A posteriori error estimates for finite volume and mixed finite element discretizations of convection–diffusion–reaction equations

We present in this paper a unified framework for a posteriori error estimation in the finite volume and lowest-order Raviart–Thomas mixed finite element methods. We consider convection–diffusion–reaction equations on simplicial meshes in two or three space dimensions and pay a particular attention to inhomogeneous and anisotropic diffusion–dispersion tensors and to convection dominance, in which case upwind-weighted schemes are considered. Our estimates are derived in the energy (semi-)norm for a locally postprocessed approximate solution preserving the finite volume/mixed finite element fluxes and are of residual type. They give a global upper bound on the error and are fully computable in the sense that all occurring constants are evaluated explicitly, so that they can serve both as indicators for adaptive refinement or for the actual control of the error. Local efficiency and semi-robustness in the sense that the local efficiency constant only depends on local variations in the inhomogeneities and anisotropies and becomes optimal as the local Peclet number gets sufficiently small can also be shown. Moreover, passing from their local to global computation, our estimates become asymptotically exact and asymptotically fully robust with respect to inhomogeneities and anisotropies. We finally present numerical experiments confirming their accuracy and briefly compare the results for the two methods.

[1]  R. Eymard,et al.  Finite volume approximation of elliptic problems and convergence of an approximate gradient , 2001 .

[2]  Jean E. Roberts,et al.  Global estimates for mixed methods for second order elliptic equations , 1985 .

[3]  M. Bebendorf.  A Note on the Poincaré Inequality for Convex Domains , 2003 .

[4]  H. Weinberger,et al.  An optimal Poincaré inequality for convex domains , 1960 .

[5]  Todd Arbogast,et al.  Enhanced Cell-Centered Finite Differences for Elliptic Equations on General Geometry , 1998, SIAM J. Sci. Comput..

[6]  J. Maître,et al.  Connection between finite volume and mixed finite element methods , 1996 .

[7]  M. Vohralík.  Equivalence between lowest-order mixed finite element and multi-point finite volume methods on simplicial meshes , 2006 .

[8]  Rüdiger Verfürth,et al.  Adaptive finite element methods for elliptic equations with non-smooth coefficients , 2000, Numerische Mathematik.

[9]  P. Raviart,et al.  A mixed finite element method for 2-nd order elliptic problems , 1977 .

[10]  Martin Vohralík,et al.  A Posteriori Error Estimates for Lowest-Order Mixed Finite Element Discretizations of Convection-Diffusion-Reaction Equations , 2007, SIAM J. Numer. Anal..

[11]  Rüdiger Verfürth,et al.  A posteriori error estimation and adaptive mesh-refinement techniques , 1994 .

[12]  Ivar Aavatsmark,et al.  Discretization on Unstructured Grids for Inhomogeneous, Anisotropic Media. Part I: Derivation of the Methods , 1998, SIAM J. Sci. Comput..

[13]  Thierry Gallouët,et al.  A cell-centred finite-volume approximation for anisotropic diffusion operators on unstructured meshes in any space dimension , 2006 .

[14]  R. Eymard,et al.  Finite Volume Methods , 2019, Computational Methods for Fluid Dynamics.

[15]  M. Vohralík.  On the Discrete Poincaré–Friedrichs Inequalities for Nonconforming Approximations of the Sobolev Space H 1 , 2005 .

[16]  Martin Petzoldt,et al.  A Posteriori Error Estimators for Elliptic Equations with Discontinuous Coefficients , 2002, Adv. Comput. Math..

[17]  M. Ohlberger,et al.  A posteriori error estimate for finite volume approximations to singularly perturbed nonlinear convection-diffusion equations , 2001, Numerische Mathematik.

[18]  Serge Nicaise,et al.  A Posteriori Error Estimations of Some Cell Centered Finite Volume Methods for Diffusion-Convection-Reaction Problems , 2006, SIAM J. Numer. Anal..

[19]  Yves Coudière,et al.  CONVERGENCE RATE OF A FINITE VOLUME SCHEME FOR A TWO DIMENSIONAL CONVECTION-DIFFUSION PROBLEM , 1999 .

[20]  Rüdiger Verfürth.  A posteriori error estimators for convection-diffusion equations , 1998, Numerische Mathematik.

[21]  Clint Dawson,et al.  Analysis of an Upwind-Mixed Finite Element Method for Nonlinear contaminant Transport Equations , 1998 .

[22]  Carsten Carstensen,et al.  A posteriori error estimate for the mixed finite element method , 1997, Math. Comput..

[23]  Serge Nicaise,et al.  A posteriori error estimations of some cell-centered finite volume methods , 2005, SIAM J. Numer. Anal..

[24]  Ivar Aavatsmark,et al.  Discretization on Unstructured Grids For Inhomogeneous, Anisotropic Media. Part II: Discussion And Numerical Results , 1998, SIAM J. Sci. Comput..