Explicit and Averaging A Posteriori Error Estimates for Adaptive Finite Volume Methods

Local mesh-refining algorithms known from adaptive finite element methods are adopted for locally conservative and monotone finite volume discretizations of boundary value problems for steady-state convection-diffusion-reaction equations. The paper establishes residual-type explicit error estimators and averaging techniques for a posteriori finite volume error control with and without upwind in global H1- and L2-norms. Reliability and efficiency are verified theoretically and confirmed empirically with experimental support for the superiority of the suggested adaptive mesh-refining algorithms over uniform mesh refining. A discussion of adaptive computations in the simulation of contaminant concentration in a nonhomogeneous water reservoir concludes the paper.

[1]  Ilya D. Mishev,et al.  FINITE VOLUME METHODS ON VORONOI MESHES , 1998 .

[2]  Randolph E. Bank,et al.  A posteriori error estimates based on hierarchical bases , 1993 .

[3]  Stanimire Tomov,et al.  A Posteriori Error Estimates for Finite Volume Element Approximations of Convection–Diffusion–Reaction Equations , 2002 .

[4]  Lutz Angermann.  Balanced a posteriori error estimates for finite-volume type discretizations of convection-dominated elliptic problems , 2005, Computing.

[5]  I. Babuska,et al.  The finite element method and its reliability , 2001 .

[6]  Ronghua Li.  Generalized Difference Methods for Differential Equations: Numerical Analysis of Finite Volume Methods , 2000 .

[7]  D. Rose,et al.  Some errors estimates for the box method , 1987 .

[8]  C. Carstensen,et al.  Constants in Clément-interpolation error and residual based a posteriori estimates in finite element methods , 2000 .

[9]  C.A.J. Appelo,et al.  Flow and Transport , 2004, Introductory Incompressible Fluid Mechanics.

[10]  Carsten Carstensen,et al.  Fully Reliable Localized Error Control in the FEM , 1999, SIAM J. Sci. Comput..

[11]  Rolf Rannacher,et al.  A Feed-Back Approach to Error Control in Finite Element Methods: Basic Analysis and Examples , 1996 .

[12]  Zhiqiang Cai,et al.  On the finite volume element method , 1990 .

[13]  Carsten Carstensen,et al.  Each averaging technique yields reliable a posteriori error control in FEM on unstructured grids. Part I: Low order conforming, nonconforming, and mixed FEM , 2002, Math. Comput..

[14]  L. R. Scott,et al.  The Mathematical Theory of Finite Element Methods , 1994 .

[15]  池田 勉.  Maximum principle in finite element models for convection-diffusion phenomena , 1983 .

[16]  G. Dagan.  Flow and transport in porous formations , 1989 .

[17]  R. Rodríguez.  Some remarks on Zienkiewicz‐Zhu estimator , 1994 .

[18]  W. Rheinboldt,et al.  Error Estimates for Adaptive Finite Element Computations , 1978 .

[19]  P. Knabner,et al.  ADAPTIVE FINITE VOLUME DISCRETIZATION OF DENSITY DRIVEN FLOWS IN POROUS MEDIA , 1998 .

[20]  Lutz Tobiska,et al.  Numerical Methods for Singularly Perturbed Differential Equations , 1996 .

[21]  Lutz Angermann.  An a posteriori estimation for the solution of elliptic boundary value problems by means of upwind FEM , 1992 .

[22]  Qian Li,et al.  Error estimates in L2, H1 and Linfinity in covolume methods for elliptic and parabolic problems: A unified approach , 1999, Math. Comput..

[23]  Carsten Carstensen,et al.  Each averaging technique yields reliable a posteriori error control in FEM on unstructured grids. Part II: Higher order FEM , 2002, Math. Comput..

[24]  Willy Dörfler,et al.  An adaptive finite element method for a linear elliptic equation with variable coefficients , 2000 .

[25]  Douglas N. Arnold,et al.  Locally Adapted Tetrahedral Meshes Using Bisection , 2000, SIAM Journal on Scientific Computing.

[26]  Kenneth Eriksson,et al.  An adaptive finite element method for linear elliptic problems , 1988 .

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

[28]  A. Samarskii.  The Theory of Difference Schemes , 2001 .

[29]  Alfio Quarteroni,et al.  Domain Decomposition Methods for Partial Differential Equations , 1999 .

[30]  O. C. Zienkiewicz,et al.  A simple error estimator and adaptive procedure for practical engineerng analysis , 1987 .

[31]  Carsten Carstensen,et al.  A posteriori error control in low-order finite element discretisations of incompressible stationary flow problems , 2001, Math. Comput..

[32]  O. C. Zienkiewicz,et al.  Adaptivity and mesh generation , 1991 .

[33]  Rolf Rannacher,et al.  Some Optimal Error Estimates for Piecewise Linear Finite Element Approximations , 1982 .

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

[35]  Peter Knabner,et al.  An error estimator for a finite volume discretization of density driven flow in porous media , 1998 .