Abstract

In this paper, we design and analyze a hybrid high-order approximation for a class of quasilinear elliptic problems of nonmonotone type. The proposed method has several advantages; for instance, it supports an arbitrary order of approximation and general polytopal meshes. The key ingredients involve local reconstruction and high-order stabilization terms. The existence of a unique discrete solution is shown by using Brouwer's fixed point theorem and the contraction principle. A priori error estimation is derived in a discrete energy norm that shows optimal order of convergence. Numerical experiments are performed to substantiate the theoretical results.

Keywords

  1. hybrid high-order methods
  2. second-order quasilinear elliptic problems
  3. Brouwer's fixed point theorem
  4. error estimates

MSC codes

  1. 65N12
  2. 65N15
  3. 65N30

Get full access to this article

View all available purchase options and get full access to this article.

References

1.
P. F. Antonietti, N. Bigoni, and M. Verani, Mimetic finite difference approximation of quasilinear elliptic problems, Calcolo, 52 (2015), pp. 45--67.
2.
P. F. Antonietti, S. Giani, and P. Houston, $hp$-version composite discontinuous Galerkin methods for elliptic problems on complicated domains, SIAM J. Sci. Comput., 35 (2013), pp. A1417--A1439.
3.
R. Araya, C. Harder, D. Paredes, and F. Valentin, Multiscale hybrid-mixed method, SIAM J. Numer. Anal., 51 (2013), pp. 3505--3531.
4.
B. Ayuso de Dios, K. Lipnikov, and G. Manzini, The nonconforming virtual element method, ESAIM Math. Model. Numer. Anal., 50 (2016), pp. 879--904.
5.
F. Bassi, L. Botti, A. Colombo, D. A. Di Pietro, and P. Tesini, On the flexibility of agglomeration based physical space discontinuous Galerkin discretizations, J. Comput. Phys., 231 (2012), pp. 45--65.
6.
L. Beira͂o da Veiga, F. Brezzi, A. Cangiani, G. Manzini, L. D. Marini, and A. Russo, Basic principles of virtual element methods, Math. Models Methods Appl. Sci., 23 (2013), pp. 199--214.
7.
L. Beira͂o da Veiga, F. Brezzi, and L. D. Marini, Virtual elements for linear elasticity problems, SIAM J. Numer. Anal., 51 (2013), pp. 794--812.
8.
C. Bi and V. Ginting, Two-grid finite volume element method for linear and nonlinear elliptic problems, Numer. Math., 108 (2007), pp. 177--198.
9.
C. Bi and V. Ginting, A residual-type a posteriori error estimate of finite volume element method for a quasi-linear elliptic problem, Numer. Math., 114 (2009), pp. 107--132.
10.
C. Bi and V. Ginting, Two-grid discontinuous Galerkin method for quasi-linear elliptic problems, J. Sci. Comput., 49 (2011), pp. 311--331.
11.
C. Bi and V. Ginting, A posteriori error estimates of discontinuous Galerkin method for nonmonotone quasi-linear elliptic problems, J. Sci. Comput., 55 (2013), pp. 659--687.
12.
C. Bi and M. Liu, A discontinuous finite volume element method for second-order elliptic problems, Numer. Methods Partial Differential Equations, 28 (2012), pp. 425--440.
13.
C. Bi, C. Wang, and Y. Lin, A posteriori error estimates of two-grid finite element methods for nonlinear elliptic problems, J. Sci. Comput., 74 (2018), pp. 23--48.
14.
J. Bonelle and A. Ern, Analysis of compatible discrete operator schemes for elliptic problems on polyhedral meshes, ESAIM Math. Model. Numer. Anal., 48 (2014), pp. 553--581.
15.
M. Botti, D. A. Di Pietro, and P. Sochala, A hybrid high-order method for nonlinear elasticity, SIAM J. Numer. Anal., 55 (2017), pp. 2687--2717.
16.
F. Brezzi, R. S. Falk, and L. D. Marini, Basic principles of mixed virtual element methods, ESAIM Math. Model. Numer. Anal., 48 (2014), pp. 1227--1240.
17.
F. Brezzi, K. Lipnikov, and M. Shashkov, Convergence of the mimetic finite difference method for diffusion problems on polyhedral meshes, SIAM J. Numer. Anal., 43 (2005), pp. 1872--1896.
18.
F. Brezzi, K. Lipnikov, M. Shashkov, and V. Simoncini, A new discretization methodology for diffusion problems on generalized polyhedral meshes, Comput. Methods Appl. Mech. Engrg., 196 (2007), pp. 3682--3692.
19.
F. Brezzi, K. Lipnikov, and V. Simoncini, A family of mimetic finite difference methods on polygonal and polyhedral meshes, Math. Models Methods Appl. Sci., 15 (2005), pp. 1533--1551.
