Anand, Akash; Ovall, Jeffrey S.; Weißer, Steffen A Nyström-based finite element method on polygonal elements. (English) Zbl 1419.65100 Comput. Math. Appl. 75, No. 11, 3971-3986 (2018). Summary: We consider families of finite elements on polygonal meshes, that are defined implicitly on each mesh cell as solutions of local Poisson problems with polynomial data. Functions in the local space on each mesh cell are evaluated via Nyström discretizations of associated integral equations, allowing for curvilinear polygons and non-polynomial boundary data. Several experiments demonstrate the approximation quality of interpolated functions in these spaces. Cited in 5 Documents MSC: 65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs 65N38 Boundary element methods for boundary value problems involving PDEs 65R20 Numerical methods for integral equations 35R09 Integro-partial differential equations 65D30 Numerical integration 35J25 Boundary value problems for second-order elliptic equations Keywords:finite element methods; Trefftz methods; polygonal meshes; Nyström methods; BEM-based FEM; virtual element methods PDFBibTeX XMLCite \textit{A. Anand} et al., Comput. Math. Appl. 75, No. 11, 3971--3986 (2018; Zbl 1419.65100) Full Text: DOI arXiv References: [1] Beirão 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, 199-214 (2013) · Zbl 1416.65433 [2] Ahmad, B.; Alsaedi, A.; Brezzi, F.; Marini, L. D.; Russo, A., Equivalent projectors for virtual element methods, Comput. Math. Appl., 66, 3, 376-391 (2013) · Zbl 1347.65172 [3] Brezzi, F.; Marini, L. D., Virtual element and discontinuous Galerkin methods, (Recent Developments in Discontinuous Galerkin Finite Element Methods for Partial Differential Equations. Recent Developments in Discontinuous Galerkin Finite Element Methods for Partial Differential Equations, IMA Math. Appl, vol.157 (2014), Springer: Springer Cham), 209-221 · Zbl 1282.65150 [4] Beirão da Veiga, L.; Brezzi, F.; Marini, L. D.; Russo, A., The hitchhiker’s guide to the virtual element method, Math. Models Methods Appl. Sci., 24, 8, 1541-1573 (2014) · Zbl 1291.65336 [5] Beirão da Veiga, L.; Manzini, G., A virtual element method with arbitrary regularity, IMA J. Numer. Anal., 34, 2, 759-781 (2014) · Zbl 1293.65146 [6] Antonietti, P. F.; Beirão da Veiga, L.; Mora, D.; Verani, M., A stream virtual element formulation of the Stokes problem on polygonal meshes, SIAM J. Numer. Anal., 52, 1, 386-404 (2014) · Zbl 1427.76198 [7] Gain, A. L.; Talischi, C.; Paulino, G. H., On the virtual element method for three-dimensional linear elasticity problems on arbitrary polyhedral meshes, Comput. Methods Appl. Mech. Engrg., 282, 132-160 (2014) · Zbl 1423.74095 [8] Beirão da Veiga, L.; Chernov, A.; Mascotto, L.; Russo, A., Basic principles of \(h p\) virtual elements on quasiuniform meshes, Math. Models Methods Appl. Sci., 26, 8, 1567-1598 (2016) · Zbl 1344.65109 [9] Benedetto, M. F.; Berrone, S.; Borio, A.; Pieraccini, S.; Scialò, S., Order preserving SUPG stabilization for the virtual element formulation of advection-diffusion problems, Comput. Methods Appl. Mech. Engrg., 311, 18-40 (2016) · Zbl 1439.76051 [10] Antonietti, P.; Bruggi, M.; Scacchi, S.; Verani, M., On the virtual element method for topology optimization on polygonal meshes :A numerical study, Comput. Math. Appl. (2017) · Zbl 1391.74205 [11] Copeland, D.; Langer, U.; Pusch, D., From the boundary element domain decomposition methods to local Trefftz finite element methods on polyhedral meshes, (Domain Decomposition Methods in Science and Engineering XVIII. Domain Decomposition Methods in Science and Engineering XVIII, Lect. Notes Comput. Sci. Eng, vol. 70 (2009), Springer: Springer Berlin), 315-322 · Zbl 1183.65158 [12] Hofreither, C.; Langer, U.; Pechstein, C., Analysis of a non-standard finite element method based on boundary integral operators, Electron. Trans. Numer. Anal., 37, 413-436 (2010) · Zbl 1205.65315 [13] Hofreither, C., \(L_2\) error estimates for a nonstandard finite element method on polyhedral meshes, J. Numer. Math., 19, 1, 27-39 (2011) · Zbl 1222.65119 [14] Weißer, S., Residual error estimate for BEM-based