{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,5,13]],"date-time":"2026-05-13T22:13:49Z","timestamp":1778710429089,"version":"3.51.4"},"reference-count":55,"publisher":"Springer Science and Business Media LLC","issue":"2","license":[{"start":{"date-parts":[[2026,2,23]],"date-time":"2026-02-23T00:00:00Z","timestamp":1771804800000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"},{"start":{"date-parts":[[2026,2,23]],"date-time":"2026-02-23T00:00:00Z","timestamp":1771804800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"}],"funder":[{"DOI":"10.13039\/100019180","name":"HORIZON EUROPE European Research Council","doi-asserted-by":"publisher","award":["101115663"],"award-info":[{"award-number":["101115663"]}],"id":[{"id":"10.13039\/100019180","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100000923","name":"Australian Research Council","doi-asserted-by":"publisher","award":["FT220100496"],"award-info":[{"award-number":["FT220100496"]}],"id":[{"id":"10.13039\/501100000923","id-type":"DOI","asserted-by":"publisher"}]},{"name":"INV-CIAS-4167","award":["Universidad Militar Nueva Granada"],"award-info":[{"award-number":["Universidad Militar Nueva Granada"]}]}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["Adv Comput Math"],"published-print":{"date-parts":[[2026,4]]},"abstract":"<jats:title>Abstract<\/jats:title>\n                  <jats:p>\n                    We propose and analyse residual-based a posteriori error estimates for the virtual element discretisation applied to the thin plate vibration problem in both two and three dimensions. Our approach involves a conforming\n                    <jats:inline-formula>\n                      <jats:alternatives>\n                        <jats:tex-math>$$C^1$$<\/jats:tex-math>\n                        <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                          <mml:msup>\n                            <mml:mi>C<\/mml:mi>\n                            <mml:mn>1<\/mml:mn>\n                          <\/mml:msup>\n                        <\/mml:math>\n                      <\/jats:alternatives>\n                    <\/jats:inline-formula>\n                    discrete formulation suitable for meshes composed of polygons and polyhedra. The reliability and efficiency of the error estimator are established through a dimension-independent proof. Finally, several numerical experiments are reported to demonstrate the optimal performance of the method in 2D and 3D.\n                  <\/jats:p>","DOI":"10.1007\/s10444-026-10288-6","type":"journal-article","created":{"date-parts":[[2026,2,23]],"date-time":"2026-02-23T10:45:37Z","timestamp":1771843537000},"update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["A posteriori error estimates for a $$C^1$$ virtual element method applied to the thin plate vibration problem."],"prefix":"10.1007","volume":"52","author":[{"given":"Franco","family":"Dassi","sequence":"first","affiliation":[]},{"given":"Andr\u00e9s E.","family":"Rubiano","sequence":"additional","affiliation":[]},{"ORCID":"https:\/\/orcid.org\/0000-0001-9761-8461","authenticated-orcid":false,"given":"Iv\u00e1n","family":"Vel\u00e1squez","sequence":"additional","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2026,2,23]]},"reference":[{"key":"10288_CR1","doi-asserted-by":"publisher","unstructured":"Gazzola, F., Grunau, H.