Bulk-surface virtual element method for systems of PDEs in two-space dimensions. (English) Zbl 1459.65182
In this paper, a coupled bulk-surface PDE in two space dimensions is solved with a bulk-surface virtual element method (BSVEM). The BSPDE consists of a PDE in the bulk coupled to another PDE on the surface through general nonlinear boundary conditions. Linear elliptic and semilinear parabolic coupled BSPDE problems are approximated numerically with the BSVEM. The proposed method extends the BSFEM for BSRDSs [A. Madzvamuse and A. H. W. Chung, “The bulk-surface finite element method for reaction-diffusion systems on stationary volumes”, Finite Elem. Anal. Des. 108, 9–21 (2016; doi:10.1016/j.finel.2015.09.002)] and the VEM for linear elliptic [B. Da Veiga et al., Math. Models Methods Appl. Sci. 23, No. 1, 199–214 (2013; Zbl 1416.65433)] and semilinear bulk parabolic problems on polygonal meshes [D. Adak et al., Numer. Methods Partial Differ. Equations 35, No. 1, 222–245 (2019; Zbl 1419.65040)].
Polygonal bulk surface meshes in two space dimensions produce geometric errors of order \({\mathcal O}(h)\) in the bulk and \({\mathcal O}(h^2)\) on the surface, with the mesh size \(h\). It was shown that suitable polygonal meshes reduce the asymptotic computational complexity of matrix assembly. Numerical examples validate the convergence rate in space and time for both the elliptic and the parabolic case and demonstrate the computational advantages using polygonal meshes.
Polygonal bulk surface meshes in two space dimensions produce geometric errors of order \({\mathcal O}(h)\) in the bulk and \({\mathcal O}(h^2)\) on the surface, with the mesh size \(h\). It was shown that suitable polygonal meshes reduce the asymptotic computational complexity of matrix assembly. Numerical examples validate the convergence rate in space and time for both the elliptic and the parabolic case and demonstrate the computational advantages using polygonal meshes.
Reviewer: Bülent Karasözen (Ankara)
MSC:
| 65M60 | Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs |
| 65N30 | Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs |
| 65M12 | Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs |
| 65M15 | Error bounds for initial value and initial-boundary value problems involving PDEs |
| 65M20 | Method of lines for initial value and initial-boundary value problems involving PDEs |
Keywords:
bulk-surface PDEs; bulk-surface finite elements; bulk-surface virtual elements; bulk-surface reaction-diffusion systems (BSRDS); virtual elementsReferences:
| [1] | Adak, D.; Natarajan, E.; Kumar, S., Convergence analysis of virtual element methods for semilinear parabolic problems on polygonal meshes, Numer. Methods PDEs, 35, 1, 222-245 (2019) · Zbl 1419.65040 · doi:10.1002/num.22298 |
| [2] | Adams, RA; Fournier, JF, Sobolev Spaces (2003), Amsterdam: Elsevier, Amsterdam · Zbl 1098.46001 |
| [3] | Ahmad, B.; Alsaedi, A.; Brezzi, F.; Marini, LD; Russo, A., Equivalent projectors for virtual element methods, CAMWA, 66, 3, 376-391 (2013) · Zbl 1347.65172 · doi:10.1016/j.camwa.2013.05.015 |
| [4] | Antonietti, PF; Beirão Da Veiga, L.; Scacchi, S.; Verani, M., A \(C^1\) virtual element method for the Cahn-Hilliard equation with polygonal meshes, SIAM J. Numer. Anal., 54, 1, 34-56 (2016) · Zbl 1336.65160 · doi:10.1137/15m1008117 |
| [5] | Beirão Da Veiga, L.; Brezzi, F.; Cangiani, A.; Manzini, G.; Marini, LD; Russo, A., Basic principles of virtual element methods, Math. Models Methods Appl. Sci., 23, 1, 199-214 (2013) · Zbl 1416.65433 · doi:10.1051/m2an/2013138 |
| [6] | Beirão Da Veiga, L.; Brezzi, F.; Marini, LD, Virtual elements for linear elasticity problems, SIAM J. Numer. Anal., 51, 2, 794-812 (2013) · Zbl 1268.74010 · doi:10.1137/120874746 |
| [7] | Beirão Da Veiga, L.; Dassi, F.; Russo, A., High-order virtual element method on polyhedral meshes, CAMWA, 74, 5, 1110-1122 (2017) · Zbl 1448.65215 · doi:10.1016/j.camwa.2017.03.021 |
| [8] | Beirão Da Veiga, L.; Lovadina, C.; Vacca, G., Divergence free virtual elements for the stokes problem on polygonal meshes, ESAIM M2AN, 51, 2, 509-535 (2017) · Zbl 1398.76094 · doi:10.1051/m2an/2016032 |
