Abstract
It is still an open problem to prove a priori error estimates for finite volume schemes of higher order MUSCL type, including limiters, on unstructured meshes, which show some improvement compared to first order schemes. In this paper we use these higher order schemes for the discretization of convection dominated elliptic problems in a convex bounded domain Ω in ℝ2 and we can prove such kind of an a priori error estimate. In the part of the estimate, which refers to the discretization of the convective term, we gain h1/2. Although the original problem is linear, the numerical problem becomes nonlinear, due to MUSCL type reconstruction/limiter technique.
Acknowledgement
The authors want to thank the unknown referees for their valuable comments.
Funding: M. Rokyta was partially supported by Prvouk P47.
References
[1] J. Bey, Finite-Volumen- und Mehrgitter-Verfahren für elliptische Randwertprobleme, Teubner Stuttgart, Leipzig, 1998.10.1007/978-3-663-10071-3Search in Google Scholar
[2] D. Bouche, J.-M. Ghidaglia, and F-P. Pascal, Error estimate for the upwind finite volume method for the nonlinear scalar conservation law, J. Comput. Appl. Math. 235 (2011), No. 18, 5394–5410.10.1016/j.cam.2011.05.050Search in Google Scholar
[3] F. Bouchut and B. Perthame, Kruzkov’s estimates for scalar conservation laws revisited, Trans. Am. Math. Soc. 350 (1998), No. 7, 2847–2870.10.1090/S0002-9947-98-02204-1Search in Google Scholar
[4] A. Bradji and J. Fuhrmann, Some abstract error estimates of a finite volume scheme for a nonstationary heat equation on general nonconforming multidimensional spatial meshes, Appl. Math. 58 (2013), No. 1, 1–38.10.1007/s10492-013-0001-ySearch in Google Scholar
[5] F. Brezzi, T.J.R. Hughes, L.D. Marini, A. Russo and E. Süli, A priori error analysis of residual-free bubbles for advection-diffusion problems, SIAM J. Numer. Anal. 36 (1999), No. 6, 1933–1948.10.1137/S0036142998342367Search in Google Scholar
[6] Z. Cai, J. Douglas Jr. and M. Park, Development and analysis of higher order finite volume methods for elliptic equations, Adv. Comput. Math. 19 (2003), 3–33.10.1023/A:1022841012296Search in Google Scholar
[7] C. Chainais-Hillairet, Second-order finite-volume schemes for a nonlinear hyperbolic equation: Error estimate, Math. Methods Appl. Sci. 23 (2000), No. 5, 467–490.10.1002/(SICI)1099-1476(20000325)23:5<467::AID-MMA124>3.0.CO;2-7Search in Google Scholar
[8] L. Chen, A new class of high order finite volume methods for second order elliptic equations, SIAM J. Numer. Anal. 47 (2010), 4021–4043.10.1137/080720164Search in Google Scholar
[9] Z. Chen, J. Wu, and Y. Xu, Higher-order finite volume methods for elliptic boundary value problems, Adv. Comput. Math. 37 (2012), 191–253.10.1007/s10444-011-9201-8Search in Google Scholar
[10] Z. Chen, Y. Xu, and Y. Zhang, A construction of higher-order finite volume methods, Math. Comp. 84 (2015), 599–628.10.1090/S0025-5718-2014-02881-0Search in Google Scholar
[11] P.G.Ciarlet, The Finite Element Method for Elliptic Problems, North-Holland, 1978.10.1115/1.3424474Search in Google Scholar
[12] B. Cockburn, F. Coquel, and P. LeFloch, An error estimate for finite volume methods for multidimensional conservation laws, Math. Comp. 63 (1994), No. 207, 77–103.10.1090/S0025-5718-1994-1240657-4Search in Google Scholar
[13] B. Cockburn, F. Coquel, and P. LeFloch, Convergence of the finite volume method for multidimensional conservation laws, SIAM J. Numer. Anal. 32 (1995), 687–705.10.1137/0732032Search in Google Scholar
[14] B. Cockburn and P.-A. Gremaud, A priori error estimates for numerical methods for scalar conservation laws. Part I: The general approach, Math. Comp. 65 (1996), No. 214, 533–573.10.1090/S0025-5718-96-00701-6Search in Google Scholar
