Document Zbl 07780864

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Algorithms</a> <a href="/?q=in%3A506373" title="Articles in this Issue">94, No. 4, 1731-1761 (2023)</a>. </div> <div class="abstract">Summary: In this work, the virtual element method (VEM) on convex polygonal meshes for the nonlinear Sobolev equations is developed, where the semi-discrete and fully discrete formulations are presented and analyzed. To overcome the complexity of nonlinear terms, the nonlinear coefficient is approximated by employing the orthogonal $\boldsymbol{L^2}$ projection operator, which is directly computable from the degrees of freedom. Under some assumptions about the nonlinear coefficient, the existence and uniqueness of the semi-discrete solution are analyzed. Furthermore, a priori error estimate showing optimal order of convergence with respect to the $\boldsymbol{H^1}$ semi-norm was derived. Finally, some numerical experiments are conducted to illustrate the theoretical convergence rate.</div> <div class="clear"></div> <br> <div class="citations"><div class="clear"><a href="/?q=rf%3A7780864">Cited in <strong>5</strong> Documents</a></div></div> <div class="classification"> <h3>MSC:</h3> <table><tr> <td> <a class="mono" href="/classification/?q=cc%3A65-XX" title="MSC2020">65-XX</a> </td> <td class="space"> Numerical analysis </td> </tr></table> </div><div class="keywords"> <h3>Keywords:</h3><a href="/?q=ut%3Avirtual+element+method">virtual element method</a>; <a href="/?q=ut%3Anonlinear+Sobolev+equations">nonlinear Sobolev equations</a>; <a href="/?q=ut%3Aconvergence+analysis">convergence analysis</a>; <a href="/?q=ut%3Apolygonal+meshes">polygonal meshes</a></div> <div class="software"> <h3>Software:</h3><a href="/software/41825">mVEM</a>; <a href="/software/23170">GitHub</a></div>  <div class="modal fade" id="metadataModal" tabindex="-1" role="dialog" aria-labelledby="myModalLabel"> <div class="modal-dialog" role="document"> <div class="modal-content"> <div class="modal-header"> <button type="button" class="close" data-dismiss="modal" aria-label="Close"><span aria-hidden="true">×</span></button> <h4 class="modal-title" id="myModalLabel">Cite</h4> </div> <div class="modal-body"> <div class="form-group"> <label for="select-output" class="control-label">Format</label> <select id="select-output" class="form-control" aria-label="Select Metadata format"></select> </div> <div class="form-group"> <label for="metadataText" class="control-label">Result</label> <textarea class="form-control" id="metadataText" rows="10" style="min-width: 100%;max-width: 100%"></textarea> </div> <div id="metadata-alert" class="alert alert-danger" role="alert" style="display: none;">  </div> </div> <div class="modal-footer"> <button type="button" class="btn btn-primary" onclick="copyMetadata()">Copy to clipboard</button> <button type="button" class="btn btn-default" data-dismiss="modal">Close</button> </div> </div> </div> </div> <div class="functions clearfix"> <div class="function">  <a type="button" class="btn btn-default btn-xs pdf" data-toggle="modal" data-target="#metadataModal" data-itemtype="Zbl" data-itemname="Zbl 07780864" data-ciurl="/ci/07780864" data-biburl="/bibtex/07780864.bib" data-amsurl="/amsrefs/07780864.bib" data-xmlurl="/xml/07780864.xml" > Cite </a> <a class="btn btn-default btn-xs pdf" data-container="body" type="button" href="/pdf/07780864.pdf" title="Zbl 07780864 as PDF">Review PDF</a> </div> <div class="fulltexts"> <span class="fulltext">Full Text:</span> <a class="btn btn-default btn-xs" type="button" href="https://doi.org/10.1007/s11075-023-01553-6" aria-label="DOI for “Virtual element method for nonlinear Sobolev equation on polygonal meshes”" title="10.1007/s11075-023-01553-6">DOI</a> </div> <div class="sfx" style="float: right;"> </div> </div> <div class="references"> <h3>References:</h3> <table><tr> <td>[1]</td> <td class="space">Barenblatt, G., Basic concepts in the theory of seepage of homogeneous liquids in fissured rocks, Prikl. Mat. Mekh., 24, 5, 852-864 (1960) · <a href="/0104.21702" class="nowrap">Zbl 0104.21702</a></td> </tr><tr> <td>[2]</td> <td class="space">Ting, TW, A cooling process according to two-temperature theory of heat conduction, J. Math. Anal. Appl., 45, 1, 23-31 (1974) · <a href="/0272.35039" class="nowrap">Zbl 0272.35039</a> · <a href="https://doi.org/10.1016/0022-247X(74)90116-4" class="nowrap">doi:10.1016/0022-247X(74)90116-4</a></td> </tr><tr> <td>[3]</td> <td class="space">Ting, TW, Certain non-steady flows of second-order fluids, Archive for Rational Mechanics and Analysis, 14, 1, 1-26 (1963) · <a href="/0139.20105" class="nowrap">Zbl 0139.20105</a> · <a href="https://doi.org/10.1007/BF00250690" class="nowrap">doi:10.1007/BF00250690</a></td> </tr><tr> <td>[4]</td> <td class="space">Davis, PL, A quasilinear parabolic and a related third order problem, J. Math. Anal. Appl., 40, 2, 327-335 (1972) · <a href="/0261.35049" class="nowrap">Zbl 0261.35049</a> · <a href="https://doi.org/10.1016/0022-247X(72)90054-6" class="nowrap">doi:10.1016/0022-247X(72)90054-6</a></td> </tr><tr> <td>[5]</td> <td class="space">Ewing, RE, Time-stepping galerkin methods for nonlinear sobolev partial differential equations, SIAM Journal on Numerical Analysis, 15, 6, 1125-1150 (1978) · <a href="/0399.65083" class="nowrap">Zbl 0399.65083</a> · <a href="https://doi.org/10.1137/0715075" class="nowrap">doi:10.1137/0715075</a></td> </tr><tr> <td>[6]</td> <td class="space">Showalter, R., Existence and representation theorems for a semilinear sobolev equation in banach space, SIAM Journal on Mathematical Analysis, 3, 3, 527-543 (1972) · <a href="/0262.34047" class="nowrap">Zbl 0262.34047</a> · <a href="https://doi.org/10.1137/0503051" class="nowrap">doi:10.1137/0503051</a></td> </tr><tr> <td>[7]</td> <td class="space">Lin, Y., Galerkin methods for nonlinear Sobolev equations, Aequationes Mathematicae, 40, 1, 54-66 (1990) · <a href="/0718.65067" class="nowrap">Zbl 0718.65067</a> · <a href="https://doi.org/10.1007/BF02112280" class="nowrap">doi:10.1007/BF02112280</a></td> </tr><tr> <td>[8]</td> <td class="space">Nakao, MT, Error estimates of a galerkin method for some nonlinear Sobolev equations in one space dimension, Numerische Mathematik, 47, 1, 139-157 (1985) · <a href="/0575.65112" class="nowrap">Zbl 0575.65112</a> · <a href="https://doi.org/10.1007/BF01389881" class="nowrap">doi:10.1007/BF01389881</a></td> </tr><tr> <td>[9]</td> <td class="space">Lin, Y.; Zhang, T., Finite element methods for nonlinear Sobolev equations with nonlinear boundary conditions, Journal of Mathematical Analysis and Applications, 165, 1, 180-191 (1992) · <a href="/0753.65080" class="nowrap">Zbl 0753.65080</a> · <a href="https://doi.org/10.1016/0022-247X(92)90074-N" class="nowrap">doi:10.1016/0022-247X(92)90074-N</a></td> </tr><tr> <td>[10]</td> <td class="space">Gu, H., Characteristic finite element methods for nonlinear Sobolev equations, Appl. Math. Comput., 102, 1, 51-62 (1999) · <a href="/0933.65111" class="nowrap">Zbl 0933.65111</a></td> </tr><tr> <td>[11]</td> <td class="space">Chen, C.; Li, K.; Chen, Y.; Huang, Y., Two-grid finite element methods combined with Crank-Nicolson scheme for nonlinear Sobolev equations, Adv. Comput. Math., 45, 2, 611-630 (2019) · <a href="/1415.65223" class="nowrap">Zbl 1415.65223</a> · <a href="https://doi.org/10.1007/s10444-018-9628-2" class="nowrap">doi:10.1007/s10444-018-9628-2</a></td> </tr><tr> <td>[12]</td> <td class="space">Dongyang, S.; Fengna, Y.; Junjun, W., Unconditional superconvergence analysis of a new mixed finite element method for nonlinear Sobolev equation, Appl. Math. Comput., 274, 182-194 (2016) · <a href="/1410.65368" class="nowrap">Zbl 1410.65368</a></td> </tr><tr> <td>[13]</td> <td class="space">Gao, F.; Rui, H., A split least-squares characteristic mixed finite element method for Sobolev equations with convection term, Math. Comput. Simulation, 80, 2, 341-351 (2009) · <a href="/1180.65126" class="nowrap">Zbl 1180.65126</a> · <a href="https://doi.org/10.1016/j.matcom.2009.07.003" class="nowrap">doi:10.1016/j.matcom.2009.07.003</a></td> </tr><tr> <td>[14]</td> <td class="space">Shi, D.; Tang, Q.; Gong, W., A low order characteristic-nonconforming finite element method for nonlinear Sobolev equation with convection-dominated term, Math. Comput. Simulation, 114, 25-36 (2015) · <a href="/1540.65389" class="nowrap">Zbl 1540.65389</a> · <a href="https://doi.org/10.1016/j.matcom.2014.03.008" class="nowrap">doi:10.1016/j.matcom.2014.03.008</a></td> </tr><tr> <td>[15]</td> <td class="space">Wang, J.; Li, Q., Superconvergence analysis of a linearized three-step backward differential formula finite element method for nonlinear Sobolev equation, Math. Methods Appl. Sci., 42, 9, 3359-3376 (2019) · <a href="/1418.65185" class="nowrap">Zbl 1418.65185</a> · <a href="https://doi.org/10.1002/mma.5589" class="nowrap">doi:10.1002/mma.5589</a></td> </tr><tr> <td>[16]</td> <td class="space">Ohm, M-R; Lee, H-Y, L 2-error analysis of fully discrete discontinuous Galerkin approximations for nonlinear Sobolev equations, Bulletin of the Korean Mathematical Society, 48, 5, 897-915 (2011) · <a href="/1227.65084" class="nowrap">Zbl 1227.65084</a> · <a href="https://doi.org/10.4134/BKMS.2011.48.5.897" class="nowrap">doi:10.4134/BKMS.2011.48.5.897</a></td> </tr><tr> <td>[17]</td> <td class="space">Sun, T.; Yang, D., A priori error estimates for interior penalty discontinuous Galerkin method applied to nonlinear Sobolev equations, Appl. Math. Comput., 200, 1, 147-159 (2008) · <a href="/1148.65069" class="nowrap">Zbl 1148.65069</a></td> </tr><tr> <td>[18]</td> <td class="space">Xie, C-M; Feng, M-F; Luo, Y.; Zhang, L., A hybrid high-order method for Sobolev equation with convection-dominated term, Computers & Mathematics with Applications, 118, 85-94 (2022) · <a href="/1524.65614" class="nowrap">Zbl 1524.65614</a> · <a href="https://doi.org/10.1016/j.camwa.2022.04.017" class="nowrap">doi:10.1016/j.camwa.2022.04.017</a></td> </tr><tr> <td>[19]</td> <td class="space">Beirão da Veiga, L., Brezzi, F., Cangiani, A., Manzini, G., Marini, L.D., Russo, A.: Basic principles of virtual element methods. Mathematical Models and Methods in Applied Sciences 23(01), 199-214 (2013) · <a href="/1416.65433" class="nowrap">Zbl 1416.65433</a></td> </tr><tr> <td>[20]</td> <td class="space">Da Veiga, LB; Brezzi, F.; Marini, LD, Virtual elements for linear elasticity problems, SIAM Journal on Numerical Analysis, 51, 2, 794-812 (2013) · <a href="/1268.74010" class="nowrap">Zbl 1268.74010</a> · <a href="https://doi.org/10.1137/120874746" class="nowrap">doi:10.1137/120874746</a></td> </tr><tr> <td>[21]</td> <td class="space">Gain, AL; Talischi, C.; Paulino, GH, On the virtual element method for three-dimensional linear elasticity problems on arbitrary polyhedral meshes, Computer Methods in Applied Mechanics and Engineering, 