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.
J. Bey, Finite-Volumen- und Mehrgitter-Verfahren für elliptische Randwertprobleme, Teubner Stuttgart, Leipzig, 1998.
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.
F. Bouchut and B. Perthame, Kruzkov’s estimates for scalar conservation laws revisited, Trans. Am. Math. Soc. 350 (1998), No. 7, 2847–2870.
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.
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.
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.
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.
L. Chen, A new class of high order finite volume methods for second order elliptic equations, SIAM J. Numer. Anal. 47 (2010), 4021–4043.
Z. Chen, J. Wu, and Y. Xu, Higher-order finite volume methods for elliptic boundary value problems, Adv. Comput. Math. 37 (2012), 191–253.
Z. Chen, Y. Xu, and Y. Zhang, A construction of higher-order finite volume methods, Math. Comp. 84 (2015), 599–628.
P.G.Ciarlet, The Finite Element Method for Elliptic Problems, North-Holland, 1978.
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.
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.
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.
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.
B. Cockburn and W. Zhang, A posteriori error estimates for HDG methods, J. Sci. Comput. 51 (2012), No. 3, 582–607.
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.
L. Cueto-Felgueroso and I. Colominas, High-order finite volume methods and multiresolution reproducing kernels, Arch. Comput. Methods. Engrg., .
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.
G. Dziuk, Theory and Numerics of Partial Differential Equations. (Theorie und Numerik partieller Differentialgleichungen), De Gruyter Studium, Berlin, 2010.
B. Engquist and S. Osher, One-sided difference approximations for nonlinear conservation laws, Math. Comp. 36 (1981), 321–351.
K. Eriksson, D. Estep, P. Hansbo, and C. Johnson, Computational Differential Equations, Cambridge University Press, 1996.
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.
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.
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.
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.
B. Heinrich, Finite Difference Methods on Irregular Networks, Birkhäuser, Basel, 1987.
R. Herbin, An error estimate for a finite volume scheme for a diffusion–convection problem on triangular mesh, Num. Meth. PDE11 (1995), 165–173.
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.
L. Ivan, Development of high-order CENO finite-volume schemes with block-based adaptive mesh refinement, Thesis, University Toronto, 2011.
C. Johnson and A. Szepessy, Convergence of a finite element method for a nonlinear hyperbolic conservation law, Math. Comp. 49 (1987), 427–444.
C. Johnson, Numerical Solution of Partial Differential Equations by the Finite Element Method, Cambridge University Press, 1994.
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.
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.
D. Kuzmin, A guide to numerical methods for transport equations, Preprint, Univ. Erlangen-Nürnberg, 2010.
G. Lube, Streamline diffusion finite element method for quasilinear elliptic problems, Num. Math. 61 (1992), 335–357.
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.
B. Riviere, Discontinuous Galerkin Methods for Solving Elliptic and Parabolic Equations. Theory and Implementation, SIAM, Philadelphia, 2008.
C. Rohde, Weakly Coupled Hyperbolic Systems, Ph.D. Thesis, Universität Freiburg, 1996.
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.
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.
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.
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.
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.
The Journal of Numerical Mathematics contains high-quality papers featuring contemporary research in all areas of Numerical Mathematics. This includes the development, analysis, and implementation of new and innovative methods in Numerical Linear Algebra, Numerical Analysis, Optimal Control/Optimization, and Scientific Computing.