-C., Sweers, G.: Polyharmonic Boundary Value Problems: Positivity Preserving and Nonlinear Higher Order Elliptic Equations in Bounded Domains, 1st edn. Springer Science & Business Media, Berlin, Heidelberg (2010). https:\/\/doi.org\/10.1007\/978-3-642-12245-3","DOI":"10.1007\/978-3-642-12245-3"},{"issue":"4","key":"10288_CR2","doi-asserted-by":"publisher","first-page":"1280","DOI":"10.1137\/0732059","volume":"32","author":"A Berm\u00fadez","year":"1995","unstructured":"Berm\u00fadez, A., Dur\u00e1n, R., Muschietti, M.A., Rodr\u00edguez, R., Solomin, J.: Finite element vibration analysis of fluid\u2013solid systems without spurious modes. SIAM J. Numer. Anal. 32(4), 1280\u20131295 (1995). https:\/\/doi.org\/10.1137\/0732059","journal-title":"SIAM J. Numer. Anal."},{"key":"10288_CR3","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1017\/S0962492910000012","volume":"19","author":"D Boffi","year":"2010","unstructured":"Boffi, D.: Finite element approximation of eigenvalue problems. Acta Numer 19, 1\u2013120 (2010). https:\/\/doi.org\/10.1017\/S0962492910000012","journal-title":"Acta Numer"},{"key":"10288_CR4","doi-asserted-by":"publisher","unstructured":"Boffi, D., Gardini, F., Gastaldi, L.: In: Blowey, J., Jensen, M. (eds.) Some Remarks on Eigenvalue Approximation by Finite Elements, pp. 1\u201377. Springer, Berlin, Heidelberg (2012). https:\/\/doi.org\/10.1007\/978-3-642-23914-4_1","DOI":"10.1007\/978-3-642-23914-4_1"},{"key":"10288_CR5","doi-asserted-by":"publisher","unstructured":"Ciarlet, P.G.: The Finite Element Method for Elliptic Problems. Classics in Applied Mathematics, vol. 40. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA (2002).https:\/\/doi.org\/10.1137\/1.9780898719208","DOI":"10.1137\/1.9780898719208"},{"key":"10288_CR6","doi-asserted-by":"publisher","unstructured":"Babu\u0161ka, I., Osborn, J.: Eigenvalue problems. In: Finite Element Methods (Part 1). Handbook of Numerical Analysis, vol. 2, pp. 641\u2013787. Elsevier, Amsterdam, Netherlands (1991). https:\/\/doi.org\/10.1016\/S1570-8659(05)80042-0","DOI":"10.1016\/S1570-8659(05)80042-0"},{"issue":"1","key":"10288_CR7","doi-asserted-by":"publisher","first-page":"23","DOI":"10.1007\/BF01396493","volume":"33","author":"R Rannacher","year":"1979","unstructured":"Rannacher, R.: Nonconforming finite element methods for eigenvalue problems in linear plate theory. Numer. Math. 33(1), 23\u201342 (1979). https:\/\/doi.org\/10.1007\/BF01396493","journal-title":"Numer. Math."},{"issue":"34","key":"10288_CR8","doi-asserted-by":"publisher","first-page":"3669","DOI":"10.1016\/S0045-7825(02)00286-4","volume":"191","author":"G Engel","year":"2002","unstructured":"Engel, G., Garikipati, K., Hughes, T.J.R., Larson, M.G., Mazzei, L., Taylor, R.L.: Continuous\/discontinuous finite element approximations of fourth-order elliptic problems in structural and continuum mechanics with applications to thin beams and plates, and strain gradient elasticity. Comput. Methods Appl. Mech. Engrg. 191(34), 3669\u20133750 (2002). https:\/\/doi.org\/10.1016\/S0045-7825(02)00286-4","journal-title":"Comput. Methods Appl. Mech. Engrg."},{"key":"10288_CR9","doi-asserted-by":"publisher","unstructured":"Giani, S.: Solving elliptic eigenvalue problems on polygonal meshes using discontinuous Galerkin composite finite element methods. Appl. Math. Comput. 267, 618\u2013631 (2015). https:\/\/doi.org\/10.1016\/j.amc.2015.01.011. The Fourth European Seminar on Computing (ESCO 2014)","DOI":"10.1016\/j.amc.2015.01.011"},{"key":"10288_CR10","doi-asserted-by":"publisher","unstructured":"Ciarlet, P.G., Raviart, P.A.: A mixed finite element method for the biharmonic equation. In: de Boor, C. (ed.) Mathematical Aspects of Finite Elements in Partial Differential Equations, pp. 125\u2013145. Academic Press, New York, NY (1974). https:\/\/doi.org\/10.1016\/B978-0-12-208350-1.50009-1","DOI":"10.1016\/B978-0-12-208350-1.50009-1"},{"issue":"4","key":"10288_CR11","doi-asserted-by":"publisher","first-page":"737","DOI":"10.1137\/0724048","volume":"24","author":"P Monk","year":"1987","unstructured":"Monk, P.: A mixed finite element method for the biharmonic equation. SIAM J. Numer. Anal. 24(4), 737\u2013749 (1987). https:\/\/doi.org\/10.1137\/0724048","journal-title":"SIAM J. Numer. Anal."