282, 132-160 (2014) · <a href="/1423.74095" class="nowrap">Zbl 1423.74095</a> · <a href="https://doi.org/10.1016/j.cma.2014.05.005" class="nowrap">doi:10.1016/j.cma.2014.05.005</a></td> </tr><tr> <td>[22]</td> <td class="space">Antonietti, PF; Da Veiga, LB; Mora, D.; Verani, M., A stream virtual element formulation of the stokes problem on polygonal meshes, SIAM Journal on Numerical Analysis, 52, 1, 386-404 (2014) · <a href="/1427.76198" class="nowrap">Zbl 1427.76198</a> · <a href="https://doi.org/10.1137/13091141X" class="nowrap">doi:10.1137/13091141X</a></td> </tr><tr> <td>[23]</td> <td class="space">da Veiga, LB; Lovadina, C.; Vacca, G., Divergence free virtual elements for the stokes problem on polygonal meshes, ESAIM: Mathematical Modelling and Numerical Analysis, 51, 2, 509-535 (2017) · <a href="/1398.76094" class="nowrap">Zbl 1398.76094</a> · <a href="https://doi.org/10.1051/m2an/2016032" class="nowrap">doi:10.1051/m2an/2016032</a></td> </tr><tr> <td>[24]</td> <td class="space">Da Veiga, LB; Lovadina, C.; Mora, D., A virtual element method for elastic and inelastic problems on polytope meshes, Comput. Methods Appl. Mechanics Eng., 295, 327-346 (2015) · <a href="/1423.74120" class="nowrap">Zbl 1423.74120</a> · <a href="https://doi.org/10.1016/j.cma.2015.07.013" class="nowrap">doi:10.1016/j.cma.2015.07.013</a></td> </tr><tr> <td>[25]</td> <td class="space">Chi, H.; Da Veiga, LB; Paulino, G., Some basic formulations of the virtual element method (vem) for finite deformations, Comput. Methods Appl. Mechanics Eng., 318, 148-192 (2017) · <a href="/1439.74397" class="nowrap">Zbl 1439.74397</a> · <a href="https://doi.org/10.1016/j.cma.2016.12.020" class="nowrap">doi:10.1016/j.cma.2016.12.020</a></td> </tr><tr> <td>[26]</td> <td class="space">Da Veiga, LB; Brezzi, F.; Dassi, F.; Marini, LD; Russo, A., Virtual element approximation of 2d magnetostatic problems, Comput. Methods Appl. Mechanics Eng., 327, 173-195 (2017) · <a href="/1439.78017" class="nowrap">Zbl 1439.78017</a> · <a href="https://doi.org/10.1016/j.cma.2017.08.013" class="nowrap">doi:10.1016/j.cma.2017.08.013</a></td> </tr><tr> <td>[27]</td> <td class="space">da Veiga, LB; Brezzi, F.; Dassi, F.; Marini, L.; Russo, A., Lowest order virtual element approximation of magnetostatic problems, Comput. Methods Appl. Mechanics Eng., 332, 343-362 (2018) · <a href="/1440.65181" class="nowrap">Zbl 1440.65181</a> · <a href="https://doi.org/10.1016/j.cma.2017.12.028" class="nowrap">doi:10.1016/j.cma.2017.12.028</a></td> </tr><tr> <td>[28]</td> <td class="space">da Veiga, LB; Dassi, F.; Manzini, G.; Mascotto, L., Virtual elements for Maxwell’s equations, Computers & Mathematics with Applications, 116, 82-99 (2022) · <a href="/1493.65191" class="nowrap">Zbl 1493.65191</a> · <a href="https://doi.org/10.1016/j.camwa.2021.08.019" class="nowrap">doi:10.1016/j.camwa.2021.08.019</a></td> </tr><tr> <td>[29]</td> <td class="space">Wriggers, P.; Rust, WT; Reddy, B., A virtual element method for contact, Computational Mechanics, 58, 6, 1039-1050 (2016) · <a href="/1398.74420" class="nowrap">Zbl 1398.74420</a> · <a href="https://doi.org/10.1007/s00466-016-1331-x" class="nowrap">doi:10.1007/s00466-016-1331-x</a></td> </tr><tr> <td>[30]</td> <td class="space">Vacca, G., Beirão da Veiga, L.: Virtual element methods for parabolic problems on polygonal meshes. Numerical Methods for Partial Differential Equations 31(6), 2110-2134 (2015) · <a href="/1336.65171" class="nowrap">Zbl 1336.65171</a></td> </tr><tr> <td>[31]</td> <td class="space">Vacca, G., Virtual element methods for hyperbolic problems on polygonal meshes, Computers & Mathematics