20.
E. Burman and A. Ern, An unfitted hybrid high-order method for elliptic interface problems, SIAM J. Numer. Anal., 56 (2018), pp. 1525--1546.
21.
G. Caloz and J. Rappaz, Numerical analysis for nonlinear and bifurcation problems, in Techniques of Scientific Computing (Part 2), P. G. Ciarlet and J. L. Lions, eds., Handb. Numer. Anal. V, North-Holland, Amsterdam, 1997, pp. 487--637.
22.
A. Cangiani, P. Chatzipantelidis, G. Diwan, and E. H. Georgoulis, Virtual element method for quasilinear elliptic problems, IMA J. Numer. Anal., 40 (2020), pp. 2450--2472.
23.
C. Carstensen, G. Mallik, and N. Nataraj, A priori and a posteriori error control of discontinuous Galerkin finite element methods for the von Kármán equations, IMA J. Numer. Anal., 39 (2019), pp. 167--200.
24.
P. Castillo, B. Cockburn, I. Perugia, and D. Schötzau, An a priori error analysis of the local discontinuous Galerkin method for elliptic problems, SIAM J. Numer. Anal., 38 (2000), pp. 1676--1706.
25.
M. Cicuttin, D. A. Di Pietro, and A. Ern, Implementation of discontinuous skeletal methods on arbitrary-dimensional, polytopal meshes using generic programming, J. Comput. Appl. Math., 344 (2018), pp. 852--874.
26.
M. Cicuttin, A. Ern, and T. Gudi, Hybrid high-order methods for the elliptic obstacle problem, J. Sci. Comput., 83 (2020), 8.
27.
B. Cockburn, D. A. Di Pietro, and A. Ern, Bridging the hybrid high-order and hybridizable discontinuous Galerkin methods, ESAIM Math. Model. Numer. Anal., 50 (2016), pp. 635--650.
28.
B. Cockburn, J. Gopalakrishnan, and R. Lazarov, Unified hybridization of discontinuous Galerkin, mixed, and continuous Galerkin methods for second order elliptic problems, SIAM J. Numer. Anal., 47 (2009), pp. 1319--1365.
29.
D. A. Di Pietro and J. Droniou, A hybrid high-order method for Leray-Lions elliptic equations on general meshes, Math. Comp., 86 (2017), pp. 2159--2191.
30.
D. A. Di Pietro and J. Droniou, The Hybrid High-Order Method for Polytopal Meshes: Design, Analysis, and Applications, Springer, Cham, 2020.
31.
D. A. Di Pietro, J. Droniou, and A. Ern, A discontinuous-skeletal method for advection-diffusion-reaction on general meshes, SIAM J. Numer. Anal., 53 (2015), pp. 2135--2157.
32.
D. A. Di Pietro, J. Droniou, and A. Harnist, Improved error estimates for hybrid high-order discretizations of Leray-Lions problems, Calcolo, 58 (2021), 19.
33.
D. A. Di Pietro, J. Droniou, and G. Manzini, Discontinuous skeletal gradient discretisation methods on polytopal meshes, J. Comput. Phys., 355 (2018), pp. 397--425.
34.
D. A. Di Pietro and A. Ern, Mathematical Aspects of Discontinuous Galerkin Methods, Math. Appl. (Berlin) 69, Springer, Heidelberg, 2012.
35.
D. A. Di Pietro and A. Ern, A hybrid high-order locking-free method for linear elasticity on general meshes, Comput. Methods Appl. Mech. Engrg., 283 (2015), pp. 1--21.
36.
D. A. Di Pietro, A. Ern, and S. Lemaire, An arbitrary-order and compact-stencil discretization of diffusion on general meshes based on local reconstruction operators, Comput. Methods Appl. Math., 14 (2014), pp. 461--472.
37.
D. A. Di Pietro, A. Ern, A. Linke, and F. Schieweck, A discontinuous skeletal method for the viscosity-dependent Stokes problem, Comput. Methods Appl. Mech. Engrg., 306 (2016), pp. 175--195.
38.
D. A. Di Pietro and S. Krell, A hybrid high-order method for the steady incompressible Navier-Stokes problem, J. Sci. Comput., 74 (2018), pp. 1677--1705.
39.
D. A. Di Pietro and S. Lemaire, An extension of the Crouzeix-Raviart space to general meshes with application to quasi-incompressible linear elasticity and Stokes flow, Math. Comp., 84 (2015), pp. 1--31.
40.
J. Douglas, Jr., and T. Dupont, A Galerkin method for a nonlinear Dirichlet problem, Math. Comp., 29 (1975), pp. 689--696.
41.
J. Douglas, Jr., T. Dupont, and J. Serrin, Uniqueness and comparison theorems for nonlinear elliptic equations in divergence form, Arch. Ration. Mech. Anal., 42 (1971), pp. 157--168.
42.
J. Droniou, Finite volume schemes for diffusion equations: Introduction to and review of modern methods, Math. Models Methods Appl. Sci., 24 (2014), pp. 1575--1619.
43.
