Skip to main content

Advertisement

SpringerLink
Log in

An \(hp\) finite element adaptive scheme to solve the Poisson problem on curved domains

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

Abstract

In this work, we introduce an \(hp\) finite element method for two-dimensional Poisson problems on curved domains using curved elements. We obtain a priori error estimates and define a local a posteriori error estimator of residual type. We show, under appropriate assumptions about the curved domain, the global reliability and the local efficiency of the estimator. More precisely, we prove that the estimator is equivalent to the energy norm of the error up to higher-order terms. The equivalence constant of the efficiency estimate depends on the polynomial degree. We also present an \(hp\) adaptive algorithm and several numerical tests which show the performance of the adaptive strategy.

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
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11
Fig. 12
Fig. 13
Fig. 14
Fig. 15
Fig. 16
Fig. 17
Fig. 18

Similar content being viewed by others

References

  • Ainsworth M, Senior B (1997) Aspects of an adaptive \(hp\) finite element method: adaptive strategy, conforming approximation and efficient solvers. Comput Methods Appl Mech Eng 150:65–87

    Article  MATH  MathSciNet  Google Scholar 

  • Ainsworth M, Senior B (1998) An adaptive refinement strategy for \(hp\)-finite element computations. Appl Numer Math 26:165–178

    Article  MATH  MathSciNet  Google Scholar 

  • Armentano MG, Padra C, Rodriguez R, Scheble M (2011) An hp finite element adaptive scheme to solve the Laplace model for fluid-solid vibrations. Comput Methods Appl Mech Eng 200(1–4):178–188

    Article  MathSciNet  Google Scholar 

  • Armentano MG, Padra C, Rodriguez R, Scheble M (2012) An hp finite element adaptive method to compute the vibration modes of a fluid-solid coupled system, CMES. Comput Model Eng Sci 84(4):359–382

    MathSciNet  Google Scholar 

  • Azaiez M, Deville MO, Gruber R, Mund EH (2008) A new \(hp\) method for the -grad(div) operator in non-Cartesian geometries. Appl Numer Math 58:985–998

    Article  MATH  MathSciNet  Google Scholar 

  • Babuška I, Suri M (1994) The P and H-P versions of the finite element method. Basci Princ Prop SIAM Rev 36:578–632

    Article  MATH  Google Scholar 

  • Babuška I, Suri M (1987) The \(h\)-\(p\) version of the finite element method with quasiuniform meshes. RAIRO Modél Math Anal Numér 21:199–238

    MATH  Google Scholar 

  • Boffi D, Costabel M, Dauge M, Demkowicz L (2006) Discrete compactness for the \(hp\) version of rectangular edge finite elements SIAM. J Numer Anal 44:979–1004

    Article  MATH  MathSciNet  Google Scholar 

  • Ciarlet PG (1978) The finite element method for elliptic problems, North Holland, Amsterdam

  • Ciarlet PG, Raviart PA (1972) Interpolation theory over curved elements with applications to finite element methods. Comput Methods Appl Mech Eng 1:217–249

    Article  MATH  MathSciNet  Google Scholar 

  • Demkowicz L (2007) Computing with \(hp\)-adaptive finite elements, Vol. 1. One and two dimensional elliptic and Maxwell problems. Chapman & Hall/CRC, Boca Raton

    Google Scholar 

  • Demkowicz L, Kurtz J, Pardo D, Paszyński M, Rachowicz W, Zdunek A (2008) Computing with \(hp\)-finite elements. Vol. 2. Frontiers: three-dimensional elliptic and maxwell problems with applications. Chapman & Hall/CRC, Boca Raton

  • Dorfler W, Heuveline V (2007) Convergence of an adaptive hp finite element strategy in one space dimension. Appl Numer Math 57:1108–1124

    Article  MathSciNet  Google Scholar 

  • García-Castillo LE, Pardo D, Gómez-Revuelto I, Demkowicz LF (2007) A two-dimensional self-adaptive \(hp\) finite element method for the characterization of waveguide discontinuities. Part I: Energy-norm based automatic \(hp\)-adaptivity. Comput Methods Appl Mech Eng 196:4823–4852

    Article  MATH  Google Scholar 

  • Girault V, Raviart PA (1986) Finite element methods for Navier–Stokes equations. Springer, Germany, Berlin