[15] B. Cockburn and C. W. Shu, TVB Runge–Kutta projection discontinuous Galerkin finite element method for conservation laws. II: General framework, Math. Comp., 52 (1989), 411–435.Search in Google Scholar
[16] B. Cockburn and W. Zhang, A posteriori error estimates for HDG methods, J. Sci. Comput. 51 (2012), No. 3, 582–607.10.1007/s10915-011-9522-2Search in Google Scholar
[17] F. Coquel and P. LeFloch, Convergence of finite difference schemes for conservation laws in several space dimensions: the corrected antidiffusive flux approach, Math. Comp. 57 (1991), 169–210.10.1090/S0025-5718-1991-1079010-2Search in Google Scholar
[18] L. Cueto-Felgueroso and I. Colominas, High-order finite volume methods and multiresolution reproducing kernels, Arch. Comput. Methods. Engrg., 10.1007/s11831-008-9017-y.Search in Google Scholar
[19] L. J. Durlofsky, B. Engquist, and S. Osher, Triangle based adaptive stencils for the solution of hyperbolic conservation laws, J. Comput. Phys. 98 (1992), 64–73.10.1016/0021-9991(92)90173-VSearch in Google Scholar
[20] G. Dziuk, Theory and Numerics of Partial Differential Equations. (Theorie und Numerik partieller Differentialgleichungen), De Gruyter Studium, Berlin, 2010.10.1515/9783110214819Search in Google Scholar
[21] B. Engquist and S. Osher, One-sided difference approximations for nonlinear conservation laws, Math. Comp. 36 (1981), 321–351.10.1090/S0025-5718-1981-0606500-XSearch in Google Scholar
[22] K. Eriksson, D. Estep, P. Hansbo, and C. Johnson, Computational Differential Equations, Cambridge University Press, 1996.Search in Google Scholar
[23] R. Eymard, T. Gallouët, and R. Herbin, A cell-centred finite-volume approximation for anisotropic diffusion operators on unstructured meshes in any space dimension, IMA J. Numer. Anal. 26 (2006), No. 2, 326–353.10.1093/imanum/dri036Search in Google Scholar
[24] R. Eymard, T. Gallouët, and R. Herbin, Cell centred discretisation of non linear elliptic problems on general multidimensional polyhedral grids, J. Numer. Math. 17 (2009), No. 3, 173–193.10.1515/JNUM.2009.010Search in Google Scholar
[25] R. Eymard, G. Henry, R, Herbin, F. Hubert, and R. Klöfkorn, 3D benchmark on discretization schemes for anisotropic diffusion problems on general grids. In: Finite Volumes for Complex Applications VI: Problems and Perspectives. (Eds. J. Fořt et al.), FVCA 6, Int. Symposium, Prague, Czech Republic, June 2011, Vol. 1 and 2. Springer Proceedings in Mathematics, Vol. 4, 2011, pp. 895–930.10.1007/978-3-642-20671-9_89Search in Google Scholar
[26] M. Feistauer, J. Felcman, and M. Lukáčová–Medvid’ová, On the convergence of a combined finite volume–finite element method for nonlinear convection–diffusion problems, Num. Meth. PDE13 (1997), 1–28.10.1002/(SICI)1098-2426(199703)13:2<163::AID-NUM3>3.0.CO;2-NSearch in Google Scholar
[27] B. Heinrich, Finite Difference Methods on Irregular Networks, Birkhäuser, Basel, 1987.10.1007/978-3-0348-7196-9Search in Google Scholar
[28] R. Herbin, An error estimate for a finite volume scheme for a diffusion–convection problem on triangular mesh, Num. Meth. PDE11 (1995), 165–173.10.1002/num.1690110205Search in Google Scholar
[29] P. Houston, J. A. Mackenzie, E. Süli, and G. Warnecke, A posteriori error analysis for numerical approximations of Friedrichs systems, Numer. Math. 82 (1999), No. 3, 433–470.10.1007/s002110050426Search in Google Scholar
[30] L. Ivan, Development of high-order CENO finite-volume schemes with block-based adaptive mesh refinement, Thesis, University Toronto, 2011.Search in Google Scholar
