Skip to main content
SpringerLink
Log in

Analysis of a goal-oriented adaptive two-grid finite-element algorithm for semilinear elliptic problems

  • Published:
Computational and Applied Mathematics Aims and scope Submit manuscript

Abstract

In this paper, we propose, analyze, and numerically validate a goal-oriented adaptive two-grid finite-element method for second-order semilinear elliptic problems. In this method, the \((k+1)\)th and the kth adaptive meshes are considered as the fine and coarse meshes. The proposed algorithm requires a one-step Newton correction for the primal problem, and applies a special treatment to the reaction term for the dual problem, which in turn leads to linear discrete primal and dual problems having the same coefficient matrix. Therefore, this algorithm is more efficient than goal-oriented adaptive finite-element methods based on the classical Newton iteration. We prove contraction properties of the primal quasi-error and the combined primal-dual quasi-error, from the latter of which the convergence theory of the proposed method is established, up to higher order primal \(L^2\)-norm error terms implicitly requiring the initial mesh to be sufficiently fine. Some numerical examples are shown to illustrate the effectiveness and efficiency of this algorithm.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4

Similar content being viewed by others

References

  • Axelsson O, Barker VA (2001) Finite element solution of boundary value problems: theory and computation. Soc Ind Appl Math 20:20

    MATH  Google Scholar 

  • Bangerth W, Rannacher R (2013) Adaptive finite element methods for differential equations. Birkhäuser, Basel

    MATH  Google Scholar 

  • Becker R, Innerberger M, Praetorius D (2021) Optimal convergence rates for goal-oriented FEM with quadratic goal functional. Comput Methods Appl Math 21(2):267–288

    Article  MathSciNet  Google Scholar 

  • Becker R, Rannacher R (2001) An optimal control approach to a posteriori error estimation in finite element methods. Acta Numer 10:1–102

    Article  MathSciNet  Google Scholar 

  • Bi C, Wang C, Lin Y (2018) A posteriori error estimates of two-grid finite element methods for nonlinear elliptic problems. J Sci Comput 74(1):23–48

    Article  MathSciNet  Google Scholar 

  • Brandts JH, Korotov S, Křížek M (2008) The discrete maximum principle for linear simplicial finite element approximations of a reaction–diffusion problem. Linear Algebra Appl 429(10):2344–2357

    Article  MathSciNet  Google Scholar 

  • Brenner S, Scott R (2007) The mathematical theory of finite element methods, vol 15. Springer, Berlin

    Google Scholar 

  • Bürg M, Nazarov M (2015) Goal-oriented adaptive finite element methods for elliptic problems revisited. J Comput Appl Math 287:125–147

    Article  MathSciNet  Google Scholar 

  • Cascon JM, Kreuzer C, Nochetto RH, Siebert KG (2008) Quasi-optimal convergence rate for an adaptive finite element method. SIAM J Numer Anal 46(5):2524–2550

    Article  MathSciNet  Google Scholar 

  • Ciarlet PG, Raviart PA (1973) Maximum principle and uniform convergence for the finite element method. Comput Methods Appl Mech Eng 2:17–31

    Article  MathSciNet  Google Scholar 

  • Ciarlet PG (2002) The finite element method for elliptic problems. Soc Ind Appl Math 20:20

    Google Scholar 

  • Feischl M, Praetorius D, Van der Zee KG (2016) An abstract analysis of optimal goal-oriented adaptivity. SIAM J Numer Anal 54(3):1423–1448

    Article  MathSciNet  Google Scholar 

  • Fidkowski KJ, Darmofal DL (2011) Review of output-based error estimation and mesh adaptation in computational fluid dynamics. AIAA J 49(4):673–694

    Article  Google Scholar 

  • Giles MB, Süli E (2002) Adjoint methods for pdes: a posteriori error analysis and postprocessing by duality. Acta Numer 11:145–236

    Article  MathSciNet  Google Scholar 

  • Heid P, Wihler TP (2020) Adaptive iterative linearization Galerkin methods for nonlinear problems. Math Comput 89(326):2707–2734

