Hybrid high-order methods for variable-diffusion problems on general meshes. (Méthodes hybrides d’ordre élevé pour des problèmes à diffusion variable sur des maillages généraux.) (English. French summary) Zbl 1308.65196

Summary: We extend the hybrid high-order method introduced by the authors and S. Lemaire [Comput. Methods Appl. Math. 14, No. 4, 461–472 (2014; Zbl 1304.65248)] for the Poisson problem to problems with heterogeneous/anisotropic diffusion. The cornerstone is a local discrete gradient reconstruction from element- and face-based polynomial degrees of freedom. Optimal error estimates are proved.


65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
35J05 Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation
65N12 Stability and convergence of numerical methods for boundary value problems involving PDEs
65N15 Error bounds for boundary value problems involving PDEs


Zbl 1304.65248
Full Text: DOI HAL


[1] Beirão da Veiga, L.; Lipnikov, K.; Manzini, G., Arbitrary-order nodal mimetic discretizations of elliptic problems on polygonal meshes, SIAM J. Numer. Anal., 49, 5, 1737-1760, (2011) · Zbl 1242.65215
[2] Beirão da Veiga, L.; Brezzi, F.; Cangiani, A.; Manzini, G.; Marini, L. D.; Russo, A., Basic principles of virtual element methods, Math. Models Methods Appl. Sci., 23, 1, 199-214, (2013) · Zbl 1416.65433
[3] Bonelle, J.; Ern, A., Analysis of compatible discrete operator schemes for elliptic problems on polyhedral meshes, Math. Model. Numer. Anal., 48, 2, 553-581, (2014) · Zbl 1297.65132
[4] Di Pietro, D. A.; Ern, A., Mathematical aspects of discontinuous Galerkin methods, Mathématiques & Applications, vol. 69, (2012), Springer-Verlag Berlin · Zbl 1231.65209
[5] Di Pietro, D. A.; Ern, A., A hybrid high-order locking-free method for linear elasticity on general meshes, Comput. Methods Appl. Mech. Eng., 283, 1-21, (2015) · Zbl 1423.74876
[6] Di Pietro, D. A.; Ern, A., A family of arbitrary-order mixed methods for heterogeneous anisotropic diffusion on general meshes, (2014), submitted for publication, preprint hal-00918482
[7] Di Pietro, D. A.; Ern, A.; Lemaire, S., An arbitrary-order and compact-stencil discretization of diffusion on general meshes based on local reconstruction operators, Comput. Methods Appl. Math., 14, 4, 461-472, (2014) · Zbl 1304.65248
[8] Droniou, J.; Eymard, R.; Gallouët, T.; Herbin, R., A unified approach to mimetic finite difference, hybrid finite volume and mixed finite volume methods, Math. Models Methods Appl. Sci., 20, 2, 265-295, (2010) · Zbl 1191.65142
[9] Manzini, G.; Lipnikov, K., A high-order mimetic method on unstructured polyhedral meshes for the diffusion equation, J. Comput. Phys., 272, 360-385, (2014) · Zbl 1349.65581
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