Abstract
In this work, we study the approximation properties of multipatch dG-IgA methods, that apply the multipatch Isogeometric Analysis discretization concept and the discontinuous Galerkin technique on the interfaces between the patches, for solving linear diffusion problems with diffusion coefficients that may be discontinuous across the patch interfaces. The computational domain is divided into non-overlapping subdomains, called patches in IgA, where B-splines, or NURBS approximations spaces are constructed. The solution of the problem is approximated in every subdomain without imposing any matching grid conditions and without any continuity requirements for the discrete solution across the interfaces. Numerical fluxes with interior penalty jump terms are applied in order to treat the discontinuities of the discrete solution on the interfaces. We provide a rigorous a priori discretization error analysis for diffusion problems in two- and three-dimensional domains, where solutions patchwise belong to \(W^{l,p}\), with some \(l\ge 2\) and \( p\in ({2d}/{(d+2(l-1))},2]\). In any case, we show optimal convergence rates of the discretization with respect to the dG - norm.
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Adams, R.A., Fournier, J.J.F.: Sobolev Spaces, Pure and Applied Mathematics, 2nd edn. ACADEMIC PRESS-imprint Elsevier Science, Netherlands (2003)
Apostolatos, A., Schmidt, R., Wüchner, R., Bletzinger, K.U.: A Nitsche-type formulation and comparison of the most common domain decomposition methods in isogeometric analysis. Int. J. Numer. Methods Eng. 97(7), 1099–1142 (2014)
Arnold, D.N., Brezzi, F., Cockburn, B., Marini, D.L.: Unified analysis of discontinuous Galerkin methods for elliptic problem. SIAM J. Numer. Anal. 39(5), 1749–1779 (2002)
Bazilevs, Y., Hughes, T.J.R.: NURBS-based isogeometric analysis for the computation of flows about rotating components. Comput. Mech. 43, 143–150 (2008)
Bazilevs, Y., da Veiga, L.B., Cottrell, J.A., Hughes, T.J.R., Sangalli, G.: Isogeometric analysis: approximation, stability and error estimates for \(h\)-refined meshes. Math. Models Methods Appl. Sci. 16(7), 1031–1090 (2006)
Brener, S.C., Scott, L.R.: The mathematical Theory of Finite Element Methods, Texts in Applied Mathematics, 3rd edn. Springer, New York (2008)
Brunero, F.: Discontinuous Galerkin methods for isogeometric analysis. Master’s thesis, Università degli Studi di Milano (2012)
Ciarlet, P.G.: The Finite Element Method for Elliptic Problems. Studies in Mathematics and its Applications. North Holland Publishing Company, Amsterdam (1978)
Cotrell, J.A., Hughes, T.J.R., Bazilevs, Y.: Isogeometric Analysis. Toward Integration of CAD and FEA. Wiley, United Kingdom (2009)
Dryja, M.: On discontinuous Galerkin methods for elliptic problems with discontinuous coefficients. Comput. Methods Appl. Math. 3(1), 76–85 (2003)
Dryja, M., Galvis, J., Sarkis, M.: BDDC methods for discontinuous Galerkin discretization of elliptic problems. J. Complex. 23, 715–739 (2007)
Evans, J.A., Hughes, T.J.R.: Isogeometric Divergence-conforming B-splines for the Darcy-Stokes-Brinkman equations. Math. Models Methods Appl. Sci. 23(4), 671–741 (2013)
Evans, L.C.: Partial Differential Equations, Graduate Studies in Mathematics, 2nd edn. American Mathematical Society, Providence (2010)
Feng, X., Karakashian, O.A.: Two-level additive Schwarz methods for a discontinuous Galerkin approximation of second order elliptic problems. SIAM J. Numer. Anal 39(4), 1343–1365 (2001)
Grisvard, P.: Elliptic Problems in Nonsmooth Domains in Classics in Applied Mathematics. SIAM, Philadelphia (2011)
Heinrich, B., Nicaise, S.: The Nitsche mortar finite-element method for transmission problems with singularities. IMA J. Numer. Anal 23, 331–358 (2003)
Hughes, T.J.R.: The Finite Element Method: Linear Static and Dynamic Finite Element Analysis. Dover Publications, New York (2000)
