×

Mesh grading in isogeometric analysis. (English) Zbl 1443.65356

Summary: This paper is concerned with the construction of graded meshes for approximating so-called singular solutions of elliptic boundary value problems by means of multipatch discontinuous Galerkin Isogeometric Analysis schemes. Such solutions appear, for instance, in domains with re-entrant corners on the boundary of the computational domain, in problems with changing boundary conditions, in interface problems, or in problems with singular source terms. Making use of the analytic behavior of the solution, we construct the graded meshes in the neighborhoods of such singular points following a multipatch approach. We prove that appropriately graded meshes lead to the same convergence rates as in the case of smooth solutions with approximately the same number of degrees of freedom. Representative numerical examples are studied in order to confirm the theoretical convergence rates and to demonstrate the efficiency of the mesh grading technology in Isogeometric Analysis.

MSC:

65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
65D07 Numerical computation using splines
65N12 Stability and convergence of numerical methods for boundary value problems involving PDEs
65N50 Mesh generation, refinement, and adaptive methods for boundary value problems involving PDEs
35J25 Boundary value problems for second-order elliptic equations

Software:

G+Smo; ISOGAT
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[1] Kondrat’ev, V. A., Boundary value problems for elliptic equations in domains with conical or angular points, Transl. Moscow Math. Soc., 16, 227-313 (1967) · Zbl 0194.13405
[2] Grisvard, P., (Elliptic Problems in Nonsmooth Domains. Elliptic Problems in Nonsmooth Domains, Monographs and Studies in Mathematics (1985), Pitman Advanced Pub. Program) · Zbl 0695.35060
[3] Grisvard, P., (Singularities in Boundary Value Problems. Singularities in Boundary Value Problems, Recherches en Mathématiques Appliquées (1992), Masson) · Zbl 0766.35001
[4] Kozlov, V. A.; Maz’ya, V. G.; Rossmann, J., (Spectral Problems Associated with Corner Singularities of Solutions to Elliptic Equations. Spectral Problems Associated with Corner Singularities of Solutions to Elliptic Equations, Mathematical Surveys and Monographs, vol. 85 (2001), American Mathematical Society: American Mathematical Society Rhode Issland, USA) · Zbl 0965.35003
[5] Strang, G.; Fix, G., An Analysis of the Finite Element Method (1973), Prentice-Hall: Prentice-Hall Englewood Cliffs, NJ · Zbl 0278.65116
[6] Apel, T.; Sändig, A.-M.; Whiteman, J. R., Graded mesh refinement and error estimates for finite element solutions of elliptic boundary value problems in non-smooth domains, Math. Methods Appl. Sci., 19, 30, 63-85 (1996) · Zbl 0838.65109
[7] Oganesjan, L. A.; Ruchovetz, L. A., Variational Difference Methods for the Solution of Elliptic Equations (1979), Isdatelstvo Akademi Nank Armjanskoj SSR: Isdatelstvo Akademi Nank Armjanskoj SSR Erevan, (in Russian) · Zbl 0496.65053
[8] Apel, T.; Milde, F., Comparison of several mesh refinement strategies near edges, Comm. Numer. Methods Engrg., 12, 373-381 (1996) · Zbl 0865.65086
[9] Feistauer, M.; Sändig, A.-M., Graded mesh refinement and error estimates of higher order for DGFE solutions of elliptic boundary value problems in polygons, Numer. Methods Partial Differential Equations, 28, 4, 1124-1151 (2012) · Zbl 1253.65193
[10] Apel, T.; Heinrich, B., Mesh refinement and windowing near edges for some elliptic problem, SIAM J. Numer. Anal., 31, 3, 695-708 (1994) · Zbl 0807.65122
