Preconditioners for the Discontinuous Galerkin time-stepping method of arbitrary order

We develop a preconditioner for systems arising from space-time finite element discretizations of parabolic equations. The preconditioner is based on a transformation of the coupled system into block diagonal form and an efficient solution strategy for the arising 2 × 2 blocks. The suggested strategy makes use of an inexact factorization of the Schur complement of these blocks, for which uniform bounds on the condition number can be proven. The main computational effort of the preconditioner lies in solving implicit Euler-like problems, which allows for the usage of efficient standard solvers. Numerical experiments are performed to corroborate our theoretical findings.

[1]  John C. Butcher,et al.  On the implementation of implicit Runge-Kutta methods , 1976 .

[2]  A. Bonito,et al.  A dG Approach to Higher Order ALE Formulations in Time , 2014 .

[3]  Laurent O. Jay,et al.  Inexact Simplified Newton Iterations for Implicit Runge-Kutta Methods , 2000, SIAM J. Numer. Anal..

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

[5]  Yvan Notay,et al.  Aggregation-Based Algebraic Multigrid for Convection-Diffusion Equations , 2012, SIAM J. Sci. Comput..

[6]  R. LeVeque.  High-resolution conservative algorithms for advection in incompressible flow , 1996 .

[7]  J. Guermond,et al.  Theory and practice of finite elements , 2004 .

[8]  Dominik Schötzau,et al.  hp-discontinuous Galerkin time stepping for parabolic problems , 2001 .

[9]  Bülent Karasözen,et al.  Variational time discretization methods for optimal control problems governed by diffusion-convection-reaction equations , 2014, J. Comput. Appl. Math..

[10]  Martin Stoll,et al.  All-at-once solution of time-dependent Stokes control , 2013, J. Comput. Phys..

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

[12]  Y. Notay.  An aggregation-based algebraic multigrid method , 2010 .

[13]  Ricardo H. Nochetto,et al.  Galerkin and Runge–Kutta methods: unified formulation, a posteriori error estimates and nodal superconvergence , 2011, Numerische Mathematik.

[14]  Kenneth Eriksson,et al.  Adaptive finite element methods for parabolic problems. I.: a linear model problem , 1991 .

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

[16]  Thomas Richter,et al.  Efficient numerical realization of discontinuous Galerkin methods for temporal discretization of parabolic problems , 2013, Numerische Mathematik.

[17]  L. Evans,et al.  Partial Differential Equations , 1941 .

[18]  R. Freund.  On conjugate gradient type methods and polynomial preconditioners for a class of complex non-hermitian matrices , 1990 .

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

[20]  Ricardo H. Nochetto,et al.  Time-Discrete Higher-Order ALE Formulations: Stability , 2013, SIAM J. Numer. Anal..

[21]  Artem Napov,et al.  An Algebraic Multigrid Method with Guaranteed Convergence Rate , 2012, SIAM J. Sci. Comput..

[22]  Roland W. Freund,et al.  Conjugate Gradient-Type Methods for Linear Systems with Complex Symmetric Coefficient Matrices , 1992, SIAM J. Sci. Comput..

[23]  Martin Stoll.  One-shot solution of a time-dependent time-periodic PDE-constrained optimization problem , 2014 .

[24]  Dominik Schötzau,et al.  Time Discretization of Parabolic Problems by the HP-Version of the Discontinuous Galerkin Finite Element Method , 2000, SIAM J. Numer. Anal..

[25]  P. Raviart,et al.  On a Finite Element Method for Solving the Neutron Transport Equation , 1974 .

[26]  Ricardo H. Nochetto,et al.  A posteriori error analysis for higher order dissipative methods for evolution problems , 2006, Numerische Mathematik.

[27]  Gene H. Golub,et al.  Matrix computations , 1983 .

[28]  R. Temam,et al.  Navier-Stokes equations: theory and numerical analysis: R. Teman North-Holland, Amsterdam and New York. 1977. 454 pp. US $45.00 , 1978 .

[29]  Laurent O. Jay,et al.  A parallelizable preconditioner for the iterative solution of implicit Runge-Kutta-type methods , 1999 .

[30]  Trygve K. Nilssen,et al.  Preconditioning of fully implicit Runge-Kutta schemes for parabolic PDEs , 2006 .

[31]  Kent-André Mardal,et al.  Order-Optimal Preconditioners for Implicit Runge-Kutta Schemes Applied to Parabolic PDEs , 2007, SIAM J. Sci. Comput..