Discontinuous Galerkin Methods for a Class of Nonvariational Problems

Expand
  • 1 Mathematics Institute, University of Warwick, Coventry CV4 7 AL, UK;
    2 Department of Mathematical Sciences, University of Bath, Bath BA2 7 AY, UK

Received date: 2020-09-22

  Revised date: 2021-03-01

  Online published: 2022-04-29

Abstract

We extend the fnite element method introduced by Lakkis and Pryer (SIAM J. Sci. Comput. 33(2):786-801, 2011) to approximate the solution of second-order elliptic problems in nonvariational form to incorporate the discontinuous Galerkin (DG) framework. This is done by viewing the "fnite element Hessian" as an auxiliary variable in the formulation. Representing the fnite element Hessian in a discontinuous setting yields a linear system of the same size and having the same sparsity pattern of the compact DG methods for variational elliptic problems. Furthermore, the system matrix is very easy to assemble; thus, this approach greatly reduces the computational complexity of the discretisation compared to the continuous approach. We conduct a stability and consistency analysis making use of the unifed frameworkset out in Arnold et al. (SIAM J. Numer. Anal. 39(5):1749-1779, 2001/2002). We also give an a posteriori analysis of the method in the case where the problem has a strong solution. The analysis applies to any consistent representation of the fnite element Hessian, and thus is applicable to the previous works making use of continuous Galerkin approximations. Numerical evidence is presented showing that the method works well also in a more general setting.

Cite this article

Andreas Dedner, Tristan Pryer . Discontinuous Galerkin Methods for a Class of Nonvariational Problems[J]. Communications on Applied Mathematics and Computation, 2022 , 4(2) : 634 -656 . DOI: 10.1007/s42967-021-00133-6