| [9] | Beirão Da Veiga, L.; Manzini, G., A virtual element method with arbitrary regularity, IMA J. Numer. Anal. (2013) · Zbl 1293.65146 · doi:10.1093/imanum/drt018 |
| [10] | Beirão Da Veiga, L.; Russo, A.; Vacca, G., The virtual element method with curved edges, ESAIM M2AN, 53, 2, 375-404 (2019) · Zbl 1426.65163 · doi:10.1051/m2an/2018052 |
| [11] | Benedetto, MF; Berrone, S.; Scialò, S., A globally conforming method for solving flow in discrete fracture networks using the virtual element method, Finite Elem. Anal. Des., 109, 23-36 (2016) · doi:10.1016/j.finel.2015.10.003 |
| [12] | Benkemoun, N.; Ibrahimbegovic, A.; Colliat, JB, Anisotropic constitutive model of plasticity capable of accounting for details of meso-structure of two-phase composite material, Comput. Struct., 90, 153-162 (2012) · doi:10.1016/j.compstruc.2011.09.003 |
| [13] | Berrone, S.; Borio, A., Orthogonal polynomials in badly shaped polygonal elements for the virtual element method, Finite Elem. Anal. Des., 129, 14-31 (2017) · doi:10.1016/j.finel.2017.01.006 |
| [14] | Bertoluzza, S.; Pennacchio, M.; Prada, D., High order VEM on curved domains, Rend. Lincei Mat. Appl., 30, 391-412 (2019) · Zbl 1422.65370 · doi:10.4171/RLM/853 |
| [15] | Bianco, S.; Tewes, F.; Tajber, L.; Caron, V.; Corrigan, OI; Healy, AM, Bulk, surface properties and water uptake mechanisms of salt/acid amorphous composite systems, Int. J. Pharm., 456, 1, 143-152 (2013) · doi:10.1016/j.ijpharm.2013.07.076 |
| [16] | Brezzi, F.; Marini, LD, Virtual element methods for plate bending problems, Comput. Methods Appl. Mech. Eng., 253, 455-462 (2013) · Zbl 1297.74049 · doi:10.1016/j.cma.2012.09.012 |
| [17] | Burman, E.; Hansbo, P.; Larson, M.; Zahedi, S., Cut finite element methods for coupled bulk-surface problems, Numer. Math., 133, 2, 203-231 (2016) · Zbl 1341.65044 · doi:10.1007/s00211-015-0744-3 |
| [18] | Cangiani, A.; Georgoulis, EH; Metcalfe, S., Adaptive discontinuous Galerkin methods for nonstationary convection-diffusion problems, IMA J. Numer. Anal., 34, 4, 1578-1597 (2014) · Zbl 1310.65122 · doi:10.1093/imanum/drt052 |
| [19] | Chen, J., A memory efficient discontinuous Galerkin finite-element time-domain scheme for simulations of finite periodic structures, Microw. Opt. Technol. Lett., 56, 8, 1929-1933 (2014) · doi:10.1002/mop.28483 |
| [20] | Chernyshenko, AY; Olshanskii, MA; Vassilevski, YV, A hybrid finite volume-finite element method for bulk-surface coupled problems, J. Comput. Phys., 352, 516-533 (2018) · Zbl 1375.76098 · doi:10.1016/j.jcp.2017.09.064 |
| [21] | Ciarlet, PG, The Finite Element Method for Elliptic Problems (2002), Philadelphia: Society for Industrial and Applied Mathematics, Philadelphia · Zbl 0999.65129 · doi:10.1137/1.9780898719208 |
| [22] | Cusseddu, D.; Edelstein-Keshet, L.; Mackenzie, JA; Portet, S.; Madzvamuse, A., A coupled bulk-surface model for cell polarisation, J. Theor. Biol., 481, 119-135 (2019) · Zbl 1422.92035 · doi:10.1016/j.jtbi.2018.09.008 |
| [23] | Dai, K.; Liu, G.; Nguyen, T., An n-sided polygonal smoothed finite element method (nSFEM) for solid mechanics, Finite Elem. Anal. Des., 43, 11, 847-860 (2007) · doi:10.1016/j.finel.2007.05.009 |
| [24] | Demlow, A., Higher-order finite element methods and pointwise error estimates for elliptic problems on surfaces, SIAM J. Numer. Anal., 47, 2, 805-827 (2009) · Zbl 1195.65168 · doi:10.1137/070708135 |
| [25] | Di Nezza, E.; Palatucci, G.; Valdinoci, E., Hitchhiker’s guide to the fractional Sobolev spaces, Bull. Sci. Math., 136, 5, 521-573 (2012) · Zbl 1252.46023 · doi:10.1016/j.bulsci.2011.12.004 |
| [26] | Dziuk, G.; Elliott, CM, Finite element methods for surface PDEs, Acta Numer., 22, 289-396 (2013) · Zbl 1296.65156 · doi:10.1017/s0962492913000056 |
| [27] | Elliott, CM; Ranner, T., Finite element analysis for a coupled bulk-surface partial differential equation, IMA J. Numer. Anal., 33, 2, 377-402 (2013) · Zbl 1271.65138 · doi:10.1093/imanum/drs022 |
