Journal of Scientific Computing

, Volume 66, Issue 3, pp 1260–1280 | Cite as

Discontinuous Galerkin Approximation of Linear Parabolic Problems with Dynamic Boundary Conditions

  • P. F. Antonietti
  • M. Grasselli
  • S. Stangalino
  • M. Verani


In this paper we propose and analyze a Discontinuous Galerkin method for a linear parabolic problem with dynamic boundary conditions. We present the formulation and prove stability and optimal a priori error estimates for the fully discrete scheme. More precisely, using polynomials of degree \(p\ge 1\) on meshes with granularity h along with a backward Euler time-stepping scheme with time-step \(\Delta t\), we prove that the fully-discrete solution is bounded by the data and it converges, in a suitable (mesh-dependent) energy norm, to the exact solution with optimal order \(h^p + \Delta t\). The sharpness of the theoretical estimates are verified through several numerical experiments.


Discontinuous Galerkin Dynamic boundary conditions Parabolic problems Error estimates 



The authors thank the anonymous Referees for their valuable comments leading to an improvement of the presentation of the results. The first and the fourth author have been partially funded by INdAM - GNCS Project 2015 “Non-standard numerical methds for geophysics”. The second author is a member of the Gruppo Nazionale per l’Analisi Matematica, la Probabilità e le loro Applicazioni (GNAMPA) of the Istituto Nazionale di Alta Matematica. The fourth author has been also partially supported by the Italian research grant Prin 2012 2012HBLYE4 “Metodologie innovative nella modellistica differenziale numerica”.


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Copyright information

© Springer Science+Business Media New York 2015

Authors and Affiliations

  • P. F. Antonietti
    • 1
  • M. Grasselli
    • 2
  • S. Stangalino
    • 1
  • M. Verani
    • 1
  1. 1.MOX-Dipartimento di MatematicaPolitecnico di MilanoMilanItaly
  2. 2.Dipartimento di MatematicaPolitecnico di MilanoMilanItaly

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