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Unified formulation and analysis of mixed and primal discontinuous skeletal methods on polytopal meshes. (English) Zbl 1402.65135

The authors consider discretizations of the Poisson equation subject to homogeneous Dirichlet boundary conditions in a bounded open connected polytopal set in \(\mathbb{R}^d\), \(d \geq 1\). Two families of discretizations are studied: the mixed and the primal formulation of discontinuous skeletal methods. The denotation “skeletal” comes from the feature that the set of degrees of freedom (short hand: DOF’s) corresponding to inside mesh elements is eliminated by static condensation and, as a consequence, only those on the mesh skeleton (which are responsible for the transmission of information) are left in the equations. The polynomials in skeletal methods are continuous in the interface points but discontinuous at vertices or edges, respectively.
Several methods existing in the literature are shown to belong to one of the skeletal discretizations. One aim of the paper is a unified formulation of the two skeletal discretizations. It is shown how to obtain from a given skeletal method in mixed formulation an equivalent method in primal formulation. The converse can be shown under an assumption on the kind of approximation of the gradients. A convergence analysis is given for the unified formulation providing optimal \(L^2\)- and energy-norm error bounds.

MSC:

65N08 Finite volume methods for boundary value problems involving PDEs
65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
65N12 Stability and convergence of numerical methods for boundary value problems involving PDEs
65N15 Error bounds for boundary value problems involving PDEs

Software:

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References:

[1] J. Aghili, S. Boyaval and D.A. Di Pietro, Hybridization of mixed high-order methods on general meshes and application to the Stokes equations. Comput. Meth. Appl. Math.15 (2015) 111-134. · Zbl 1311.76038
[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) A1417-A1439. · Zbl 1284.65163
[3] R. Araya, C. Harder, D. Paredes and F. Valentin, Multiscale hybrid-mixed method. SIAM J. Numer. Anal.51 (2013) 3505-3531. · Zbl 1296.65152
[4] T. Arbogast and Z. Chen, On the implementation of mixed methods as nonconforming methods for second-order elliptic problems. Math. Comput.64 (1995) 943-972. · Zbl 0829.65127
[5] T. Arbogast and M.R. Correa, Two Families of H(div) Mixed Finite Elements on Quadrilaterals of Minimal Dimension. SIAM J. Numer. Anal.54 (2016) 3332-3356. · Zbl 1353.65118
[6] D.N. Arnold, D. Boffi and R.S. Falk, Quadrilateral H(div) finite elements. SIAM J. Numer. Anal.42 (2005) 2429-2451. · Zbl 1086.65105
[7] D.N. Arnold and F. Brezzi, Mixed and nonconforming finite element methods: implementation, postprocessing and error estimates. RAIRO: M2AN19 (1985) 7-32. · Zbl 0567.65078
[8] B. Ayuso de Dios, K. Lipnikov and G. Manzini, The nonconforming virtual element method. ESAIM: M2AN50 (2016) 879-904. · Zbl 1343.65140
[9] C. Bahriawati and C. Carstensen, Three Matlab implementations of the lowest-order Raviart-Thomas MFEM with a posteriori error control. Comput. Meth. Appl. Math.5 (2005) 333-361. · Zbl 1086.65107
[10] 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) 45-65. · Zbl 1457.65178
[11] L. Beirão 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.199 (2013) 199-214. · Zbl 1416.65433
[12] L. Beirão da Veiga, F. Brezzi and L.D. Marini, Virtual elements for linear elasticity problems. SIAM J. Numer. Anal.2 (2013) 794-812. · Zbl 1268.74010
[13] L. Beirão da Veiga, F. Brezzi, L.D. Marini and A. Russo, H(div) and H(curl)-conforming VEM. Numer. Math.133 (2016) 303-332. · Zbl 1343.65133
[14] L. Beirão da Veiga, F. Brezzi, L.D. Marini and A. Russo, Mixed virtual element methods for general second order elliptic problems on polygonal meshes. ESAIM: M2AN50 (2016) 727-747. · Zbl 1343.65134
[15] L. Beirão da Veiga, K. Lipnikov and G. Manzini, The Mimetic Finite Difference Method for Elliptic Problems. Vol. 11 of Modeling, Simulation and Applications. Springer (2014). · Zbl 1286.65141
[16] D. Boffi, F. Brezzi and M. Fortin, Mixed finite element methods and applications. Vol. 44 of Springer Series in Computational Mathematics. Springer, Heidelberg (2013). · Zbl 1277.65092
[17] J. Bonelle and A. Ern, Analysis of compatible discrete operator schemes for elliptic problems on polyhedral meshes. ESAIM: M2AN48 (2014) 553-581. · Zbl 1297.65132
[18] F. Brezzi, A. Buffa and K. Lipnikov, Mimetic finite difference for elliptic problem. ESAIM: M2AN43 (2009) 277-295. · Zbl 1177.65164
[19] F. Brezzi, J. Douglas and L.D. Marini, Two families of mixed finite elements for second order elliptic problems. Numer. Math.47 (1985) 217-235. · Zbl 0599.65072
[20] F. Brezzi, R.S. Falk and L.D. Marini, Basic principles of mixed virtual element methods. ESAIM: M2AN48 (2014) 1227-1240. · Zbl 1299.76130
[21] 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) 1872-1896. · Zbl 1108.65102
[22] A. Cangiani, E. H. Georgoulis and P. Houston, hp-version discontinuous Galerkin methods on polygonal and polyhedral meshes. Math. Models Methods Appl. Sci.24 (2014) 2009-2041. · Zbl 1298.65167
[23] 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) 1676-1706. · Zbl 0987.65111
[24] Z. Chen, Equivalence between and multigrid algorithms for nonconforming and mixed methods for second-order elliptic problems. East-West J. Numer. Math.4 (1996) 1-33. · Zbl 0932.65126
[25] B. Cockburn, D.A. Di Pietro and A. Ern, Bridging the Hybrid High-Order and Hybridizable Discontinuous Galerkin methods. ESAIM: M2AN50 (2016) 635-650. · Zbl 1341.65045
[26] B. Cockburn and G. Fu, Superconvergence by M-decompositions. Part II: construction of two-dimensional finite elements. ESAIM: M2AN51 (2017) 165-186. · Zbl 1412.65205
[27] B. Cockburn and G. Fu, Superconvergence by M-decompositions. Part III: construction of three-dimensional finite elements. ESAIM: M2AN51 (2017) 365-398. · Zbl 1412.65137
[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) 1319-1365. · Zbl 1205.65312
[29] L. Codecasa, R. Specogna and F. Trevisan, A new set of basis functions for the discrete geometric approach. J. Comput. Phys.19 (2010) 7401-7410. · Zbl 1196.78027