},{"issue":"268","key":"10288_CR12","doi-asserted-by":"publisher","first-page":"1891","DOI":"10.1090\/S0025-5718-09-02228-5","volume":"78","author":"D Mora","year":"2009","unstructured":"Mora, D., Rodr\u00edguez, R.: A piecewise linear finite element method for the buckling and the vibration problems of thin plates. Math. Comp. 78(268), 1891\u20131917 (2009). https:\/\/doi.org\/10.1090\/S0025-5718-09-02228-5","journal-title":"Math. Comp."},{"key":"10288_CR13","doi-asserted-by":"publisher","unstructured":"Beir\u00e3o da Veiga, L., Brezzi, F., Cangiani, A., Manzini, G., Marini, L.D., Russo, A.: Basic principles of virtual element methods. Math. Models Methods Appl. Sci. 23(1), 199\u2013214 (2013). https:\/\/doi.org\/10.1142\/S0218202512500492","DOI":"10.1142\/S0218202512500492"},{"key":"10288_CR14","doi-asserted-by":"publisher","unstructured":"Beir\u00e3o da Veiga, L., Brezzi, F., Marini, L.D., Russo, A.: The hitchhiker\u2019s guide to the virtual element method. Math. Models Methods Appl. Sci. 24(8), 1541\u20131573 (2014). https:\/\/doi.org\/10.1142\/S021820251440003X","DOI":"10.1142\/S021820251440003X"},{"key":"10288_CR15","doi-asserted-by":"publisher","unstructured":"Beir\u00e3o da Veiga, L., Manzini, G.: A virtual element method with arbitrary regularity. IMA J. Numer. Anal. 34(2), 759\u2013781 (2014). https:\/\/doi.org\/10.1093\/imanum\/drt018","DOI":"10.1093\/imanum\/drt018"},{"issue":"1","key":"10288_CR16","doi-asserted-by":"publisher","first-page":"34","DOI":"10.1137\/15M1008117","volume":"54","author":"PF Antonietti","year":"2016","unstructured":"Antonietti, P.F., Veiga, L.B., Scacchi, S., Verani, M.: A C$$^1$$ virtual element method for the Cahn-Hilliard equation with polygonal meshes. SIAM J. Numer. Anal. 54(1), 34\u201356 (2016). https:\/\/doi.org\/10.1137\/15M1008117","journal-title":"SIAM J. Numer. Anal."},{"key":"10288_CR17","doi-asserted-by":"publisher","unstructured":"Beir\u00e3o da Veiga, L., Dassi, F., Russo, A.: A C$$^1$$ virtual element method on polyhedral meshes 79(7), 1936\u20131955 (2020). https:\/\/doi.org\/10.1016\/j.camwa.2019.06.019. Advanced Computational methods for PDEs","DOI":"10.1016\/j.camwa.2019.06.019"},{"key":"10288_CR18","doi-asserted-by":"publisher","first-page":"455","DOI":"10.1016\/j.cma.2012.09.012","volume":"253","author":"F Brezzi","year":"2013","unstructured":"Brezzi, F., Marini, L.D.: Virtual element methods for plate bending problems. Comput. Methods Appl. Mech. Engrg. 253, 455\u2013462 (2013). https:\/\/doi.org\/10.1016\/j.cma.2012.09.012","journal-title":"Comput. Methods Appl. Mech. Engrg."},{"issue":"4","key":"10288_CR19","doi-asserted-by":"publisher","first-page":"3632","DOI":"10.1093\/imanum\/drab078","volume":"42","author":"D Mora","year":"2021","unstructured":"Mora, D., Reales, C., Silgado, A.: A $$C^1$$-virtual element method of high order for the brinkman equations in stream function formulation with pressure recovery. IMA J. Numer. Anal. 42(4), 3632\u20133674 (2021). https:\/\/doi.org\/10.1093\/imanum\/drab078","journal-title":"IMA J. Numer. Anal."},{"key":"10288_CR20","doi-asserted-by":"publisher","unstructured":"Mora, D., Silgado, A.: A $$C^1$$ virtual element method for the stationary quasi-geostrophic equations of the ocean. Comput. Math. Appl. 116, 212\u2013228 (2022). https:\/\/doi.org\/10.1016\/j.camwa.2021.05.022. New trends in Computational Methods for PDEs","DOI":"10.1016\/j.camwa.2021.05.022"},{"issue":"3","key":"10288_CR21","doi-asserted-by":"publisher","first-page":"333","DOI":"10.21136\/am.2018.0093-18","volume":"63","author":"O \u010cert\u00edk","year":"2018","unstructured":"\u010cert\u00edk, O., Gardini, F., Manzini, G., Vacca, G.: The virtual element method for eigenvalue problems with potential terms on polytopic meshes. Appl. Math. 63(3), 333\u2013365 (2018). https:\/\/doi.org\/10.21136\/am.2018.0093-18","journal-title":"Appl. Math."