References

1.Agouzal, A., Vassilevski, Y.:On a discrete Hessian recovery for P1 fnite elements.J.Numer.Math.10(1), 1-12 (2002)
2.Aguilera, N.E., Morin, P.:On convex functions and the fnite element method.SIAM J.Numer.Anal.47(4), 3139-3157 (2009)
3.Alnæs, M.S., Logg, A., Ølgaard, K.B., Rognes, M.E., Wells, G.N.:Unifed form language:a domain-specifc language for weak formulations of partial diferential equations.CoRR (2012).arXiv:1211.4047
4.Arnold, D.N., Brezzi, F., Cockburn, B., Marini, L.D.:Unifed analysis of discontinuous Galerkin methods for elliptic problems.SIAM J.Numer.Anal.39(5), 1749-1779 (2002)
5.Barles, G., Souganidis, P.E.:Convergence of approximation schemes for fully nonlinear second order equations.Asymptot.Anal.4(3), 271-283 (1991)
6.Bastian, P., Blatt, M., Dedner, A., Engwer, C., Klöfkorn, R., Ohlberger, M., Sander, O.:A generic grid interface for parallel and adaptive scientifc computing.I.Abstract framework.Computing 82(2/3), 103-119 (2008)
7.Bastian, P., Blatt, M., Dedner, A., Engwer, C., Klöfkorn, R., Kornhuber, R., Ohlberger, M., Sander, O.:A generic grid interface for parallel and adaptive scientifc computing.II.Implementation and tests in DUNE.Computing 82(2/3), 121-138 (2008)
8.Böhmer, K.:On fnite element methods for fully nonlinear elliptic equations of second order.SIAM J.Numer.Anal.46(3), 1212-1249 (2008)
9.Brenner, S.C., Sung, L.-Y.:Virtual enriching operators.Calcolo 56(4), 1-25 (2019)
10.Bufa, A., Ortner, C.:Compact embeddings of broken Sobolev spaces and applications.IMA J.Numer.Anal.29(4), 827-855 (2009)
11.Burman, E., Ern, A.:Discontinuous Galerkin approximation with discrete variational principle for the nonlinear Laplacian.C.R.Math.Acad.Sci.Paris 346(17/18), 1013-1016 (2008)
12.Cordes, H.O.:Über die erste randwertaufgabe bei quasilinearen diferentialgleichungen zweiter ordnung in mehr als zwei variablen.Math.Ann.131(3), 278-312 (1956)
13.Dedner, A., Klöfkorn, R., Nolte, M.:Python bindings for the DUNE-FEM module.Zenodo (2020).https://doi.org/10.5281/zenodo.3706993
14.Dedner, A., Klöfkorn, R., Nolte, M., Ohlberger, M.:A generic interface for parallel and adaptive scientifc computing:abstraction principles and the DUNE-FEM module.Computing 90, 165-196 (2010)
15.Dedner, A., Nolte, M.:The Dune-Python Module.CoRR (2018).arXiv:1807.05252
16.Douglas, J.Jr., Dupont, T.:Interior penalty procedures for elliptic and parabolic Galerkin methods.In:Computing Methods in Applied Sciences (Second Internat.Sympos., Versailles, 1975).Lecture Notes in Phys., vol.58, pp.207-216.Springer, Berlin (1976)
17.Di Pietro, D.A., Ern, A.:Discrete functional analysis tools for discontinuous Galerkin methods with application to the incompressible Navier-Stokes equations.Math.Comput.79(271), 1303-1330 (2010)
18.Elman, H.C., Silvester, D.J., Wathen, A.J.:Finite elements and fast iterative solvers:with applications in incompressible fuid dynamics.In:Numerical Mathematics and Scientifc Computation.Oxford University Press, New York (2005)
19.Ern, A., Guermond, J.-L.:Theory and practice of fnite elements.In:Antman, S.S., Marsden, J.E., Sirovich, L.(eds) Applied Mathematical Sciences, vol.159.Springer, New York (2004)
20.Feng, X., Neilan, M.:Mixed fnite element methods for the fully nonlinear Monge-Ampère equation based on the vanishing moment method.SIAM J.Numer.Anal.47(2), 1226-1250 (2009)
21.Feng, X., Neilan, M.:Vanishing moment method and moment solutions for fully nonlinear second order partial diferential equations.J.Sci.Comput.38(1), 74-98 (2009)
22.Feng, X., Hennings, L., Neilan, M.:Finite element methods for second order linear elliptic partial differential equations in non-divergence form.Math.Comput.86(307), 2025-2051 (2017)
23.Feng, X., Neilan, M., Schnake, S.:Interior penalty discontinuous Galerkin methods for second order linear non-divergence form elliptic PDES.J.Sci.Comput.74(3), 1651-1676 (2018)
24.Gallistl, D.:Variational formulation and numerical analysis of linear elliptic equations in nondivergence form with Cordes coefcients.SIAM J.Numer.Anal.55(2), 737-757 (2017)
25.Gallistl, D.:Numerical approximation of planar oblique derivative problems in nondivergence form.Math.Comput.88(317), 1091-1119 (2019)
26.Georgoulis, E.H., Houston, P., Virtanen, J.:An a posteriori error indicator for discontinuous Galerkin approximations of fourth-order elliptic problems.IMA J.Numer.Anal.31(1), 281-298 (2011)
27.Gilbarg, D., Trudinger, N.S.:Elliptic Partial Diferential Equations of Second Order, 2nd edn.Springer, Berlin (1983)
28.Jensen, M., Smears, I.:On the convergence of fnite element methods for Hamilton-Jacobi-Bellman equations.Technical report, 01 (2011)
29.Kawecki, E.L.:A DGFEM for nondivergence form elliptic equations with Cordes coefcients on curved domains.Numer.Methods Partial Difer.Equations 35(5), 1717-1744 (2019)
30.Kawecki, E.L., Smears, I.:Convergence of adaptive discontinuous Galerkin and c0-interior penalty fnite element methods for Hamilton-Jacobi-Bellman and Isaacs equations (2020).arXiv:2006.07215
31.Lakkis, O., Mousavi, A.:A least-squares Galerkin approach to gradient and Hessian recovery for nondivergence-form elliptic equations (2019).arXiv:1909.00491
32.Lakkis, O., Pryer, T.:A fnite element method for second order nonvariational elliptic problems.SIAM J.Sci.Comput.33(2), 786-801 (2011)
33.Lakkis, O., Pryer, T.:A nonvariational fnite element method for fully nonlinear elliptic problems.Submitted-Tech report (2012).arXiv:1103.2970
34.Miranda, C.:Sulle equazioni ellittiche del secondo ordine di tipo non variazionale, a coefcienti discontinui.Ann.Mat.63(1), 353-386 (1963)
35.Mu, L., Ye, X.:A simple fnite element method for non-divergence form elliptic equations.Int.J.Numer.Anal.Model.14(2), 306-311 (2017)
36.Oberman, A.M.:Convergent diference schemes for degenerate elliptic and parabolic equations:Hamilton-Jacobi equations and free boundary problems.SIAM J.Numer.Anal.44(2), 879-895 (2006) (electronic)
37.Pryer, T.:Discontinuous Galerkin methods for the p-biharmonic equation from a discrete variational perspective.Electron.Trans.Numer.Anal.41, 328-349 (2014)
38.Smears, I., Süli, E.:Discontinuous Galerkin fnite element approximation of nondivergence form elliptic equations with Cordès coefcients.SIAM J.Numer.Anal.51(4), 2088-2106 (2013)
39.Talenti, G.:Sopra una classe di equazioni ellittiche a coefcienti misurabili.Ann.Mat.69(1), 285-304 (1965)
40.Vallet, M.-G., Manole, C.-M., Dompierre, J., Dufour, S., Guibault, F.:Numerical comparison of some Hessian recovery techniques.Int.J.Numer.Methods Eng.72(8), 987-1007 (2007)
Options
Outlines

/