J. Droniou and R. Eymard, A mixed finite volume scheme for anisotropic diffusion problems on any grid, Numer. Math., 105 (2006), pp. 35--71.
44.
J. Droniou, R. Eymard, T. Gallouët, C. Guichard, and R. Herbin, The Gradient Discretisation Method, Math. Appl. (Berlin) 82, Springer, Cham, 2018.
45.
J. Droniou, R. Eymard, T. Gallouët, and R. Herbin, A unified approach to mimetic finite difference, hybrid finite volume and mixed finite volume methods, Math. Models Methods Appl. Sci., 20 (2010), pp. 265--295.
46.
J. Droniou, R. Eymard, and R. Herbin, Gradient schemes: Generic tools for the numerical analysis of diffusion equations, ESAIM Math. Model. Numer. Anal., 50 (2016), pp. 749--781.
47.
J. Droniou and L. Yemm, Robust hybrid high-order method on polytopal meshes with small faces, Comput. Methods Appl. Math., 22 (2022), pp. 44--71.
48.
R. Eymard, T. Gallouët, and R. Herbin, Discretization of heterogeneous and anisotropic diffusion problems on general nonconforming meshes SUSHI: A scheme using stabilization and hybrid interfaces, IMA J. Numer. Anal., 30 (2010), pp. 1009--1043.
49.
D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, Classics Math., Springer, Berlin, 2001. Reprint of the 1998 edition.
50.
P. Grisvard, Elliptic Problems in Nonsmooth Domains, Monogr. Stud. Math. 24, Pitman, Boston, MA, 1985.
51.
P. Grisvard, Singularities in Boundary Value Problems, Rech. Math. Appl. 22, Masson, Paris, 1992.
52.
T. Gudi, N. Nataraj, and A. K. Pani, $hp$-discontinuous Galerkin methods for strongly nonlinear elliptic boundary value problems, Numer. Math., 109 (2008), pp. 233--268.
53.
T. Gudi, N. Nataraj, and A. K. Pani, An $hp$-local discontinuous Galerkin method for some quasilinear elliptic boundary value problems of nonmonotone type, Math. Comp., 77 (2008), pp. 731--756.
54.
T. Gudi and A. K. Pani, Discontinuous Galerkin methods for quasi-linear elliptic problems of nonmonotone type, SIAM J. Numer. Anal., 45 (2007), pp. 163--192.
55.
R. Herbin and F. Hubert, Benchmark on discretization schemes for anisotropic diffusion problems on general grids, in Finite Volumes for Complex Applications V, John Wiley & Sons, 2008, pp. 659--692.
56.
I. Hlaváček, M. Křížek, and J. Malý, On Galerkin approximations of a quasilinear nonpotential elliptic problem of a nonmonotone type, J. Math. Anal. Appl., 184 (1994), pp. 168--189.
57.
H. B. Keller, Approximation methods for nonlinear problems with application to two-point boundary value problems, Math. Comp., 29 (1975), pp. 464--474.
58.
Y. Kuznetsov, K. Lipnikov, and M. Shashkov, The mimetic finite difference method on polygonal meshes for diffusion-type problems, Comput. Geosci., 8 (2004), pp. 301--324.
59.
S. Lemaire, Bridging the hybrid high-order and virtual element methods, IMA J. Numer. Anal., 41 (2021), pp. 549--593.
60.
K. Lipnikov and G. Manzini, A high-order mimetic method on unstructured polyhedral meshes for the diffusion equation, J. Comput. Phys., 272 (2014), pp. 360--385.
61.
L. Liu, M. Křížek, and P. Neittaanmäki, Higher order finite element approximation of a quasilinear elliptic boundary value problem of a non-monotone type, Appl. Math., 41 (1996), pp. 467--478.
62.
G. Mallik and N. Nataraj, A nonconforming finite element approximation for the von Karman equations, ESAIM Math. Model. Numer. Anal., 50 (2016), pp. 433--454.
63.
F. A. Milner, Mixed finite element methods for quasilinear second-order elliptic problems, Math. Comp., 44 (1985), pp. 303--320.
64.
L. Mu, J. Wang, and X. Ye, Weak Galerkin finite element methods on polytopal meshes, Int. J. Numer. Anal. Model., 12 (2015), pp. 31--53.
65.
L. Song and Z. Zhang, Superconvergence property of an over-penalized discontinuous Galerkin finite element gradient recovery method, J. Comput. Phys., 299 (2015), pp. 1004--1020.
66.
J. Wang and X. Ye, A weak Galerkin finite element method for second-order elliptic problems, J. Comput. Appl. Math., 241 (2013), pp. 103--115.
67.
J. Wang and X. Ye, A weak Galerkin mixed finite element method for second order elliptic problems, Math. Comp., 83 (2014), pp. 2101--2126.
68.
J. Xu, Two-grid discretization techniques for linear and nonlinear PDEs, SIAM J. Numer. Anal., 33 (1996), pp. 1759--1777.

