Issue |
ESAIM: M2AN
Volume 50, Number 3, May-June 2016
Special Issue – Polyhedral discretization for PDE
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Page(s) | 851 - 877 | |
DOI | https://doi.org/10.1051/m2an/2015088 | |
Published online | 23 May 2016 |
- I. Aavatsmark, T. Barkve, O. Bœ and T. Mannseth, Discretization on unstructured grids for inhomogeneous, anisotropic media. Part i: Derivation of the methods. SIAM J. Sci. Comput. 19 (1998) 1700–1716. [CrossRef] [MathSciNet] [Google Scholar]
- I. Aavatsmark, T. Barkve, O. Bœ and T. Mannseth, Discretization on unstructured grids for inhomogeneous, anisotropic media. Part ii: Discussion and numerical results. SIAM J. Sci. Comput. 19 (1998) 1717–1736. [CrossRef] [MathSciNet] [Google Scholar]
- P. Antonietti, L. Beirão da Veiga, N. Bigoni and M. Verani, Mimetic finite differences for nonlinear and control problems. Math. Models Methods Appl. Sci. 24 (2014) 1457–1493. [Google Scholar]
- L. Beirão da Veiga, A mimetic discretization method for linear elasticity. ESAIM: M2AN 44 (2010) 231–250. [CrossRef] [EDP Sciences] [Google Scholar]
- 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. 23 (2013) 119–214. [Google Scholar]
- L. Beirão da Veiga, J. Droniou and G. Manzini, A unified approach to handle convection term in finite volumes and mimetic discretization methods for elliptic problems. IMA J. Num. Anal. 31 (2011) 1357–1401. [CrossRef] [MathSciNet] [Google Scholar]
- L. Beirão da Veiga, V. Gyrya, K. Lipnikov and G. Manzini, Mimetic finite difference method for the Stokes problem on polygonal meshes. J. Comput. Phys. 228 (2009) 7215–7232. [CrossRef] [MathSciNet] [Google Scholar]
- L. Beirão da Veiga, K. Lipnikov and G. Manzini, Convergence analysis of the high-order mimetic finite difference method. Numer. Math. 113 (2009) 325–356. [CrossRef] [MathSciNet] [Google Scholar]
- L. Beirão da Veiga, K. Lipnikov and G. Manzini, Arbitrary-order nodal mimetic discretizations of elliptic problems on polygonal meshes. SIAM J. Numer. Anal. 49 (2011) 1737–1760. [CrossRef] [Google Scholar]
- L. Beirão da Veiga, K. Lipnikov and G. Manzini, Error analysis for a mimetic discretization of the steady Stokes problem on polyhedral meshes. SIAM J. Numer. Anal. 48 (2011) 1419–1443. [CrossRef] [Google Scholar]
- L. Beirão da Veiga, K. Lipnikov and G. Manzini, The Mimetic Finite Difference Method. Vol. 11 of Model. Simul. Appl. 1st edition. Springer-Verlag, New York (2014). [Google Scholar]
- L. Beirão da Veiga, and G. Manzini, A higher-order formulation of the mimetic finite difference method. SIAM J. Sci. Comput. 31 (2008) 732–760. [CrossRef] [MathSciNet] [Google Scholar]
- L. Beirão da Veiga, and D. Mora, A mimetic discretization of the Reissner–Mindlin plate bending problem. Numer. Math. 117 (2011) 425–462. [CrossRef] [MathSciNet] [Google Scholar]
- P. Bochev and J.M. Hyman, Principle of mimetic discretizations of differential operators. Compatible discretizations. In Proc. of IMA hot topics workshop on compatible discretizations, edited by D. Arnold, P. Bochev, R. Lehoucq, R. Nicolaides and M. Shashkov. IMA. Springer-Verlag 142 (2006) 89–120. [Google Scholar]
- D. Boffi, F. Brezzi and M. Fortin, Mixed finite element methods and applications. Springer Series Comput. Math. Springer, Berlin, Heidelberg (2013). [Google Scholar]
- S. Brenner and L. Scott, The Mathematical Theory of Finite Element Methods. Springer-Verlag, Berlin/Heidelberg (1994). [Google Scholar]
- F. Brezzi, A. Buffa and K. Lipnikov, Mimetic finite differences for elliptic problems. ESAIM: M2AN 43 (2009) 277–295. [CrossRef] [EDP Sciences] [MathSciNet] [Google Scholar]
- F. Brezzi, A. Buffa and G. Manzini, Mimetic inner products for discrete differential forms. J. Comput. Phys. B 257 (2014) 1228–1259. [CrossRef] [MathSciNet] [Google Scholar]