[31] C. Johnson and A. Szepessy, Convergence of a finite element method for a nonlinear hyperbolic conservation law, Math. Comp. 49 (1987), 427–444.10.1090/S0025-5718-1987-0906180-5Search in Google Scholar
[32] C. Johnson, Numerical Solution of Partial Differential Equations by the Finite Element Method, Cambridge University Press, 1994.Search in Google Scholar
[33] D. Kröner, S. Noelle, and M. Rokyta, Convergence of higher order upwind finite volume schemes on unstructured grids for scalar conservation laws in two space dimensions, Num. Math. 71 (1995), No. 4, 527–560.10.1007/s002110050156Search in Google Scholar
[34] D. Kröner and M. Rokyta, Convergence of upwind finite volume schemes for scalar conservation laws in 2D, SIAM J. Numer. Anal. 31 (1994), No. 2, 324–343.10.1137/0731017Search in Google Scholar
[35] D. Kuzmin, A guide to numerical methods for transport equations, Preprint, Univ. Erlangen-Nürnberg, 2010.Search in Google Scholar
[36] G. Lube, Streamline diffusion finite element method for quasilinear elliptic problems, Num. Math. 61 (1992), 335–357.10.1007/BF01385513Search in Google Scholar
[37] M. Lukáčová–Meďviová, Combined finite element–finite volume method (convergence analysis), Comment. Math. Univ. Carolinae38 (1997), No. 3, 717–741.Search in Google Scholar
[38] M. Oevermann and R. Klein, A Cartesian grid finite volume method for elliptic equations with variable coefficients and embedded interfaces, J. Comput. Phys. 219 (2006), 749–769.10.1016/j.jcp.2006.04.010Search in Google Scholar
[39] B. Riviere, Discontinuous Galerkin Methods for Solving Elliptic and Parabolic Equations. Theory and Implementation, SIAM, Philadelphia, 2008.10.1137/1.9780898717440Search in Google Scholar
[40] C. Rohde, Weakly Coupled Hyperbolic Systems, Ph.D. Thesis, Universität Freiburg, 1996.Search in Google Scholar
[41] E. Süli, A posteriori error analysis and adaptivity for finite element approximations of hyperbolic problems. In: An Introduction to Recent Developments in Theory and Numerics for Conservation Laws, (Eds. D. Kröner et al.), Proc. of the Int. School, Freiburg/Littenweiler, Germany, October 20–24, 1997. Springer Lect. Notes Comput. Sci. Eng. 5 (1999), 123–194.10.1007/978-3-642-58535-7_4Search in Google Scholar
[42] J.-P. Vila, Convergence and error estimates in finite volume schemes for general multi-dimensional scalar conservation laws. I: Explicit monotone schemes, M2 AN28 (1994), No. 3, 267–295.10.1051/m2an/1994280302671Search in Google Scholar
[43] M. Vlasák, V. Dolejší, and J. Hájek, A priori error estimates of an extrapolated space–time discontinuous Galerkin method for nonlinear convection–diffusion problems, Numer. Meth. Part. D. E. 27 (2011), No. 6, 1456–1482.10.1002/num.20591Search in Google Scholar
[44] W. Wang, J. Guzman, and C.-W. Shu, The multiscale discontinuous Galerkin method for solving a class of second order elliptic problems with rough coefficients, Int.J. Numer. Anal. Model. 8 (2011), 28–47.Search in Google Scholar
[45] M. Wierse, Higher order upwind schemes on unstructured grids for the compressible Euler equations in time dependent geometries in 3D, Ph.D. Thesis, Universität Freiburg, 1994.Search in Google Scholar
[46] C. Zamfirescu, Survey of two-dimensional acute triangulations, Discrete Math. 313 (2013), No. 1, 35–49 (see also http://czamfirescu.tricube.de/CTZamfirescu-08.pdf).10.1016/j.disc.2012.09.016Search in Google Scholar
[47] Q. Zhang and C.-W. Shu, Error estimates to smooth solutions of Runge–Kutta discontinuous Galerkin method for scalar conservation laws, SIAM J. Numer. Anal. 42 (2004), No. 2, 641–666.10.1137/S0036142902404182Search in Google Scholar
© 2018 Walter de Gruyter Berlin/Boston