[11] Hughes, T. J.R.; Cottrell, J. A.; Bazilevs, Y., Isogeometric analysis: CAD, finite elements, NURBS, exact geometry and mesh refinement, Comput. Methods Appl. Mech. Engrg., 194, 4135-4195 (2005) · Zbl 1151.74419
[12] Cotrell, J. A.; Hughes, T. J.R.; Bazilevs, Y., Isogeometric Analysis, Toward Integration of CAD and FEA (2009), John Wiley and Sons · Zbl 1378.65009
[13] Langer, U.; Toulopoulos, I., Analysis of multipatch discontinuous Galerkin IgA approximations to elliptic boundary value problems. RICAM Reports 2014-08 (2014), Johann Radon Institute for Computational and Applied Mathematics, Austrian Academy of Sciences: Johann Radon Institute for Computational and Applied Mathematics, Austrian Academy of Sciences Linz
[14] Langer, U.; Moore, S. E., Discontinuous Galerkin isogeometric analysis of elliptic PDEs on surfaces. NFN Technical Report 12 (2014), Johannes Kepler University Linz, NFN Geometry and Simulation: Johannes Kepler University Linz, NFN Geometry and Simulation Linz, http://arxiv.org/abs/1402.1185 and (in press) in the DD22 proceedings
[15] Langer, U.; Mantzaflaris, A.; Moore, S. E.; Toulopoulos, I., Multipatch discontinuous Galerkin isogeometric analysis. RICAM Reports 2014-18 (2014), Johann Radon Institute for Computational and Applied Mathematics, Austrian Academy of Sciences: Johann Radon Institute for Computational and Applied Mathematics, Austrian Academy of Sciences Linz
[16] Dryja, M., On discontinuous Galerkin methods for elliptic problems with discontinuous coeffcients, Comput. Methods Appl. Math., 3, 76-85 (2003) · Zbl 1039.65079
[17] Rivière, B., Discontinuous Galerkin Methods for Solving Elliptic and Parabolic Equations: Theory and Implementation (2008), SIAM: SIAM Philadelphia · Zbl 1153.65112
[18] Di Pietro, Daniele A.; Ern, Alexandre, (Mathematical Aspects of Discontinuous Galerkin Methods. Mathematical Aspects of Discontinuous Galerkin Methods, Mathématiques et Applications, vol. 69 (2012), Springer-Verlag: Springer-Verlag Heidelberg, Dordrecht, London, New York) · Zbl 1231.65209
[19] Apel, T., Interpolation of non-smooth functions on anisotropic finite element meshes, M2AN, 33, 6, 1149-1185 (1999) · Zbl 0984.65113
[20] Oh, H. S.; Kim, H.; Jeong, J. W., Enriched isogeometric analysis of elliptic boundary value problems in domains with cracks and/or corners, Internat. J. Numer. Methods Engrg., 97, 3 (2014) · Zbl 1352.74419
[21] Jeong, J. W.; Oh, H. S.; Kang, S.; Kim, H., Mapping techniques for isogeometric analysis of elliptic boundary value problems containing singularities, Comput. Methods Appl. Mech. Engrg., 254, 0, 334-352 (2013) · Zbl 1297.65156
[22] Beirão da Veiga, L.; Cho, D.; Sangalli, G., Anisotropic NURBS approximation in isogeometric analysis, Comput. Methods Appl. Mech. Engrg., 209-212, 1-11 (2012) · Zbl 1243.65027
[23] De Luycker, E.; Benson, D. J.; Belytschko, T.; Bazilevs, Y.; Hsu, M. C., X-FEM in isogeometric analysis for linear fracture mechanics, Internat. J. Numer. Methods Engrg., 87, 6, 541-565 (2011) · Zbl 1242.74105
[24] Ghorashi, S. Sh.; Valizadeh, N.; Mohammadi, S., Extended isogeometric analysis for simulation of stationary and propagating cracks, Internat. J. Numer. Methods Engrg., 89, 9, 1069-1101 (2012) · Zbl 1242.74119
[25] Ghorashi, S. Sh.; Valizadeh, N.; Mohammadi, S.; Rabczuk, T., T-spline based XIGA for fracture analysis of orthotropic media, Comput. Struct., 147, 0, 138-146 (2015)
