Variational time discretization for mixed finite element approximations of nonstationary diffusion problems

We develop and study numerically two families of variational time discretization schemes for mixed finite element approximations applied to nonstationary diffusion problems. Continuous and discontinuous approximations of the time variable are encountered. The solution of the arising algebraic block system of equations by a Schur complement technique is described and an efficient preconditioner for the iterative solution process is constructed. The expected higher order rates of convergence are demonstrated in numerical experiments. Moreover, superconvergence properties are verified. Further, the efficiency and stability of the approaches are illustrated for a more sophisticated three-dimensional application of practical interest with discontinuous and anisotropic material properties.

[1]  Steffen Basting,et al.  Efficient preconditioning of variational time discretization methods for parabolic Partial Differential Equations , 2015 .

[2]  Todd Arbogast,et al.  A Nonlinear Mixed Finite Eelement Method for a Degenerate Parabolic Equation Arising in Flow in Porous Media , 1996 .

[3]  Peter Knabner,et al.  Order of Convergence Estimates for an Euler Implicit, Mixed Finite Element Discretization of Richards' Equation , 2004, SIAM J. Numer. Anal..

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

[5]  Peter Knabner,et al.  An Improved Optimal Order Mixed Finite Element Method for Semilinear Transport Problems , 2013 .

[6]  Stefan Turek,et al.  Higher order Galerkin time discretizations and fast multigrid solvers for the heat equation , 2011, J. Num. Math..

[7]  Stefan Turek,et al.  Higher Order Galerkin Time Discretization for Nonstationary Incompressible Flow , 2013 .

[8]  Yuanle Ma,et al.  Computational methods for multiphase flows in porous media , 2007, Math. Comput..

[9]  W. D. Evans,et al.  PARTIAL DIFFERENTIAL EQUATIONS , 1941 .

[10]  M. Bause.  Higher and lowest order mixed finite element approximation of subsurface flow problems with solutions of low regularity , 2008 .

[11]  Peter Knabner,et al.  Optimal order convergence of a modified BDM1 mixed finite element scheme for reactive transport in porous media , 2012 .

[12]  Zhangxin Chen.  Finite Element Methods And Their Applications , 2005 .

[13]  P. Knabner,et al.  Computation of variably saturated subsurface flow by adaptive mixed hybrid finite element methods , 2004 .

[14]  Ricardo H. Nochetto,et al.  Preconditioning a class of fourth order problems by operator splitting , 2011, Numerische Mathematik.

[15]  Carsten Burstedde,et al.  p4est: Scalable Algorithms for Parallel Adaptive Mesh Refinement on Forests of Octrees , 2011, SIAM J. Sci. Comput..

[16]  M. Celia,et al.  A General Mass-Conservative Numerical Solution for the Unsaturated Flow Equation , 1990 .

[17]  E. Hairer,et al.  Solving Ordinary Differential Equations II: Stiff and Differential-Algebraic Problems , 2010 .

[18]  Stefan Turek,et al.  A Note on Accurate and Efficient Higher Order Galerkin Time Stepping Schemes for the Nonstationary Stokes Equations , 2012 .

[19]  Harold A. Buetow,et al.  Adaptive finite element methods for differential equations , 2003, Lectures in mathematics.

[20]  Markus Bause,et al.  Variational Space–Time Methods for the Wave Equation , 2014, J. Sci. Comput..

[21]  Florin A. Radu,et al.  Error analysis for discretizations of parabolic problems using continuous finite elements in time and mixed finite elements in space , 2015, Numerische Mathematik.

[22]  Yousef Saad,et al.  Iterative methods for sparse linear systems , 2003 .

[23]  R. Helmig.  Multiphase Flow and Transport Processes in the Subsurface: A Contribution to the Modeling of Hydrosystems , 2011 .

[24]  Uwe Köcher,et al.  Variational Space-Time Methods for the Elastic Wave Equation and the Diffusion Equation , 2015 .

[25]  V. Thomée.  Galerkin Finite Element Methods for Parabolic Problems (Springer Series in Computational Mathematics) , 2010 .

[26]  Christopher E. Kees,et al.  Mixed finite element methods and higher-order temporal approximations , 2002 .

[27]  G. Matthies,et al.  Numerical studies of variational-type time-discretization techniques for transient Oseen problem , 2015 .

[28]  E. Hairer,et al.  Solving Ordinary Differential Equations II , 2010 .

[29]  Rolf Rannacher,et al.  Adaptive Galerkin Finite Element Methods for the Wave Equation , 2010, Comput. Methods Appl. Math..

[30]  G. Matthies,et al.  Numerical Studies of Galerkin-type Time-discretizations Applied to Transient Convection-diffusion-reaction Equations , 2012 .

[31]  Carol S. Woodward,et al.  Analysis of Expanded Mixed Finite Element Methods for a Nonlinear Parabolic Equation Modeling Flow into Variably Saturated Porous Media , 2000, SIAM J. Numer. Anal..

[32]  Friedhelm Schieweck,et al.  A-stable discontinuous Galerkin–Petrov time discretization of higher order , 2010, J. Num. Math..

[33]  Mark Frederick Hoemmen,et al.  An Overview of Trilinos , 2003 .

[34]  Joachim Hoffmann,et al.  First-order convergence of multi-point flux approximation on triangular grids and comparison with mixed finite element methods , 2010, Numerische Mathematik.

[35]  Peter Monk,et al.  Continuous finite elements in space and time for the heat equation , 1989 .

[36]  Sabine Attinger,et al.  Analysis of an Euler implicit‐mixed finite element scheme for reactive solute transport in porous media , 2009 .