A posteriori error estimation for an augmented mixed-primal method applied to sedimentation-consolidation systems

Abstract In this paper we develop the a posteriori error analysis of an augmented mixed-primal finite element method for the 2D and 3D versions of a stationary flow and transport coupled system, typically encountered in sedimentation–consolidation processes. The governing equations consist in the Brinkman problem with concentration-dependent viscosity, written in terms of Cauchy pseudo-stresses and bulk velocity of the mixture; coupled with a nonlinear advection – nonlinear diffusion equation describing the transport of the solids volume fraction. We derive two efficient and reliable residual-based a posteriori error estimators for a finite element scheme using Raviart–Thomas spaces of order k for the stress approximation, and continuous piecewise polynomials of degree ≤ k + 1 for both velocity and concentration. For the first estimator we make use of suitable ellipticity and inf–sup conditions together with a Helmholtz decomposition and the local approximation properties of the Clement interpolant and Raviart–Thomas operator to show its reliability, whereas the efficiency follows from inverse inequalities and localisation arguments based on triangle-bubble and edge-bubble functions. Next, we analyse an alternative error estimator, whose reliability can be proved without resorting to Helmholtz decompositions. Finally, we provide some numerical results confirming the reliability and efficiency of the estimators and illustrating the good performance of the associated adaptive algorithm for the augmented mixed-primal finite element method.

[1]  A note on stable Helmholtz decompositions in 3D , 2018, Applicable Analysis.

[2]  Martin Vohralík,et al.  A posteriori error estimates, stopping criteria, and adaptivity for two-phase flows , 2013, Computational Geosciences.

[3]  Raimund Bürger,et al.  Discontinuous approximation of viscous two-phase flow in heterogeneous porous media , 2016, J. Comput. Phys..

[4]  Jean E. Roberts,et al.  Mixed and hybrid methods , 1991 .

[5]  V. Girault,et al.  Vector potentials in three-dimensional non-smooth domains , 1998 .

[6]  C. Bernardi,et al.  A posteriori analysis of a space and time discretization of a nonlinear model for the flow in partially saturated porous media , 2014 .

[7]  Raimund Bürger,et al.  Convection-diffusion-reaction and transport-flow problems motivated by models of sedimentation: some recent advances , 2019 .

[8]  Gabriel N. Gatica,et al.  An augmented mixed-primal finite element method for a coupled flow-transport problem , 2015 .

[9]  Mary F. Wheeler,et al.  Local problem-based a posteriori error estimators for discontinuous Galerkin approximations of reactive transport , 2007 .

[10]  G. Gatica.  A note on the efficiency of residual-based a posteriori error estimators for some mixed finite element methods. , 2004 .

[11]  Gabriel N. Gatica,et al.  A Residual-Based A Posteriori Error Estimator for the Stokes-Darcy Coupled Problem , 2010, SIAM J. Numer. Anal..

[12]  G. Gatica.  A Simple Introduction to the Mixed Finite Element Method: Theory and Applications , 2014 .

[13]  A. I. Garralda-Guillem,et al.  A posteriori error analysis of twofold saddle point variational formulations for nonlinear boundary value problems , 2014 .

[14]  Vivette Girault,et al.  Finite Element Methods for Navier-Stokes Equations - Theory and Algorithms , 1986, Springer Series in Computational Mathematics.

[15]  M. Alvarez,et al.  A posteriori error analysis of a fully-mixed formulation for the Brinkman–Darcy problem , 2017 .

[16]  U. Schaflinger.  Experiments on sedimentation beneath downward-facing inclined walls , 1985 .

[17]  Thomas Richter,et al.  Solving Multidimensional Reactive Flow Problems with Adaptive Finite Elements , 2007 .

[18]  Gabriel N. Gatica,et al.  A priori and a posteriori error analyses of a velocity-pseudostress formulation for a class of quasi-Newtonian Stokes flows , 2011 .

[19]  Raimund Bürger,et al.  Discontinuous finite volume element discretization for coupled flow-transport problems arising in models of sedimentation , 2015, J. Comput. Phys..

[20]  Mats G. Larson,et al.  Goal oriented adaptivity for coupled flow and transport problems with applications in oil reservoir simulations , 2007 .

[21]  G. Gatica,et al.  Analysis of a velocity–pressure–pseudostress formulation for the stationary Stokes equations ☆ , 2010 .

[22]  Gabriel N. Gatica,et al.  Analysis of a pseudostress-based mixed finite element method for the Brinkman model of porous media flow , 2013, Numerische Mathematik.

[23]  Michel Fortin,et al.  Mixed and Hybrid Finite Element Methods , 2011, Springer Series in Computational Mathematics.

[24]  P. Clément.  Approximation by finite element functions using local regularization , 1975 .

[25]  Carsten Carstensen,et al.  A posteriori error estimates for mixed FEM in elasticity , 1998, Numerische Mathematik.

[26]  G. Gatica,et al.  A priori error analysis of a fully-mixed finite element method for a two-dimensional fluid-solid interaction problem , 2013 .

[27]  Ricardo Ruiz-Baier,et al.  Numerical solution of a multidimensional sedimentation problem using finite volume-element methods , 2015 .

[28]  Ricardo Ruiz-Baier,et al.  Partitioned coupling of advection-diffusion-reaction systems and Brinkman flows , 2017, J. Comput. Phys..

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

[30]  M. Alvarez,et al.  A mixed-primal finite element approximation of a sedimentation-consolidation system , 2016 .

[31]  Ricardo Ruiz-Baier,et al.  On a vorticity-based formulation for reaction-diffusion-Brinkman systems , 2018, Networks Heterog. Media.

[32]  Mats G. Larson,et al.  Adaptive finite element approximation of coupled flow and transport problems with applications in heat transfer , 2008 .

[33]  M. Alvarez,et al.  A posteriori error analysis for a viscous flow-transport problem , 2016 .

[34]  Gabriel N. Gatica,et al.  A priori and a posteriori error analyses of a pseudostress-based mixed formulation for linear elasticity , 2016, Comput. Math. Appl..

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

[36]  Ricardo Ruiz-Baier,et al.  Mixed finite element - discontinuous finite volume element discretization of a general class of multicontinuum models , 2016, J. Comput. Phys..

[37]  Martin Vohralík,et al.  A posteriori error estimates, stopping criteria, and adaptivity for multiphase compositional Darcy flows in porous media , 2014, J. Comput. Phys..

[38]  Carlos A. Vega,et al.  On Numerical Methods for Hyperbolic Conservation Laws and Related Equations Modelling Sedimentation of Solid-Liquid Suspensions , 2014 .

[39]  Raimund Bürger,et al.  A Stabilized Finite Volume Element Formulation for Sedimentation-Consolidation Processes , 2012, SIAM J. Sci. Comput..