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Math."],"published-print":{"date-parts":[[2024,6]]},"abstract":"Abstract<\/jats:title>The higher-order guaranteed lower eigenvalue bounds of the Laplacian in the recent work by Carstensen et al. (Numer Math 149(2):273\u2013304, 2021) require a parameter $$C_{\\text {st},1}$$<\/jats:tex-math>\n \n C<\/mml:mi>\n \n st<\/mml:mtext>\n ,<\/mml:mo>\n 1<\/mml:mn>\n <\/mml:mrow>\n <\/mml:msub>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula> that is found not<\/jats:italic> robust as the polynomial degree p<\/jats:italic> increases. This is related to the $$H^1$$<\/jats:tex-math>\n \n H<\/mml:mi>\n 1<\/mml:mn>\n <\/mml:msup>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula> stability bound of the $$L^{2}$$<\/jats:tex-math>\n \n L<\/mml:mi>\n 2<\/mml:mn>\n <\/mml:msup>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula> projection onto polynomials of degree at most p<\/jats:italic> and its growth $$C_{\\textrm{st, 1}}\\propto (p+1)^{1\/2}$$<\/jats:tex-math>\n \n \n C<\/mml:mi>\n \n st, 1<\/mml:mtext>\n <\/mml:mrow>\n <\/mml:msub>\n \u221d<\/mml:mo>\n \n \n (<\/mml:mo>\n p<\/mml:mi>\n +<\/mml:mo>\n 1<\/mml:mn>\n )<\/mml:mo>\n <\/mml:mrow>\n \n 1<\/mml:mn>\n \/<\/mml:mo>\n 2<\/mml:mn>\n <\/mml:mrow>\n <\/mml:msup>\n <\/mml:mrow>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula> as $$p \\rightarrow \\infty $$<\/jats:tex-math>\n \n p<\/mml:mi>\n \u2192<\/mml:mo>\n \u221e<\/mml:mi>\n <\/mml:mrow>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula>. A similar estimate for the Galerkin projection holds with a p<\/jats:italic>-robust constant $$C_{\\text {st},2}$$<\/jats:tex-math>\n \n C<\/mml:mi>\n \n st<\/mml:mtext>\n ,<\/mml:mo>\n 2<\/mml:mn>\n <\/mml:mrow>\n <\/mml:msub>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula> and $$C_{\\text {st},2} \\le 2$$<\/jats:tex-math>\n \n \n C<\/mml:mi>\n \n st<\/mml:mtext>\n ,<\/mml:mo>\n 2<\/mml:mn>\n <\/mml:mrow>\n <\/mml:msub>\n \u2264<\/mml:mo>\n 2<\/mml:mn>\n <\/mml:mrow>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula> for right-isosceles triangles. This paper utilizes the new inequality with the constant $$C_{\\text {st},2}$$<\/jats:tex-math>\n \n C<\/mml:mi>\n \n st<\/mml:mtext>\n ,<\/mml:mo>\n 2<\/mml:mn>\n <\/mml:mrow>\n <\/mml:msub>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula> to design a modified hybrid high-order eigensolver that directly computes guaranteed lower eigenvalue bounds under the idealized hypothesis of exact solve of the generalized algebraic eigenvalue problem and a mild explicit condition on the maximal mesh-size in the simplicial mesh. A key advance is a p<\/jats:italic>-robust parameter selection. The analysis of the new method with a different fine-tuned volume stabilization allows for a\u00a0priori quasi-best approximation and improved $$L^{2}$$<\/jats:tex-math>\n \n L<\/mml:mi>\n 2<\/mml:mn>\n <\/mml:msup>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula> error estimates as well as a stabilization-free reliable and efficient a posteriori error control. The associated adaptive mesh-refining algorithm performs superior in computer benchmarks with striking numerical evidence for optimal higher empirical convergence rates.<\/jats:p>","DOI":"10.1007\/s00211-024-01407-w","type":"journal-article","created":{"date-parts":[[2024,5,6]],"date-time":"2024-05-06T03:17:31Z","timestamp":1714965451000},"page":"813-851","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":5,"title":["Adaptive hybrid high-order method for guaranteed lower eigenvalue bounds"],"prefix":"10.1007","volume":"156","author":[{"given":"Carsten","family":"Carstensen","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Benedikt","family":"Gr\u00e4\u00dfle","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Ngoc Tien","family":"Tran","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"297","published-online":{"date-parts":[[2024,5,6]]},"reference":[{"issue":"4","key":"1407_CR1","doi-asserted-by":"crossref","first-page":"909","DOI":"10.1007\/s00466-018-1538-0","volume":"62","author":"M Abbas","year":"2018","unstructured":"Abbas, M., Ern, A., Pignet, N.: Hybrid high-order methods for finite deformations of hyperelastic materials. 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