[30] M. Crouzeix and P.-A. Raviart, Conforming and nonconforming finite element methods for solving the stationary Stokes equations. RAIRO: M2AN7 (1973) 33-75.
[31] D.A. Di Pietro, Cell centered Galerkin methods for diffusive problems. ESAIM: M2AN46 (2012) 111-144. · Zbl 1279.65125
[32] D.A. Di Pietro, On the conservativity of cell centered Galerkin methods. C. R. Acad. Sci Paris, Ser. I351 (2013) 155-159. · Zbl 1269.65123
[33] D.A. Di Pietro and J. Droniou, A Hybrid High-Order method for Leray-Lions elliptic equations on general meshes. Math. Comput.86 (2016) 2159-2191. · Zbl 1364.65224
[34] D.A. Di Pietro and J. Droniou, W\^{}{s,p}-approximation properties of elliptic projectors on polynomial spaces, with application to the error analysis of a Hybrid High-Order discretisation of Leray-Lions problems. Math. Models Methods Appl. Sci.27 (2017) 879-908. · Zbl 1365.65251
[35] 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) 2135-2157. · Zbl 1457.65194
[36] D.A. Di Pietro and A. Ern, Mathematical aspects of discontinuous Galerkin methods. Vol. 69 of Math. Appl. Springer-Verlag, Berlin (2012). · Zbl 1231.65209
[37] D.A. Di Pietro and A. Ern, A hybrid high-order locking-free method for linear elasticity on general meshes. Comput. Meth. Appl. Mech. Engrg.283 (2015) 1-21. · Zbl 1423.74876
[38] D.A. Di Pietro and A. Ern, Arbitrary-order mixed methods for heterogeneous anisotropic diffusion on general meshes. IMA J. Numer. Anal.37 (2016) 40-63. · Zbl 1433.65285
[39] 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. Meth. Appl. Math.14 (2014) 461-472. · Zbl 1304.65248
[40] D.A. Di Pietro, A. Ern and S. Lemaire, Building bridges: Connections and challenges in modern approaches to numerical partial differential equations, chapter A review of Hybrid High-Order methods: formulations, computational aspects, comparison with other methods. No 114 in Lect. Notes in Comput. Sci. Eng. Springer (2016) 205-236. · Zbl 1357.65258
[41] D.A. Di Pietro and R. Tittarelli, Numerical methods for PDEs. Lectures from the fall 2016 thematic quarter at Institut Henri Poincaré, chapter An introduction to Hybrid High-Order methods. SEMA SIMAI series. Springer (2017). Preprint [math.NA].
[42] J. Droniou and R. Eymard, A mixed finite volume scheme for anisotropic diffusion problems on any grid. Numer. Math.105 (2006) 35-71. · Zbl 1109.65099
[43] 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) 1-31. · Zbl 1191.65142
[44] J. Droniou, R. Eymard, T. Gallouet and R. Herbin, Gradient schemes: a generic framework for the discretisation of linear, nonlinear and nonlocal elliptic and parabolic equations. Math. Models Methods Appl. Sci.23 (2013) 2395-2432. · Zbl 1281.65136
[45] J. Droniou and N. Nataraj, Improved L\^{}{2} estimate for gradient schemes and super-convergence of the TPFA finite volume scheme. To appear in IMA J. Numer. Anal. (2017). Preprint [math.NA]. · Zbl 1477.65174
[46] T. Dupont and R. Scott, Polynomial approximation of functions in Sobolev spaces. Math. Comput.34 (1980) 441-463. · Zbl 0423.65009
[47] 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) 1009-1043. · Zbl 1202.65144
[48] R. Eymard, C. Guichard and R. Herbin, Small-stencil 3D schemes for diffusive flows in porous media. ESAIM: M2AN46 (2012) 265-290. · Zbl 1271.76324
[49] C. Lehrenfeld, Hybrid Discontinuous Galerkin methods for solving incompressible flow problems. Ph.D. thesis, Rheinisch-Westfälischen Technischen Hochschule Aachen (2010).
[50] K. Lipnikov and G. Manzini, A high-order mimetic method on unstructured polyhedral meshes for the diffusion equation. J. Comput. Phys.272 (2014) 360-385. · Zbl 1349.65581
[51] L.D. Marini, An inexpensive method for the evaluation of the solution of the lowest order Raviart-Thomas mixed method. SIAM J. Numer. Anal.22 (1985) 493-496. · Zbl 0573.65082
[52] P.A. Raviart and J.M. Thomas, A mixed finite element method for 2nd order elliptic problems. In Mathematical Aspects of the Finite Element Method, edited by I. Galligani and E. Magenes. Springer, New York (1977). · Zbl 0362.65089
[53] E. Tonti, On the formal structure of physical theories. Istituto di Matematica del Politecnico di Milano (1975).
[54] M. Vohralík and B.I. Wohlmuth, Mixed finite element methods: implementation with one unknown per element, local flux expressions, positivity, polygonal meshes, and relations to other methods. Math. Models Methods Appl. Sci.23 (2013) 803-838. · Zbl 1264.65198
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