FEM on polygonal meshes, Numer. Math., 118, 4, 765-788 (2011) · Zbl 1227.65103 [15] Rjasanow, S.; Weißer, S., Higher order BEM-based FEM on polygonal meshes, SIAM J. Numer. Anal., 50, 5, 2357-2378 (2012) · Zbl 1264.65196 [16] Rjasanow, S.; Weißer, S., FEM with Trefftz trial functions on polyhedral elements, J. Comput. Appl. Math., 263, 202-217 (2014) · Zbl 1301.65125 [17] Weißer, S., Arbitrary order Trefftz-like basis functions on polygonal meshes and realization in BEM-based FEM, Comput. Math. Appl., 67, 7, 1390-1406 (2014) · Zbl 1350.65131 [18] Hofreither, C.; Langer, U.; Weißer, S., Convection-adapted BEM-based FEM, ZAMM Z. Angew. Math. Mech., 96, 12, 1467-1481 (2016) [19] Weißer, S., Residual based error estimate and quasi-interpolation on polygonal meshes for high order BEM-based FEM, Comput. Math. Appl., 73, 2, 187-202 (2017) · Zbl 1368.65238 [20] Weißer, S.; Wick, T., The dual-weighted residual estimator realized on polygonal meshes, Comput. Methods Appl. Math. (2017) · Zbl 1422.65417 [21] Gillette, A.; Rand, A.; Bajaj, C., Construction of scalar and vector finite element families on polygonal and polyhedral meshes, Comput. Methods Appl. Math., 16, 4, 667-683 (2016) · Zbl 1348.65163 [22] Floater, M.; Gillette, A.; Sukumar, N., Gradient bounds for Wachspress coordinates on polytopes, SIAM J. Numer. Anal., 52, 1, 515-532 (2014) · Zbl 1292.65006 [23] Rand, A.; Gillette, A.; Bajaj, C., Quadratic serendipity finite elements on polygons using generalized barycentric coordinates, Math. Comp., 83, 290, 2691-2716 (2014) · Zbl 1300.65091 [24] Gillette, A.; Rand, A.; Bajaj, C., Error estimates for generalized barycentric interpolation, Adv. Comput. Math., 37, 3, 417-439 (2012) · Zbl 1259.65013 [25] Manzini, G.; Russo, A.; Sukumar, N., New perspectives on polygonal and polyhedral finite element methods, Math. Models Methods Appl. Sci., 24, 8, 1665-1699 (2014) · Zbl 1291.65322 [26] Spring, D. W.; Leon, S. E.; Paulino, G. H., Unstructured polygonal meshes with adaptive refinement for the numerical simulation of dynamic cohesive fracture, Int. J. Fract., 189, 1, 33-57 (2014), 9 [27] Floater, M. S., Generalized barycentric coordinates and applications, Acta Numer., 24, 161-214 (2015) · Zbl 1317.65065 [28] Bonelle, J.; Ern, A., Analysis of compatible discrete operator schemes for elliptic problems on polyhedral meshes, ESAIM Math. Model. Numer. Anal., 48, 2, 553-581 (2014) · Zbl 1297.65132 [29] Bonelle, J.; Ern, A., Analysis of compatible discrete operator schemes for the Stokes equations on polyhedral meshes, IMA J. Numer. Anal., 35, 4, 1672-1697 (2015) · Zbl 1329.76249 [30] Bonelle, J.; Di Pietro, D. A.; Ern, A., Low-order reconstruction operators on polyhedral meshes: application to compatible discrete operator schemes, Comput. Aided Geom. Design, 35, 36, 27-41 (2015) · Zbl 1417.65197 [31] Di Pietro, D. A.; Ern, A.; Lemaire, S., An arbitrary-order and compact-stencil discretization of diffusion on general meshes based on local reconstruction operators, Comput. Methods Appl. Math., 14, 4, 461-472 (2014) · Zbl 1304.65248 [32] Di Pietro, D. A.; Ern, A., A hybrid high-order locking-free method for linear elasticity on general meshes, Comput. Methods Appl. Mech. Engrg., 283, 1-21 (2015) · Zbl 1423.74876 [33] Di Pietro, D. A.; Ern, A., Hybrid high-order methods for variable-diffusion problems on general meshes, C. R. Math. Acad. Sci. Paris, 353, 1, 31-34 (2015) · Zbl 1308.65196 [34] Wang, J.; Ye, X., A weak Galerkin finite element method for second-order elliptic problems, J. Comput. Appl. Math., 241, 103-115 (2013) · Zbl 1261.65121 [35] Wang, J.; Ye, X., A weak Galerkin mixed finite element method for second order elliptic problems, Math. Comp., 83, 289, 2101-2126 (2014) · Zbl 1308.65202 [36] Wang, C.; Wang, J., An efficient numerical scheme for the biharmonic equation by weak Galerkin finite element methods on polygonal or polyhedral meshes, Comput. Math. Appl., 68, 12, part B, 2314-2330 (2014) · Zbl 1361.35058 [37] Mu, L.; Wang, J.; Ye, X., Weak Galerkin finite element methods on polytopal meshes, Int. J. Numer. Anal. Model., 12, 1, 31-53 (2015) · Zbl 1332.65172 [38] Mu, L.; Wang, J.; Ye, X., A