Hughes, T.J.R., Cottrell, J.A., Bazilevs, Y.: Isogeometric analysis: CAD, finite elements, NURBS, exact geometry and mesh refinement. Comput. Methods Appl. Mech. Eng. 194, 4135–4195 (2005)
Kellogg, R.B.: On the Poisson equation with intersecting interfaces. Appl. Anal. 4, 101–129 (1975)
Kleiss, S.K., Pechstein, C., Jüttler, B., Tomar, S.: IETI - Isogeometric Tearing and Interconnecting. Comput. Methods Appl. Mech. Eng. 247–248, 201–215 (2012)
Knees, D.: On the regularity of weak solutions of quasi-linear elliptic transmission problems on polyhedral domains. Z. Anal. Anwendungen 23(3), 509–546 (2004)
Korneev, V.: Skhemy metoda konechnykh elementov vysokikh poryadkov tochnosti (Finite Element Schemes of high-order Accuracy). Izd-vo Leningradskogo gos. universiteta, Leningrad (1977). (in Russian)
Langer, U., Mantzaflaris, A., Moore, S.E., Toulopoulos, I.: Mesh grading in isogeometric analysis. Computers Math. Appl. 70(7), 1685–1700 (2015)
Langer, U., Moore, S.E.: Discontinuous Galerkin isogeometric analysis of elliptic PDEs on surfaces. NFN Technical Report 12, Johannes Kepler University Linz, NFN Geometry and Simulation, Linz (2014). Also available at http://arxiv.org/abs/1402.1185, and accepted for publication in the proceedings of the 22nd International Domain Decomposition Conference held at Lugano, Switzerland, September 2013
Li, B.Q.: Discontinuous Finite Element in Fluid Dynamics and Heat Transfer. Computational Fluid and Solid Mechanics. Springer, New York (2006)
Pietro, D.A.D., Ern, A.: Analysis of a discontinuous Galerkin method for heterogeneous diffusion problems with low-regularity solutions. Numer. Methods Partial Differ. Equ. 28(4), 1161–1177 (2012)
Pietro, D.A.D., Ern, A.: Mathematical Aspects of Discontinuous Galerkin Methods (Mathmatiques et Applications), Mathmatiques et Applications. Springer, Berlin (2012)
Pigl, L.A., Tiller, W.: The NURBS Book, Monographs in Visual Communication, 2nd edn. Springer, Berlin (1997)
Riviere, B.: Discontinuous Galerkin methods for Solving Elliptic and Parabolic Equations. SIAM, Society for Industrial and Applied Mathematics, Philadelphia (2008)
Schumaker, L.L.: Spline Functions: Basic Theory, 3rd edn. University Press, Cambridge, Cambridge (2007)
Toulopoulos, I.: An interior penalty discontinuous Galerkin finite element method for quasilinear parabolic problems. Finite Elements in Analysis and Design 95, 42–50 (2015)
Turek, S.: Efficient Solvers for Incompressible Flow Problems: An Algorithmic and Computational Approach. Lecture Notes in Computational Science and Engineering. Springer, Berlin (2013)
da Veiga, L.B., Buffa, A., Rivas, J., Sangalli, G.: Some estimates for \(hpk-\) refinement in Isogeometric Analysis. Numer. Math. 118(7), 271–305 (2011)
da Veiga, L.B., Cho, D., Pavarino, L., Scacchi, S.: BDDC preconditioners for isogeometric analysis. Math. Models Methods Appl. Sci. 23(6), 1099–1142 (2013)
Wihler, T.P., Riviere, B.: Discontinuous Galerkin methods for second-order elliptic PDE with low-regularity solutions. J. Sci. Comput. 46(2), 151–165 (2011)
Zlamál, M.: The finite element method in domains with curved boundaries. Int. J. Numer. Methods Eng. 5(3), 367–373 (1973)
Acknowledgments
The authors thank A. Mantzaflaris, S. Moore and C. Hofer for their help on performing the numerical tests. This work was supported by Austrian Science Fund (FWF) under the grant NFN S117-03.
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Communicated by Gabriel Wittum.
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Langer, U., Toulopoulos, I. Analysis of multipatch discontinuous Galerkin IgA approximations to elliptic boundary value problems. Comput. Visual Sci. 17, 217–233 (2015). https://doi.org/10.1007/s00791-016-0262-6
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DOI: https://doi.org/10.1007/s00791-016-0262-6
Keywords
- Linear elliptic problems
- Discontinuous coefficients
- Discontinuous Galerkin discretization
- Isogeometric analysis
- Non-matching meshes
- Low regularity solutions
- A priori discretization error estimates