},{"key":"10288_CR22","doi-asserted-by":"publisher","first-page":"85","DOI":"10.1016\/j.camwa.2022.07.001","volume":"121","author":"F Dassi","year":"2022","unstructured":"Dassi, F., Vel\u00e1squez, I.: Virtual element method on polyhedral meshes for bi-harmonic eigenvalues problems. Comput. Math. Appl. 121, 85\u2013101 (2022). https:\/\/doi.org\/10.1016\/j.camwa.2022.07.001","journal-title":"Comput. Math. Appl."},{"issue":"4","key":"10288_CR23","doi-asserted-by":"publisher","first-page":"2026","DOI":"10.1093\/imanum\/drx063","volume":"38","author":"F Gardini","year":"2017","unstructured":"Gardini, F., Vacca, G.: Virtual element method for second-order elliptic eigenvalue problems. IMA J. Numer. Anal. 38(4), 2026\u20132054 (2017). https:\/\/doi.org\/10.1093\/imanum\/drx063","journal-title":"IMA J. Numer. Anal."},{"issue":"5","key":"10288_CR24","doi-asserted-by":"publisher","first-page":"68","DOI":"10.1007\/s10444-020-09810-1","volume":"46","author":"J Meng","year":"2020","unstructured":"Meng, J., Mei, L.: A mixed virtual element method for the vibration problem of clamped Kirchhoff plate. Adv. Comput. Math. 46(5), 68 (2020). https:\/\/doi.org\/10.1007\/s10444-020-09810-1","journal-title":"Adv. Comput. Math."},{"issue":"9","key":"10288_CR25","doi-asserted-by":"publisher","first-page":"1821","DOI":"10.1080\/00207160.2020.1849635","volume":"98","author":"J Meng","year":"2021","unstructured":"Meng, J., Mei, L.: A $$C^0$$ virtual element method for the biharmonic eigenvalue problem. Int. J. Comput. Math. 98(9), 1821\u20131833 (2021). https:\/\/doi.org\/10.1080\/00207160.2020.1849635","journal-title":"Int. J. Comput. Math."},{"issue":"4","key":"10288_CR26","doi-asserted-by":"publisher","first-page":"325","DOI":"10.1515\/apam-2018-0072","volume":"10","author":"G Monz\u00f3n","year":"2019","unstructured":"Monz\u00f3n, G.: A virtual element method for a biharmonic Steklov eigenvalue problem. Adv. Pure Appl. Math. 10(4), 325\u2013337 (2019). https:\/\/doi.org\/10.1515\/apam-2018-0072","journal-title":"Adv. Pure Appl. Math."},{"issue":"8","key":"10288_CR27","doi-asserted-by":"publisher","first-page":"1421","DOI":"10.1007\/s10915-021-01555-3","volume":"25","author":"D Mora","year":"2015","unstructured":"Mora, D., Rivera, G., Rodr\u00edguez, R.: A virtual element method for the Steklov eigenvalue problem. Math. Models Methods Appl. Sci. 25(8), 1421\u20131445 (2015). https:\/\/doi.org\/10.1007\/s10915-021-01555-3","journal-title":"Math. Models Methods Appl. Sci."},{"issue":"14","key":"10288_CR28","doi-asserted-by":"publisher","first-page":"2803","DOI":"10.1142\/S0218202518500616","volume":"28","author":"D Mora","year":"2018","unstructured":"Mora, D., Vel\u00e1squez, I.: A virtual element method for the transmission eigenvalue problem. Math. Models Methods Appl. Sci. 28(14), 2803\u20132831 (2018). https:\/\/doi.org\/10.1142\/S0218202518500616","journal-title":"Math. Models Methods Appl. Sci."},{"issue":"4","key":"10288_CR29","doi-asserted-by":"publisher","first-page":"2425","DOI":"10.1137\/20M1347887","volume":"43","author":"D Mora","year":"2021","unstructured":"Mora, D., Vel\u00e1squez, I.: Virtual elements for the transmission eigenvalue problem on polytopal meshes. SIAM J. Sci. Comput. 43(4), 2425\u20132447 (2021). https:\/\/doi.org\/10.1137\/20M1347887","journal-title":"SIAM J. Sci. Comput."