    Book  MATH  Google Scholar 

  • Grisvard P (1985) Elliptic problems in nonsmooth domain. Pitman, Boston

  • Guo BQ, Babuška I (1986) The h-p-version of the finite element method. Part 1. Comput Mech 1:21–41

    Article  MATH  Google Scholar 

  • Guo BQ, Babuška I (1986) The h-p-version of the finite element method. Part 1. Comput Mech 1:203–226

    Article  MATH  Google Scholar 

  • Melenk JM (2005) \(hp\)-interpolation of nonsmooth functions and an application to \(hp\)-a posteriori error estimation. SIAM J Numer Anal 43(1):127–155

    Article  MATH  MathSciNet  Google Scholar 

  • Melenk JM, Wohlmuth BI (2001) On residual-based a posteriori error estimation in hp-FEM. Adv Comput Math 15:311–331

    Article  MATH  MathSciNet  Google Scholar 

  • Oden JT, Wu W, Ainsworth M (1995) Three-step \(h\)-\(p\) adaptive strategy for the incompressible Navier–Stokes equations, modeling, mesh generation, and adaptive numerical methods for partial differential equations. IMA Math Appl 75:347–366

    MathSciNet  Google Scholar 

  • Oden JT, Demkowicz L, Rachowicz W, Westermann TA (1989) Toward a universal \(hp\) adaptive finite element strategy, part 2. A posteriori error estimation. Comput Methods Appl Mech Eng 77:113–180

    Article  MATH  MathSciNet  Google Scholar 

  • Oden JT, Patra A, Feng YS (1992) An \(hp\) adaptive strategy. In: Noor AK (ed) adaptive, multilevel and hierarchical computational strategies, AMD vol. 157, pp 23–46. ASME Publications

  • Pardo D, García-Castillo LE, Demkowicz LF, Torres-Verdín C (2007) A two-dimensional self-adaptive \(hp\) finite element method for the characterization of waveguide discontinuities. Part II: Goal-oriented \(hp\)-adaptivity. Comput Methods Appl Mech Eng 196:4811–4822

    Article  MATH  Google Scholar 

  • Scott R (1975) Interpolated boundary conditions in the finite element method. SIAM J Numer Anal 12:404–427

    Article  MATH  MathSciNet  Google Scholar 

  • Schwab Ch (1998) p- and hp- finite element methods, Theory and applications in solid and fluid mechanics (Numerical Mathematics and Scientific Computation). Oxford University Press

  • Tarancon JE, Fuenmayor FJ, Baeza L (2005) An a posteriori error estimator for the \(p\)- and \(hp\)-versions of the finite element method. Intern J Numer Methods Eng 62:1–18

    Article  MATH  MathSciNet  Google Scholar 

  • Verfürth R (1996) A review of a posteriori error estimation and adaptive mesh-refinement techniques. Wiley & Teubner, New York

  • Zlâmal M (1973) Curved elements in the finite element method I. SIAM J Numer Anal 10(1):229–240

    Article  MATH  MathSciNet  Google Scholar 

Download references

Acknowledgments

This work was partially supported by ANPCyT under grant PICT 2010-01675. The first author was partially supported by ANPCyT under grant PICT-2007-00910 and by Universidad de Buenos Aires under grant 20020100100143. The first and second authors are members of CONICET, Argentina. The authors thank Dr. Mariano Cantero for helpful discussions concerning this work.

Author information

Authors and Affiliations

  1. Departamento de Matemática, Facultad de Ciencias Exactas y Naturales, Universidad de Buenos Aires, IMAS-Conicet, 1428 , Buenos Aires, Argentina

    M. G. Armentano

  2. Centro Atómico Bariloche, 4800 , Bariloche, Argentina

    C. Padra & M. Scheble

Authors
  1. M. G. Armentano
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. C. Padra
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. M. Scheble
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to M. G. Armentano.

Additional information

Communicated by André Nachbin.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Armentano, M.G., Padra, C. & Scheble, M. An \(hp\) finite element adaptive scheme to solve the Poisson problem on curved domains. Comp. Appl. Math. 34, 705–727 (2015). https://doi.org/10.1007/s40314-014-0133-z

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s40314-014-0133-z

Keywords

Mathematics Subject Classification

Search

Navigation