Information & Authors

Information

Published In

cover image SIAM Journal on Numerical Analysis
SIAM Journal on Numerical Analysis
Pages: 2318 - 2344
ISSN (online): 1095-7170

History

Submitted: 14 April 2021
Accepted: 6 June 2022
Published online: 30 August 2022

Keywords

  1. hybrid high-order methods
  2. second-order quasilinear elliptic problems
  3. Brouwer's fixed point theorem
  4. error estimates

MSC codes

  1. 65N12
  2. 65N15
  3. 65N30

Authors

Affiliations

Funding Information

Department of Science and Technology, Ministry of Science and Technology https://doi.org/10.13039/501100001409 : MATRICS, R(IA)/CVR- PDF/2020
National Board for Higher Mathematics https://doi.org/10.13039/501100005276 : 0204/58/2018/R&D-II/14746

Metrics & Citations

Metrics

Citations

If you have the appropriate software installed, you can download article citation data to the citation manager of your choice. Simply select your manager software from the list below and click Download.

Cited By

There are no citations for this item

Media

Figures

Other

Tables

Share

Share

Copy the content Link

Share with email

Email a colleague

Share on social media

The SIAM Publications Library now uses SIAM Single Sign-On for individuals. If you do not have existing SIAM credentials, create your SIAM account https://my.siam.org.