},{"issue":"4","key":"10288_CR30","doi-asserted-by":"publisher","first-page":"1437","DOI":"10.1051\/m2an\/2017041","volume":"52","author":"D Mora","year":"2018","unstructured":"Mora, D., Rivera, G., Vel\u00e1squez, I.: A virtual element method for the vibration problem of kirchhoff plates. ESAIM: M2AN 52(4), 1437\u20131456 (2018). https:\/\/doi.org\/10.1051\/m2an\/2017041","journal-title":"A virtual element method for the vibration problem of kirchhoff plates. ESAIM: M2AN"},{"key":"10288_CR31","doi-asserted-by":"publisher","DOI":"10.1016\/j.cma.2024.116931","volume":"425","author":"C Carstensen","year":"2024","unstructured":"Carstensen, C., Gr\u00e4\u00dfle, B.: Rate-optimal higher-order adaptive conforming fem for biharmonic eigenvalue problems on polygonal domains. Comput. Methods Appl. Mech. Engrg. 425, 116931 (2024). https:\/\/doi.org\/10.1016\/j.cma.2024.116931","journal-title":"Comput. Methods Appl. Mech. Engrg."},{"key":"10288_CR32","doi-asserted-by":"publisher","unstructured":"Feng, J., Wang, S., Bi, H., Yang, Y.: An hp-mixed discontinuous Galerkin method for the biharmonic eigenvalue problem. Appl. Math. Comput. 450(C) (2023). https:\/\/doi.org\/10.1016\/j.amc.2023.127969","DOI":"10.1016\/j.amc.2023.127969"},{"issue":"4","key":"10288_CR33","doi-asserted-by":"publisher","first-page":"1779","DOI":"10.1093\/imanum\/dru054","volume":"35","author":"D Gallistl","year":"2014","unstructured":"Gallistl, D.: Morley finite element method for the eigenvalues of the biharmonic operator 35(4), 1779\u20131811 (2014). https:\/\/doi.org\/10.1093\/imanum\/dru054","journal-title":"Morley finite element method for the eigenvalues of the biharmonic operator"},{"issue":"1","key":"10288_CR34","doi-asserted-by":"publisher","first-page":"55","DOI":"10.1186\/s13660-018-1643-9","volume":"2018","author":"H Li","year":"2018","unstructured":"Li, H., Yang, Y.: Adaptive morley element algorithms for the biharmonic eigenvalue problem. J. Inequal. Appl. 2018(1), 55 (2018). https:\/\/doi.org\/10.1186\/s13660-018-1643-9","journal-title":"J. Inequal. Appl."},{"key":"10288_CR35","doi-asserted-by":"publisher","first-page":"857","DOI":"10.1007\/s00211-017-0891-9","volume":"137","author":"A Cangiani","year":"2017","unstructured":"Cangiani, A., Georgoulis, E.H., Pryer, T., Sutton, O.J.: A posteriori error estimates for the virtual element method. Numer. Math. 137, 857\u2013893 (2017). https:\/\/doi.org\/10.1007\/s00211-017-0891-9","journal-title":"Numer. Math."},{"key":"10288_CR36","unstructured":"Dassi, F., Khot, R., Rubiano, A.E., Ruiz-Baier, R.: A posteriori error analysis of a robust virtual element method for stress-assisted diffusion problems (2025). Available at arXiv:2504.00648"},{"issue":"9","key":"10288_CR37","doi-asserted-by":"publisher","first-page":"2172","DOI":"10.1016\/j.camwa.2017.05.016","volume":"74","author":"D Mora","year":"2017","unstructured":"Mora, D., Rivera, G., Rodriguez, R.: A posteriori error estimates for a virtual element method for the Steklov eigenvalue problem. Comput. Math. Appl. 74(9), 2172\u20132190 (2017). https:\/\/doi.org\/10.1016\/j.camwa.2017.05.016","journal-title":"Comput. Math. Appl."},{"key":"10288_CR38","doi-asserted-by":"publisher","first-page":"182","DOI":"10.1016\/j.camwa.2024.05.011","volume":"166","author":"M Munar","year":"2024","unstructured":"Munar, M., Cangiani, A., Vel\u00e1squez, I.: Residual-based a posteriori error estimation for mixed virtual element methods. Comput. Math. Appl. 166, 182\u2013197 (2024). https:\/\/doi.org\/10.1016\/j.camwa.2024.05.011","journal-title":"Comput. Math. Appl."