A posteriori analysis for space-time, discontinuous in time Galerkin approximations for parabolic equations in a variable domain

Antonopoulou, Dimitra; Plexousakis, Michael

ESAIM: Mathematical Modelling and Numerical Analysis , Tome 53 (2019) no. 2, pp. 523-549.

Suite au passage du modèle économique de la revue en S20, le texte intégral des articles des années 2019 et 2020 est accessible uniquement sur le site de l'éditeur et est réservé aux abonnés.

This paper presents an a posteriori error analysis for the discontinuous in time space–time scheme proposed by Jamet for the heat equation in multi-dimensional, non-cylindrical domains Jamet (SIAM J. Numer. Anal. 15 (1978) 913–928). Using a Clément-type interpolant, we prove abstract a posteriori error bounds for the numerical error. Furthermore, in the case of two-dimensional spatial domains we transform the problem into an equivalent one, of parabolic type, with space-time dependent coefficients but posed on a cylindrical domain. We formulate a discontinuous in time space–time scheme and prove a posteriori error bounds of optimal order. The a priori estimates of Evans (American Mathematical Society (1998)) for general parabolic initial and boundary value problems are used in the derivation of the upper bound. Our lower bound coincides with that of Picasso (Comput. Meth. Appl. Mech. Eng. 167 (1998) 223–237), proposed for adaptive, Runge-Kutta finite element methods for linear parabolic problems. Our theoretical results are verified by numerical experiments.

[1] G. Akrivis, V.A. Dougalis and G.E. Zouraris, Finite difference schemes for the “Parabolic” equation in a variable depth environment with a rigid bottom boundary condition. SIAM J. Numer. Anal. 39 (2001) 539–565. | DOI | MR | Zbl

[2] G. Akrivis and C. Makridakis, Galerkin time-stepping methods for nonlinear parabolic equations. ESAIM: M2AN 38 (2004) 261–289. | DOI | Numdam | MR | Zbl

[3] R. Alexander, Diagonally Implicit Runge-Kutta methods for stiff O.D.E’.S. SINUM 14 (1977) 1006–1021. | DOI | MR | Zbl

[4] D.C. Antonopoulou, Theory and numerical analysis of parabolic approximations. Ph.D. thesis, Department of Mathematics, University of Athens, Greece (2006).

[5] D.C. Antonopoulou, V.A. Dougalis and G.E. Zouraris, Galerkin methods for Parabolic and Schrödinger Equations with dynamical boundary conditions and applications to underwater acoustics. SIAM J. Numer. Anal. 47 (2009) 2752–2781. | DOI | MR | Zbl

[6] D.C. Antonopoulou, G.D. Karali, M. Plexousakis and G.E. Zouraris, Crank-Nicolson finite element discretizations for a 2D linear Schrödinger-type equation posed in a noncylindrical domain. Math. Comput. 84 (2015) 1571–1598. | DOI | MR | Zbl

[7] D.C. Antonopoulou and M. Plexousakis, Discontinuous Galerkin methods for the linear Schrödinger equation in non-cylindrical domains. Numer. Math. 115 (2010) 585–608. | DOI | MR | Zbl

[8] I. Babuska, R. Duran and R. Rodriguez, Analysis of the efficiency of an a posteriori error estimator for linear triangular finite elements. SIAM J. Numer. Anal. 29 (1992) 947–964. | DOI | MR | Zbl

[9] M.J. Baines, Moving Finite Elements, Monographs on Numerical Analysis. The Clarendon Press, Oxford University Press, Oxford Science Publications, New York, NY (1994). | DOI | MR | Zbl

[10] J. Baranger and H. El-Amri, Estimateurs a posteriori d’ erreur pour le calcul adaptif d’écoulements quasi-newtoniens. RAIRO Model. Math. Anal. Numer. 25 (1991) 31–48. | DOI | Numdam | MR | Zbl

[11] C. Bernardi, Optimal finite element interpolation on curved domains. SIAM J. Numer. Anal. 26 (1989) 1212–1240. | DOI | MR | Zbl

[12] K.S. Bey, A. Patra and J.T. Oden, $hp$-version discontinuous Galerkin methods for hyperbolic conservation laws: a parallel strategy. Internat. J. Numer. Methods Eng. 38 (1995) 3889–3908. | DOI | MR | Zbl

[13] K.S. Bey, A. Patra and J.T. Oden, hp-version discontinuous Galerkin methods for hyperbolic conservation laws. Comput. Methods Appl. Mech. Eng. 133 (1996) 259–286. | DOI | MR | Zbl

[14] K.S. Bey, A. Patra and J.T. Oden, A parallel hp-adaptive discontinuous Galerkin method for hyperbolic conservation laws. Appl. Numer. Math. 20 (1996) 321–386. | DOI | MR | Zbl

[15] S.C. Brenner and L. Ridgway Scott, The Mathematical Theory of Finite Element Methods, Springer-Verlag, New York, NY (1991). | MR | Zbl

[16] P. Clément, Approximation by finite element functions using local regularization. RAIRO Anal. Numér. 9 (1975) 77–84. | Numdam | MR | Zbl

[17] B. Cockburn and C.W. Shu, The local discontinuous Galerkin method for time-dependent convection-diffusion systems. SIAM J. Numer. Anal. 35 (1998) 2440–2463. | DOI | MR | Zbl

[18] M. Crouzeix, Sur l’approximation des equations diffeŕentielles opeŕationnelles lineáires par des methodes de Runge Kutta, These, University Paris VI, Paris (1975).