weak Galerkin finite element method with polynomial reduction, J. Comput. Appl. Math., 285, 45-58 (2015) · Zbl 1315.65099 [39] Mu, L.; Wang, J.; Ye, X., A new weak Galerkin finite element method for the Helmholtz equation, IMA J. Numer. Anal., 35, 3, 1228-1255 (2015) · Zbl 1323.65116 [40] Wang, J.; Ye, X., A weak Galerkin finite element method for the stokes equations, Adv. Comput. Math., 42, 1, 155-174 (2016) · Zbl 1382.76178 [41] A.V. Astaneh, F. Fuentes, J. Mora, L. Demkowicz, High-order polygonal discontinuous Petrov-Galerkin (PolyDPG) methods using ultraweak formulations. ArXiv e-prints, arXiv:1706.06754; A.V. Astaneh, F. Fuentes, J. Mora, L. Demkowicz, High-order polygonal discontinuous Petrov-Galerkin (PolyDPG) methods using ultraweak formulations. ArXiv e-prints, arXiv:1706.06754 [42] Kirsch, A.; Monk, P., An analysis of the coupling of finite-element and Nyström methods in acoustic scattering, IMA J. Numer. Anal., 14, 4, 523-544 (1994) · Zbl 0816.65104 [43] Grisvard, P., (Elliptic Problems in Nonsmooth Domains. Elliptic Problems in Nonsmooth Domains, Monographs and Studies in Mathematics, vol. 24 (1985), Pitman (Advanced Publishing Program): Pitman (Advanced Publishing Program) Boston, MA) · Zbl 0695.35060 [44] Grisvard, P., (Singularities in Boundary Value Problems. Singularities in Boundary Value Problems, Recherches en Mathématiques Appliquées [Research in Applied Mathematics], vol. 22 (1992), Masson: Masson Paris) · Zbl 0766.35001 [45] Wigley, N. M., Asymptotic expansions at a corner of solutions of mixed boundary value problems, J. Math. Mech., 13, 549-576 (1964) · Zbl 0178.45902 [46] Zargaryan, S. S.; Maz’ya, V. G., The asymptotic form of the solutions of integral equations of potential theory in the neighbourhood of the corner points of a contour, Prikl. Mat. Mekh., 48, 1, 169-174 (1984) [47] Karachik, V. V.; Antropova, N. A., On the solution of a nonhomogeneous polyharmonic equation and the nonhomogeneous Helmholtz equation, Differ. Uravn., 46, 3, 384-395 (2010) · Zbl 1194.31007 [48] Kress, R., (Linear Integral Equations. Linear Integral Equations, Applied Mathematical Sciences, vol. 82 (2014), Springer: Springer New York) · Zbl 1328.45001 [49] Szabó, B.; Babuška, I., Finite element analysis, (A Wiley-Interscience Publication (1991), John Wiley & Sons, Inc.: John Wiley & Sons, Inc. New York) [50] Nyström, E. J., Über Die Praktische Auflösung von Integralgleichungen mit Anwendungen auf Randwertaufgaben der Potentialtheorie, Soc. Sci. Fenn. Comment. Phys.-Math., 4, 15, 1-52 (1928) [51] Nyström, E. J., Über Die Praktische Auflösung von Integralgleichungen mit Anwendungen auf Randwertaufgaben, Acta Math., 54, 1, 185-204 (1930) · JFM 56.0342.01 [52] Kress, R., A Nyström method for boundary integral equations in domains with corners, Numer. Math., 58, 2, 145-161 (1990) · Zbl 0707.65078 [53] Melenk, J. M.; Babuška, I., Approximation with harmonic and generalized harmonic polynomials in the partition of unity method, Comput. Assist. Methods Eng. Sci., 4, 3/4, 607-632 (1997) · Zbl 0951.65128 [54] Melenk, J., Operator adapted spectral element methods i: harmonic and generalized harmonic polynomials, Numer. Math., 84, 1, 35-69 (1999) · Zbl 0941.65112 [55] Talischi, C.; Paulino, G. H.; Pereira, A.; Menezes, I. F.M., : a general-purpose mesh generator for polygonal elements written in Matlab, Struct. Multidiscip. Optim., 45, 3, 309-328 (2012) · Zbl 1274.74401 [56] Scott, R., Finite Element Techniques for Curved Boundaries (1973), Massachusetts Institute of Technology, (Ph.D. thesis) [57] Lenoir, M., Optimal isoparametric finite elements and error estimates for domains involving curved boundaries, SIAM J. Numer. Anal., 23, 3, 562-580 (1986) · Zbl 0605.65071 [58] Bernardi, C., Optimal finite-element interpolation on curved domains, SIAM J. Numer. Anal., 26, 5, 1212-1240 (1989) · Zbl 0678.65003 [59] Ern, A.; Guermond, J.-L., (Theory and Practice of Finite Elements. Theory and Practice of Finite Elements, Applied Mathematical Sciences, vol. 159 (2004), Springer-Verlag: Springer-Verlag New York) · Zbl 1059.65103 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.