    Article  MathSciNet  Google Scholar 

  • Holst M, Pollock S (2016) Convergence of goal-oriented adaptive finite element methods for nonsymmetric problems. Numer Methods Partial Differ Equ 32(2):479–509

    Article  MathSciNet  Google Scholar 

  • Holst M, Pollock S, Zhu Y (2015) Convergence of goal-oriented adaptive finite element methods for semilinear problems. Comput Vis Sci 17(1):43–63

    Article  MathSciNet  Google Scholar 

  • Guanghui H, Meng X, Yi N (2016) Adjoint-based an adaptive finite volume method for steady Euler equations with non-oscillatory k-exact reconstruction. Comput Fluids 139:174–183

    Article  MathSciNet  Google Scholar 

  • Innerberger M, Praetorius D (2021) Instance-optimal goal-oriented adaptivity. Comput Methods Appl Math 21(1):109–126

    Article  MathSciNet  Google Scholar 

  • Kerkhoven T, Jerome JW (1990) \(L_\infty \) stability of finite element approximations to elliptic gradient equations. Numer Math 57(6–7):561–575

    Article  MathSciNet  Google Scholar 

  • Key K, Ovall J (2011) A parallel goal-oriented adaptive finite element method for 2.5-D electromagnetic modelling. Geophys J Int 186(1):137–154

    Article  Google Scholar 

  • Korotov S, Neittaanmäki P, Repin S (2003) A posteriori error estimation of goal-oriented quantities by the superconvergence patch recovery. J Numer Math 11(1):33–59

    Article  MathSciNet  Google Scholar 

  • Lakshmikantham V, Vatsala AS (1977) Elliptic partial differential equations of second order. Springer, Berlin

    Google Scholar 

  • Larsson F, Hansbo P, Runesson K (2002) Strategies for computing goal-oriented a posteriori error measures in non-linear elasticity. Int J Numer Methods Eng 55(8):879–894

    Article  MathSciNet  Google Scholar 

  • Li Y, Zhang Y (2021) Analysis of adaptive two-grid finite element algorithms for linear and nonlinear problems. SIAM J Sci Comput 43(2):A908–A928

    Article  MathSciNet  Google Scholar 

  • Mommer MS, Stevenson R (2009) A goal-oriented adaptive finite element method with convergence rates. SIAM J Numer Anal 47(2):861–886

    Article  MathSciNet  Google Scholar 

  • Oden JT, Prudhomme S (2001) Goal-oriented error estimation and adaptivity for the finite element method. Comput Math Appl 41(5–6):735–756

    Article  MathSciNet  Google Scholar 

  • Prudhomme JS, Oden T (1999) On goal-oriented error estimation for elliptic problems: application to the control of pointwise errors. Comput Methods Appl Mech Eng 176(1–4):313–331

    Article  MathSciNet  Google Scholar 

  • Scott LR, Zhang S (1990) Finite element interpolation of nonsmooth functions satisfying boundary conditions. Math Comput 54(190):483–493

    Article  MathSciNet  Google Scholar 

Download references

Acknowledgements

Li’s research was supported by Hunan Provincial Innovation Foundation for Postgraduate (CX20190462). Yi’s research was partially supported by NSFC Project (12071400), China’s National Key R&D Programs (2020YFA0713500), and Hunan Provincial NSF Project (2019JJ20016).

Author information

Authors and Affiliations

  1. Hunan Key Laboratory for Computation and Simulation in Science and Engineering, School of Mathematics and Computational Science, Xiangtan University, Xiangtan, 411105, Hunan, People’s Republic of China

    Fei Li & Nianyu Yi

Authors
  1. Fei Li
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Nianyu Yi
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Nianyu Yi.

Additional information

Communicated by Forrest Carpenter.

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Li, F., Yi, N. Analysis of a goal-oriented adaptive two-grid finite-element algorithm for semilinear elliptic problems. Comp. Appl. Math. 41, 108 (2022). https://doi.org/10.1007/s40314-022-01815-4

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s40314-022-01815-4

Keywords

Mathematics Subject Classification

Search

Navigation