[19] L.C. Evans, Partial Differential Equations, American Mathematical Society, Providence, RI (1998). | MR | Zbl

[20] G. Fichera, On a unified theory of boundary value problems for elliptic–parabolic equations of second order. In: Boundary Problems in Differential Equations. University of Wisconsin Press, Madison, WI (1960). | MR | Zbl

[21] A. Hiltebrand and S. Mishra, Entropy stable shock capturing space-time discontinuous Galerkin schemes for systems of Conservation Laws. Research Report No. 2012-07, ETHZ (2012). | MR | Zbl

[22] A. Hiltebrand, S. Mishra and C. Parés, Entropy-stable space-time DG schemes for non-conservative hyperbolic systems. ESAIM: M2AN 52 (2018) 995–1022. | DOI | Numdam | MR | Zbl

[23] P. Jamet, Numerical methods and existence theorems for parabolic differential equations whose coefficients are singular on the boundary. Math. Comput. 22 (1968) 721–743. | DOI | MR | Zbl

[24] P. Jamet, Estimation d’erreur pour des éléments finis droits presque dégénérés. Rev. Francaise Automat. Informat. Recherche Opérationelle Sér. Rouge Anal. Numér. 10 (1976) 43–61. | Numdam | MR | Zbl

[25] P. Jamet, Galerkin-type approximations which are discontinuous in time for parabolic equations in a variable domain. SIAM J. Numer. Anal. 15 (1978) 913–928. | DOI | MR | Zbl

[26] O. Karakashian and C. Makridakis, A space-time finite element method for the nonlinear Schrödinger equation: the discontinuous galerkin method. Math. Comput. 67 (1998) 479–499. | DOI | MR | Zbl

[27] O.A. Karakashian and F. Pascal, A posteriori error estimates for a Discontinuous Galerkin approximation of second-order Elliptic problems. SIAM J. Numer. Anal. 41 (2003) 2374–2399. | DOI | MR | Zbl

[28] C. Klaij, J.J.W.V. Der Vegt and H.V. Der Ven, Space-time discontinuous Galerkin method for the compressible Navier-Stokes equations. J. Comput. Phys. 217 (2006) 589–611. | DOI | MR | Zbl

[29] P. Lesaint and P.A. Raviart, On a finite element method for solving the neutron transport equation, edited by C. Deboor. In: Mathematical Aspects of Finite Elements in Partial Differential Equations. Academic Press, New York, NY (1974). | DOI | MR | Zbl

[30] J.L. Lions, Sur les problemes mixtes pour certains systemes paraboliques dans des ouverts non cylindriques. Ann. Inst. Fourier 7 (1957) 143–182. | DOI | Numdam | MR | Zbl

[31] C. Makridakis and R.H. Nochetto, A posteriori error analysis for higher order dissipative methods for evolution problems. Numer. Math. 104 (2006) 489–514. | DOI | MR | Zbl

[32] G. May and M. Zakerzadeh, On the convergence of space-time discontinuous galerkin schemes for scalar conservation laws. SIAM J. Numer. Anal. 54 (2016) 2452–2465. | DOI | MR | Zbl

[33] A.L. Mignot, Methodes d’approximation des solutions de certains problemes aux limites lineaires. Rend. Sem. Mat. Univ. Padova, XL 40 (1968) 1–138. | Numdam | MR | Zbl

[34] K. Miller, Moving finite elements, II. SINUM 18 (1981) 1033–1057. | DOI | MR | Zbl

[35] L. Pareschi and G. Russo, Implicit-explicit Runge-Kutta schemes and applications to hyperbolic systems with relaxation. J. Sci. Comput. 25 (2005) 129–155. | MR | Zbl

[36] M. Picasso, Adaptive finite elements for a linear parabolic problem. Comput. Meth. Appl. Mech. Eng. 167 (1998) 223–237. | DOI | MR | Zbl

[37] W.H. Reed and T.R. Hill, Triangular mesh methods for the neutron transport equation. Tech. Report LA-UR-73-479. Los Alamos Scientific Laboratory (1973).

[38] D. Schötzau and T.P. Wihler, A posteriori error estimation for $h$p-version time-stepping methods for parabolic partial differential equations. Numer. Math. 115 (2010) 475–509. | DOI | MR | Zbl

[39] M. Tavelli and M. Dumbser, A staggered space-time discontinuous Galerkin method for the three-dimensional incompressible Navier-Stokes equations on unstructured tetrahedral meshes. J. Comput. Phys. 319 (2016) 294–323. | DOI | MR | Zbl

[40] M. Tavelli and M. Dumbser, Arbitrary high order accurate space-time discontinuous Galerkin finite element schemes on staggered unstructured meshes for linear elasticity. J. Comput. Phy. 366 (2018) 386–414. | DOI | MR | Zbl

[41] J.J.W. Van Der Vegt and H. Van Der Ven, Space-time discontinuous Galerkin finite element method with dynamic grid motion for inviscid compressible flows I. general formulation. J. Comp. Phys. 182 (2002) 546–585. | MR | Zbl

[42] H. Van Der Ven and J.J.W. Van Der Vegt, Space-time discontinuous Galerkin finite element method with dynamic grid motion for inviscid compressible flows II. efficient flux quadrature. Comput. Methods Appl. Mech. Eng. 191 (2002) 4747–4780. | DOI | MR | Zbl

[43] R. Verfürth, error estimators for the Stokes equations. Numer. Math. 55 (1989) 309–325. | DOI | MR | Zbl