| [28] | Elliott, CM; Ranner, T., Evolving surface finite element method for the Cahn-Hilliard equation, Numer. Math., 129, 3, 483-534 (2014) · Zbl 1312.65159 · doi:10.1007/s00211-014-0644-y |
| [29] | Elliott, CM; Ranner, T.; Venkataraman, C., Coupled bulk-surface free boundary problems arising from a mathematical model of receptor-ligand dynamics, SIAM J. Math. Anal., 49, 1, 360-397 (2017) · Zbl 1366.35244 · doi:10.1137/15m1050811 |
| [30] | Frittelli, M.; Madzvamuse, A.; Sgura, I.; Venkataraman, C., Preserving invariance properties of reaction-diffusion systems on stationary surfaces, IMA J. Numer. Anal., 39, 1, 235-270 (2017) · Zbl 1483.65155 · doi:10.1093/imanum/drx058 |
| [31] | Frittelli, M.; Sgura, I., Virtual element method for the Laplace-Beltrami equation on surfaces, ESAIM M2AN, 52, 3, 965-993 (2018) · Zbl 1456.65160 · doi:10.1051/m2an/2017040 |
| [32] | Gross, S.; Olshanskii, MA; Reusken, A., A trace finite element method for a class of coupled bulk-interface transport problems, ESAIM M2AN, 49, 5, 1303-1330 (2015) · Zbl 1329.76171 · doi:10.1051/m2an/2015013 |
| [33] | Kovács, B.; Lubich, C., Numerical analysis of parabolic problems with dynamic boundary conditions, IMA J. Numer. Anal., 37, 1, 1-39 (2017) · Zbl 1433.65216 · doi:10.1093/imanum/drw015 |
| [34] | Lee, AA; Münch, A.; Süli, E., Degenerate mobilities in phase field models are insufficient to capture surface diffusion, Appl. Phys. Lett., 107, 8, 081603 (2015) · doi:10.1063/1.4929696 |
| [35] | MacDonald, G.; Mackenzie, JA; Nolan, M.; Insall, RH, A computational method for the coupled solution of reaction-diffusion equations on evolving domains and manifolds: application to a model of cell migration and chemotaxis, J. Comput. Phys., 309, 207-226 (2016) · Zbl 1352.65359 · doi:10.1016/j.jcp.2015.12.038 |
| [36] | Mackenzie, JA; Nolan, M.; Insall, RH, Local modulation of chemoattractant concentrations by single cells: dissection using a bulk-surface computational model, Interface Focus, 6, 5, 20160036 (2016) · doi:10.1098/rsfs.2016.0036 |
| [37] | Madzvamuse, A.; Chung, AHW, The bulk-surface finite element method for reaction-diffusion systems on stationary volumes, Finite Elem. Anal. Des., 108, 9-21 (2016) · doi:10.1016/j.finel.2015.09.002 |
| [38] | Madzvamuse, A.; Chung, AHW; Venkataraman, C., Stability analysis and simulations of coupled bulk-surface reaction-diffusion systems, Proc. R. Soc. A Math. Phys. Eng. Sci., 471, 2175, 20140546 (2015) · Zbl 1371.35147 · doi:10.1098/rspa.2014.0546 |
| [39] | Mora, D.; Rivera, G.; Rodríguez, R., A virtual element method for the Steklov eigenvalue problem, Math. Models Methods Appl. Sci., 25, 8, 1421-1445 (2015) · Zbl 1330.65172 · doi:10.1142/s0218202515500372 |
| [40] | Paquin-Lefebvre, F.; Nagata, W.; Ward, MJ, Pattern formation and oscillatory dynamics in a two-dimensional coupled bulk-surface reaction-diffusion system, SIAM J. Appl. Dyn. Syst., 18, 3, 1334-1390 (2019) · Zbl 1428.37071 · doi:10.1137/18m1213737 |
| [41] | Rätz, A.; Röger, M., Symmetry breaking in a bulk-surface reaction-diffusion model for signalling networks, Nonlinearity, 27, 8, 1805-1827 (2014) · Zbl 1301.92025 · doi:10.1088/0951-7715/27/8/1805 |
| [42] | Savaré, G., Regularity results for elliptic equations in Lipschitz domains, J. Funct. Anal., 152, 1, 176-201 (1998) · Zbl 0889.35018 · doi:10.1006/jfan.1997.3158 |
| [43] | Sobolev, S., Partial Differential Equations of Mathematical Physics (1964), Amsterdam: Elsevier, Amsterdam · Zbl 0123.06508 · doi:10.1016/c2013-0-01785-9 |
| [44] | Stein, EM, Singular Integrals and Differentiability Properties of Functions (PMS-30) (1971), Princeton: Princeton University Press, Princeton · doi:10.1515/9781400883882 |
| [45] | Vacca, G., Virtual element methods for hyperbolic problems on polygonal meshes, CAMWA (2016) · Zbl 1448.65177 · doi:10.1016/j.camwa.2016.04.029 |
| [46] | Vacca, G.; Beirão Da Veiga, L., Virtual element methods for parabolic problems on polygonal meshes, Numer. Methods PDEs, 31, 6, 2110-2134 (2015) · Zbl 1336.65171 · doi:10.1002/num.21982 |
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. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