},{"key":"10288_CR39","doi-asserted-by":"publisher","unstructured":"Chen, M., Huang, J., Lin, S.: A posteriori error estimation for a C$$^1$$ virtual element method of kirchhoff plates 120, 132\u2013150 (2022). https:\/\/doi.org\/10.1016\/j.camwa.2022.05.001","DOI":"10.1016\/j.camwa.2022.05.001"},{"key":"10288_CR40","doi-asserted-by":"publisher","unstructured":"Dassi, F.: VEM++, a C++ library to handle and play with the virtual element method. Numer. Algorithms, 1\u201343 (2025). https:\/\/doi.org\/10.1007\/s11075-025-02059-z","DOI":"10.1007\/s11075-025-02059-z"},{"key":"10288_CR41","volume-title":"Sobolev Spaces","author":"RA Adams","year":"2003","unstructured":"Adams, R.A., Fournier, J.J.F.: Sobolev Spaces, 2nd edn. Elsevier\/Academic Press, Amsterdam, Netherlands (2003)","edition":"2"},{"key":"10288_CR42","doi-asserted-by":"publisher","DOI":"10.1093\/imanum\/dry063","author":"D Mora","year":"2019","unstructured":"Mora, D., Rivera, G.: A priori and a posteriori error estimates for a virtual element spectral analysis for the elasticity equations. IMA J. Numer. Anal. (2019). https:\/\/doi.org\/10.1093\/imanum\/dry063","journal-title":"IMA J. Numer. Anal."},{"key":"10288_CR43","doi-asserted-by":"publisher","unstructured":"Grisvard, P.: Elliptic Problems in Nonsmooth Domains. SIAM, Philadelphia, PA (2011). https:\/\/doi.org\/10.1137\/1.9781611972030","DOI":"10.1137\/1.9781611972030"},{"issue":"9","key":"10288_CR44","doi-asserted-by":"publisher","first-page":"2172","DOI":"10.1016\/j.camwa.2017.05.016","volume":"74","author":"D Mora","year":"2017","unstructured":"Mora, D., Rivera, G., Rodr\u00edguez, R.: A posteriori error estimates for a virtual element method for the Steklov eigenvalue problem. Comput. Math. Appl. 74(9), 2172\u20132190 (2017). https:\/\/doi.org\/10.1016\/j.camwa.2017.05.016","journal-title":"Comput. Math. Appl."},{"key":"10288_CR45","volume-title":"A Review of A Posteriori Error Estimation and Adaptive Mesh-Refinement Techniques","author":"R Verf\u00fcrth","year":"1996","unstructured":"Verf\u00fcrth, R.: A Review of A Posteriori Error Estimation and Adaptive Mesh-Refinement Techniques. Advances in Numerical Mathematics. Wiley-Teubner, Stuttgart (1996)"},{"issue":"342","key":"10288_CR46","doi-asserted-by":"publisher","first-page":"1543","DOI":"10.1090\/mcom\/3828","volume":"92","author":"J Zhao","year":"2023","unstructured":"Zhao, J., Mao, S., Zhang, B., Wang, F.: The interior penalty virtual element method for the biharmonic problem. Math. Comput. 92(342), 1543\u20131574 (2023). https:\/\/doi.org\/10.1090\/mcom\/3828","journal-title":"Math. Comput."},{"issue":"3","key":"10288_CR47","doi-asserted-by":"publisher","first-page":"1273","DOI":"10.1137\/20M1364114","volume":"59","author":"Z Dong","year":"2021","unstructured":"Dong, Z., Mascotto, L., Sutton, O.J.: Residual-based a posteriori error estimates for \\$hp\\$-discontinuous Galerkin discretizations of the biharmonic problem. SIAM J. Numer. Anal. 59(3), 1273\u20131298 (2021). https:\/\/doi.org\/10.1137\/20M1364114","journal-title":"SIAM J. Numer. Anal."},{"key":"10288_CR48","doi-asserted-by":"publisher","unstructured":"Verf\u00fcrth, R.: A Posteriori Error Estimation Techniques for Finite Element Methods. Oxford University Press, Oxford (2013). https:\/\/doi.org\/10.1093\/acprof:oso\/9780199679423.001.0001","DOI":"10.1093\/acprof:oso\/9780199679423.001.0001"},{"key":"10288_CR49","doi-asserted-by":"publisher","unstructured":"Hern\u00e1ndez, V., Rom\u00e1n, J.E., Vidal, V.: SLEPc: Scalable library for eigenvalue problem computations. In: Palma, J.M.L.M., Sousa, A.A., Dongarra, J., Hern\u00e1ndez, V. (eds.), High Performance Computing for Computational Science \u2014 VECPAR 2002, pp. 377\u2013391. Springer, Berlin, Heidelberg (2003). https:\/\/doi.org\/10.1007\/3-540-36569-9_25","DOI":"10.1007\/3-540-36569-9_25"},{"key":"10288_CR50","unstructured":"Yu, Y.: Implementation of polygonal mesh refinement in MATLAB (2021). Available at arXiv:2101.03456"},{"issue":"3","key":"10288_CR51","doi-asserted-by":"publisher","first-page":"1103","DOI":"10.1137\/100791634","volume":"33","author":"C Burstedde","year":"2011","unstructured":"Burstedde, C., Wilcox, L.C., Ghattas, O.: p4est: scalable algorithms for parallel adaptive mesh refinement on forests of octrees. SIAM J. Sci. Comput. 