On Efficient Numerical Solution of Linear Algebraic Systems Arising in Goal-Oriented Error Estimates

We deal with the numerical solution of linear partial differential equations (PDEs) with focus on the goal-oriented error estimates including algebraic errors arising by an inaccurate solution of the corresponding algebraic systems. The goal-oriented error estimates require the solution of the primal as well as dual algebraic systems. We solve both systems simultaneously using the bi-conjugate gradient method which allows to control the algebraic errors of both systems. We develop a stopping criterion which is cheap to evaluate and guarantees that the estimation of the algebraic error is smaller than the estimation of the discretization error. Using this criterion and an adaptive mesh refinement technique, we obtain an efficient and robust method for the numerical solution of PDEs, which is demonstrated by several numerical experiments.

[1]  Thomas Richter,et al.  Variational localizations of the dual weighted residual estimator , 2015, J. Comput. Appl. Math..

[2]  J. Oden,et al.  A Posteriori Error Estimation in Finite Element Analysis: Oden/A Posteriori , 2000 .

[3]  M. Arioli,et al.  A stopping criterion for the conjugate gradient algorithm in a finite element method framework , 2000, Numerische Mathematik.

[4]  S.,et al.  " Goal-Oriented Error Estimation and Adaptivity for the Finite Element Method , 1999 .

[5]  Pierre Gosselet,et al.  A strict error bound with separated contributions of the discretization and of the iterative solver in non-overlapping domain decomposition methods , 2014 .

[6]  Ralf Hartmann,et al.  Symmetric Interior Penalty DG Methods for the Compressible Navier-Stokes Equations II: Goal--Oriented A Posteriori Error Estimation , 2005 .

[7]  Sergey Korotov,et al.  A posteriori error estimation of goal-oriented quantities for elliptic type BVPs , 2006, International Conference of Computational Methods in Sciences and Engineering 2004 (ICCMSE 2004).

[8]  Rodolfo Bermejo,et al.  Anisotropic "Goal-Oriented" Mesh Adaptivity for Elliptic Problems , 2013, SIAM J. Sci. Comput..

[9]  Michael Woopen,et al.  Adjoint-based hp-adaptivity on anisotropic meshes for high-order compressible flow simulations , 2016 .

[10]  Vít Dolejší,et al.  Discontinuous Galerkin Method: Analysis and Applications to Compressible Flow , 2015 .

[11]  M. Picasso.  A stopping criterion for the conjugate gradient algorithm in the framework of anisotropic adaptive finite elements , 2009 .

[12]  Ricardo H. Nochetto,et al.  A safeguarded dual weighted residual method , 2008 .

[13]  M. Arioli,et al.  Interplay between discretization and algebraic computation in adaptive numerical solutionof elliptic PDE problems , 2013 .

[14]  J. Brandts,et al.  [Review of: W. Bangerth, R. Rannacher (2003) Adaptive finite element methods for solving differential equations.] , 2005 .

[15]  A. Greenbaum.  Estimating the Attainable Accuracy of Recursively Computed Residual Methods , 1997, SIAM J. Matrix Anal. Appl..

[16]  R. Fletcher.  Conjugate gradient methods for indefinite systems , 1976 .

[17]  Georg May,et al.  A Goal-Oriented High-Order Anisotropic Mesh Adaptation Using Discontinuous Galerkin Method for Linear Convection-Diffusion-Reaction Problems , 2019, SIAM J. Sci. Comput..

[18]  Rolf Rannacher,et al.  Goal-oriented error control of the iterative solution of finite element equations , 2009, J. Num. Math..

[19]  Vít Dolejsí,et al.  Efficient solution strategy for the semi-implicit discontinuous Galerkin discretization of the Navier-Stokes equations , 2011, J. Comput. Phys..

[20]  Dmitri Kuzmin,et al.  Goal-oriented mesh adaptation for flux-limited approximations to steady hyperbolic problems , 2010, J. Comput. Appl. Math..

[21]  Vít Dolejší,et al.  Goal-oriented error estimates including algebraic errors in discontinuous Galerkin discretizations of linear boundary value problems , 2017 .

[22]  Martin Vohralík,et al.  A Posteriori Error Estimates Including Algebraic Error and Stopping Criteria for Iterative Solvers , 2010, SIAM J. Sci. Comput..

[23]  L. Formaggia,et al.  Anisotropic mesh adaptation in computational fluid dynamics: application to the advection-diffusion-reaction and the Stokes problems , 2004 .

[24]  D. Darmofal,et al.  Review of Output-Based Error Estimation and Mesh Adaptation in Computational Fluid Dynamics , 2011 .

[25]  Pierre Gosselet,et al.  Strict bounding of quantities of interest in computations based on domain decomposition , 2015, ArXiv.

[26]  Paul Houston,et al.  Discontinuous Galerkin methods on hp-anisotropic meshes II: a posteriori error analysis and adaptivity , 2009 .

[27]  Leszek Demkowicz,et al.  Goal-oriented hp-adaptivity for elliptic problems , 2004 .

[28]  J. Oden,et al.  A Posteriori Error Estimation in Finite Element Analysis , 2000 .

[29]  Ralf Hartmann,et al.  Multitarget Error Estimation and Adaptivity in Aerodynamic Flow Simulations , 2008, SIAM J. Sci. Comput..

[30]  Frédéric Alauzet,et al.  Fully anisotropic goal-oriented mesh adaptation for 3D steady Euler equations , 2010, J. Comput. Phys..

[31]  Georg May,et al.  A goal-oriented anisotropic hp-mesh adaptation method for linear convection-diffusion-reaction problems , 2019, Comput. Math. Appl..

[32]  Richard Barrett,et al.  Templates for the Solution of Linear Systems: Building Blocks for Iterative Methods , 1994, Other Titles in Applied Mathematics.

[33]  Zdenek Strakos,et al.  On Efficient Numerical Approximation of the Bilinear Form c*A-1b , 2011, SIAM J. Sci. Comput..

[34]  Rüdiger Verfürth,et al.  A Posteriori Error Estimation Techniques for Finite Element Methods , 2013 .

[35]  Endre Süli,et al.  Adaptive finite element methods for differential equations , 2003, Lectures in mathematics.

[36]  Dmitri Kuzmin,et al.  Goal-oriented a posteriori error estimates for transport problems , 2010, Math. Comput. Simul..

[37]  Rolf Rannacher,et al.  An optimal control approach to a posteriori error estimation in finite element methods , 2001, Acta Numerica.

[38]  M. Giles,et al.  Adjoint methods for PDEs: a posteriori error analysis and postprocessing by duality , 2002, Acta Numerica.

[39]  Thomas Richter,et al.  A posteriori error estimation and anisotropy detection with the dual‐weighted residual method , 2010 .

[40]  Mark Ainsworth,et al.  Guaranteed computable bounds on quantities of interest in finite element computations , 2012 .

[41]  Martin Vohralík,et al.  Goal-oriented a posteriori error estimation for conforming and nonconforming approximations with inexact solvers , 2020, J. Comput. Appl. Math..