with Applications, 74, 5, 882-898 (2017) · <a href="/1448.65177" class="nowrap">Zbl 1448.65177</a> · <a href="https://doi.org/10.1016/j.camwa.2016.04.029" class="nowrap">doi:10.1016/j.camwa.2016.04.029</a></td> </tr><tr> <td>[32]</td> <td class="space">Zhang, B., Zhao, J., Chen, S.: Virtual element method for the sobolev equations. Mathematical Methods in the Applied Sciences (2022)</td> </tr><tr> <td>[33]</td> <td class="space">Adak, D.; Natarajan, S., Virtual element method for a nonlocal elliptic problem of Kirchhoff type on polygonal meshes, Computers & Mathematics with Applications, 79, 10, 2856-2871 (2020) · <a href="/1447.65126" class="nowrap">Zbl 1447.65126</a> · <a href="https://doi.org/10.1016/j.camwa.2019.12.018" class="nowrap">doi:10.1016/j.camwa.2019.12.018</a></td> </tr><tr> <td>[34]</td> <td class="space">Da Veiga, LB; Lovadina, C.; Vacca, G., Virtual elements for the Navier-Stokes problem on polygonal meshes, SIAM Journal on Numerical Analysis, 56, 3, 1210-1242 (2018) · <a href="/1397.65302" class="nowrap">Zbl 1397.65302</a> · <a href="https://doi.org/10.1137/17M1132811" class="nowrap">doi:10.1137/17M1132811</a></td> </tr><tr> <td>[35]</td> <td class="space">Cangiani, A.; Chatzipantelidis, P.; Diwan, G.; Georgoulis, EH, Virtual element method for quasilinear elliptic problems, IMA Journal of Numerical Analysis, 40, 4, 2450-2472 (2020) · <a href="/1466.65182" class="nowrap">Zbl 1466.65182</a> · <a href="https://doi.org/10.1093/imanum/drz035" class="nowrap">doi:10.1093/imanum/drz035</a></td> </tr><tr> <td>[36]</td> <td class="space">Adak, D.; Natarajan, E.; Kumar, S., Convergence analysis of virtual element methods for semilinear parabolic problems on polygonal meshes, Numerical Methods for Partial Differential Equations, 35, 1, 222-245 (2019) · <a href="/1419.65040" class="nowrap">Zbl 1419.65040</a> · <a href="https://doi.org/10.1002/num.22298" class="nowrap">doi:10.1002/num.22298</a></td> </tr><tr> <td>[37]</td> <td class="space">Adak, D.; Natarajan, E.; Kumar, S., Virtual element method for semilinear hyperbolic problems on polygonal meshes, Int. J. Comput. Math., 96, 5, 971-991 (2019) · <a href="/1499.65478" class="nowrap">Zbl 1499.65478</a> · <a href="https://doi.org/10.1080/00207160.2018.1475651" class="nowrap">doi:10.1080/00207160.2018.1475651</a></td> </tr><tr> <td>[38]</td> <td class="space">Cangiani, A., Manzini, G., Sutton, O.J.: Conforming and nonconforming virtual element methods for elliptic problems. IMA Journal of Numerical Analysis 37(3), 1317-1354 (2017) · <a href="/1433.65282" class="nowrap">Zbl 1433.65282</a></td> </tr><tr> <td>[39]</td> <td class="space">Ahmad, B.; Alsaedi, A.; Brezzi, F.; Marini, LD; Russo, A., Equivalent projectors for virtual element methods, Comput. Math. Appl., 66, 3, 376-391 (2013) · <a href="/1347.65172" class="nowrap">Zbl 1347.65172</a> · <a href="https://doi.org/10.1016/j.camwa.2013.05.015" class="nowrap">doi:10.1016/j.camwa.2013.05.015</a></td> </tr><tr> <td>[40]</td> <td class="space">Chen, Y., Hu, H.: Two-grid method for miscible displacement problem with dispersion by finite element method of characteristics. ZAMM-Journal of Applied Mathematics and Mechanics/Zeitschrift für Angewandte Mathematik und Mechanik 101(3), 201900275 (2021) · <a href="/07809787" class="nowrap">Zbl 07809787</a></td> </tr><tr> <td>[41]</td> <td class="space">Yu, Y.: mVEM: MATLAB Programming for Virtual Element Methods (2019-2022). https://github.com/Terenceyuyue/mVEM</td> </tr></table> <div class="reference_disclaimer"> This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. 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