33(3), 1103\u20131133 (2011). https:\/\/doi.org\/10.1137\/100791634","journal-title":"SIAM J. Sci. Comput."},{"issue":"52","key":"10288_CR52","doi-asserted-by":"publisher","first-page":"2520","DOI":"10.21105\/joss.02520","volume":"5","author":"S Badia","year":"2020","unstructured":"Badia, S., Verdugo, F.: Gridap: an extensible finite element toolbox in Julia. J. Open Source Softw. 5(52), 2520 (2020). https:\/\/doi.org\/10.21105\/joss.02520","journal-title":"J. Open Source Softw."},{"issue":"2","key":"10288_CR53","doi-asserted-by":"publisher","first-page":"97","DOI":"10.1007\/s006070050053","volume":"63","author":"PE Bj\u00f8rstad","year":"1999","unstructured":"Bj\u00f8rstad, P.E., Tj\u00f8stheim, B.P.: High precision solutions of two fourth order eigenvalue problems. Computing 63(2), 97\u2013107 (1999). https:\/\/doi.org\/10.1007\/s006070050053","journal-title":"Computing"},{"issue":"2","key":"10288_CR54","doi-asserted-by":"publisher","first-page":"3332","DOI":"10.3934\/math.2024163","volume":"9","author":"J Feng","year":"2024","unstructured":"Feng, J., Wang, S., Bi, H., Yang, Y.: The a posteriori error estimates of the Ciarlet-Raviart mixed finite element method for the biharmonic eigenvalue problem. AIMS Mathematics 9(2), 3332\u20133348 (2024). https:\/\/doi.org\/10.3934\/math.2024163","journal-title":"AIMS Mathematics"},{"issue":"5","key":"10288_CR55","doi-asserted-by":"publisher","first-page":"1623","DOI":"10.1002\/num.21964","volume":"31","author":"J Hu","year":"2015","unstructured":"Hu, J., Yang, X.: Lower bounds of eigenvalues of the biharmonic operators by the rectangular Morley element methods. Numer. Methods Partial Differential Equations 31(5), 1623\u20131644 (2015). https:\/\/doi.org\/10.1002\/num.21964","journal-title":"Numer. Methods Partial Differential Equations"}],"container-title":["Advances in Computational Mathematics"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s10444-026-10288-6.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/article\/10.1007\/s10444-026-10288-6","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s10444-026-10288-6.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,5,13]],"date-time":"2026-05-13T22:03:04Z","timestamp":1778709784000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/s10444-026-10288-6"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2026,2,23]]},"references-count":55,"journal-issue":{"issue":"2","published-print":{"date-parts":[[2026,4]]}},"alternative-id":["10288"],"URL":"https:\/\/doi.org\/10.1007\/s10444-026-10288-6","relation":{},"ISSN":["1019-7168","1572-9044"],"issn-type":[{"value":"1019-7168","type":"print"},{"value":"1572-9044","type":"electronic"}],"subject":[],"published":{"date-parts":[[2026,2,23]]},"assertion":[{"value":"31 July 2025","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"29 January 2026","order":2,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"23 February 2026","order":3,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}},{"order":1,"name":"Ethics","group":{"name":"EthicsHeading","label":"Declarations"}},{"value":"The authors declare no competing interests.","order":2,"name":"Ethics","group":{"name":"EthicsHeading","label":"Conflict of interest"}}],"article-number":"17"}}