[26] Nguyen-Thanh, N.; Valizadeh, N.; Nguyen, M. N.; Nguyen-Xuan, H.; Zhuang, X.; Areias, P.; Zi, G.; Bazilevs, Y.; De Lorenzis, L.; Rabczuk, T., An extended isogeometric thin shell analysis based on Kirchhoff-love theory, Comput. Methods Appl. Mech. Engrg., 284, 265-291 (2015), Isogeometric Analysis Special Issue · Zbl 1423.74811
[27] Nguyen-Thanh, N.; Nguyen-Xuan, H.; Bordas, S. P.A.; Rabczuk, T., Isogeometric analysis using polynomial splines over hierarchical t-meshes for two-dimensional elastic solids, Comput. Methods Appl. Mech. Engrg., 200, 21-22, 1892-1908 (2011) · Zbl 1228.74091
[28] Bazilevs, Y.; Calo, V. M.; Cottrell, J. A.; Evans, J. A.; Hughes, T. J.R.; Lipton, S.; Scott, M. A.; Sederberg, T. W., Isogeometric analysis using T-splines, Comput. Methods Appl. Mech. Engrg., 199, 229-263 (2010) · Zbl 1227.74123
[29] Nguyen-Thanh, N.; Kiendl, J.; Nguyen-Xuan, H.; Wüchner, R.; Bletzinger, K. U.; Bazilevs, Y.; Rabczuk, T., Rotation free isogeometric thin shell analysis using PHT-splines, Comput. Methods Appl. Mech. Engrg., 200, 47-48, 3410-3424 (2011) · Zbl 1230.74230
[30] Vuong, A.-V.; Giannelli, C.; Jüttler, B.; Simeon, B., A hierarchical approach to adaptive local refinement in isogeometric analysis, Comput. Methods Appl. Mech. Engrg., 200, 49-52, 3554-3567 (2011) · Zbl 1239.65013
[31] Kiss, G.; Giannelli, C.; Zore, U.; Jüttler, B.; Grossmann, D.; Barner, J., Adaptive {CAD} model (re-)construction with THB-splines, Graph. Models, 76, 5, 273-288 (2014)
[33] Adams, R. A.; Fournier, J. J.F., (Sobolev Spaces. Sobolev Spaces, Pure and Applied Mathematics, vol. 140 (2003), ACADEMIC PRESS-imprint Elsevier Science) · Zbl 1098.46001
[34] Evans, L. C., (Partial Differential Equestions. Partial Differential Equestions, Graduate Studies in Mathematics, vol. 19 (1998), American Mathematical Society) · Zbl 0902.35002
[35] Bazilevs, Y.; Beirão da Veiga, L.; Cottrell, J. A.; Hughes, T. J.R.; Sangalli, G., Isogeometric analysis: approximation, stability and error estimates for \(h\)-refined meshes, M3AS, 16, 07, 1031-1090 (2006) · Zbl 1103.65113
[36] Schumaker, L. L., Spline Functions: Basic Theory (2007), University Press: University Press Cambridge · Zbl 1123.41008
[37] Jüttler, B.; Langer, U.; Mantzaflaris, A.; Moore, S. E.; Zulehner, W., Geometry + simulation modules: implementing isogeometric analysis, PAMM, 14, 1, 961-962 (2014)
[38] Falkand, S. R.; Osborn, J. E., Remarks on mixed finite element methods for problems with rough coefficients, Math. Comp., 62, 205, 1-19 (1994) · Zbl 0801.65108
[39] Kellogg, R. B., On the Poisson equation with intersecting interfaces, Appl. Anal., 4, 101-129 (1975) · Zbl 0307.35038
[40] Kröner, D.; Růžička, M.; Toulopoulos, I., Numerical solutions of systems with \((p, \delta)\)-structure using local discontinuous Galerkin finite element methods, Internat. J. Numer. Methods Fluids, 76, 11, 855-874 (2014)
[41] Apel, T.; Heinrich, B., The finite element method with anisotropic mesh grading for elliptic problems in domains with corner and edges, SIAM J. Numer. Anal., 31, 3, 695-708 (1998) · Zbl 0807.65122
[42] Jung, M., On adaptive grids in multilevel methods, (Hengst, S., GAMM-Seminar on Multigrid-Methods, Gosen, Germany, 21-25 September 1992. GAMM-Seminar on Multigrid-Methods, Gosen, Germany, 21-25 September 1992, WIAS Report, vol. 5 (1993), WIAS), 67-80
[43] Scheidurov, V. V., Multigrid Methods for Finite Elements (1989), Nauka: Nauka Moscow, (in Russian) · Zbl 0682.65065
[44] Jung, M.; Nicaise, S.; Tabka, J., Some multilevel methods on graded meshes, J. Comput. Appl. Math., 138, 1, 151-171 (2002) · Zbl 0996.65136
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.