id":"https:\/\/ror.org\/00p4k0j84","id-type":"ROR","asserted-by":"publisher"}],"name":"Kyushu University","department":["Research Institute for Information Technology"]}]},{"ORCID":"http:\/\/orcid.org\/0000-0001-5228-0591","authenticated-orcid":false,"given":"Mitsuhiro","family":"T. Nakao","sequence":"additional","affiliation":[{"id":[{"id":"https:\/\/ror.org\/00ntfnx83","id-type":"ROR","asserted-by":"publisher"}],"name":"Waseda University","department":["Faculty of Science and Engineering"]}]}],"member":"5401","published-online":{"date-parts":[[2022,9,2]]},"reference":[{"key":"ref1","doi-asserted-by":"crossref","DOI":"10.1007\/978-1-4757-3658-8","volume-title":"The Mathematical Theory of Finite Element Methods","author":"Brenner","year":"2002","unstructured":"Brenner, Susanne C. and Scott, L. Ridgway. The Mathematical Theory of Finite Element Methods. Springer, New York, second edition, 2002."},{"key":"ref2","volume-title":"Handbook of Numerical Analysis Volume II, Finite Element Methods (Part 1)","author":"Ciarlet","year":"2003","unstructured":"Ciarlet, P.G. and Lions, J.L. Handbook of Numerical Analysis Volume II, Finite Element Methods (Part 1). 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An introduction to the mathematical theory of finite elements. John Wiley & Sons, New York, 1976."},{"key":"ref9","doi-asserted-by":"publisher","DOI":"10.1016\/0377-0427(94)00090-N"},{"key":"ref10","doi-asserted-by":"publisher","DOI":"10.1007\/BF00945108"},{"key":"ref11","doi-asserted-by":"crossref","unstructured":"Rump, Siegfried M. INTLAB --- INTerval LABoratory. In Csendes, Tibor, editor, Developments in Reliable Computing, pages 77-104. 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