- 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. [Google Scholar]
- J. Campbell and M. Shashkov, A tensor artificial viscosity using a mimetic finite difference algorithm. J. Comput. Phys. 172 (2001) 739–765. [CrossRef] [MathSciNet] [Google Scholar]
- A. Cangiani, F. Gardini and G. Manzini, Convergence of the mimetic finite difference method for eigenvalue problems in mixed form. Comput. Methods Appl. Mech. Engrg. 200 (2011) 1150–1160. [CrossRef] [MathSciNet] [Google Scholar]
- A. Cangiani and G. Manzini, Flux reconstruction and pressure post-processing in mimetic finite difference methods. Comput. Methods Appl. Mech. Engrg. 197 (2008) 933–945. [CrossRef] [MathSciNet] [Google Scholar]
- 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. [CrossRef] [MathSciNet] [Google Scholar]
- Y. Coudière and G. Manzini, The discrete duality finite volume method for convection-diffusion problems. SIAM J. Numer. Anal. 47 (2010) 4163–4192. [CrossRef] [MathSciNet] [Google Scholar]
- D.A. Di Pietro and A. Ern, Mathematical Aspects of Discontinuous Galerkin Methods. Mathématiques et Applications. Springer (2011). [Google Scholar]
- D.A. Di Pietro and A. Ern, Arbitrary-order mixed methods for heterogeneous anisotropic diffusion on general meshes, hal-00918482-v3 (2013). [Google Scholar]
- D.A. Di Pietro and A. Ern, Hybrid high-order methods for variable diffusion problems on general meshes. C. R. Math. 353 (2014) 31–34. [Google Scholar]
- K. Domelevo and P. Omnes, A finite volume method for the Laplace equation on almost arbitrary two-dimensional grids. ESAIM: M2AN 39 (2005) 1203–1249. [CrossRef] [EDP Sciences] [Google Scholar]
- J. Droniou, Finite volume schemes for diffusion equations: introduction to and review of modern methods. Math. Models Methods Appl. Sci. 24 (2014) 1575–1619. [CrossRef] [MathSciNet] [Google Scholar]
- 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) 265–295. [CrossRef] [MathSciNet] [Google Scholar]
- J. Droniou, R. Eymard, T. Gallouët 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. [Google Scholar]
- R. Eymard, T. Gallouët and R. Herbin, The finite volume method. In Handbook for Numerical Analysis, edited by P. Ciarlet and J.L. Lions. North Holland (2000) 715–1022. [Google Scholar]
- R. Eymard, T. Gallouet and R. Herbin, Discretization of heterogeneous and anisotropic diffusion problems on general non-conforming meshes. SUSHI: a scheme using stabilization and hybrid interface. IMA J. Numer. Anal. 30 (2010) 1009–1043. [CrossRef] [MathSciNet] [Google Scholar]
- V. Gyrya and K. Lipnikov, High-order mimetic finite difference method for diffusion problems on polygonal meshes. J. Comput. Phys. 227 (2008) 8841–8854. [CrossRef] [MathSciNet] [Google Scholar]
- F. Hermeline, Approximation of diffusion operators with discontinuous tensor coefficients on distorted meshes. Comput. Methods Appl. Mech. Engrg. 192 (2003) 1939–1959. [Google Scholar]
- J. Hyman and M. Shashkov, Mimetic discretizations for Maxwell’s equations and the equations of magnetic diffusion. Progress Electromagn. Res. 32 (2001) 89–121. [CrossRef] [Google Scholar]
- J.M. Hyman, M.J. Shashkov and S. Steinberg, The numerical solution of diffusion problems in strongly heterogeneous non-isotropic materials. J. Comput. Phys. 132 (1997) 130–148. [CrossRef] [MathSciNet] [Google Scholar]
- G. Lin, J. Liu and F. Sadre-Marandi, A comparative study on the weak Galerkin, discontinuous Galerkin, and mixed finite element methods. J. Comput. Appl. Math. 273 (2015) 346–362. [CrossRef] [MathSciNet] [Google Scholar]
- K. Lipnikov and G. Manzini, A high-order mimetic method on unstructured polyhedral meshes for the diffusion equation. J. Comput. Phys. 227 (2014) 360–385. [CrossRef] [Google Scholar]
- K. Lipnikov, G. Manzini, F. Brezzi and A. Buffa, The mimetic finite difference method for 3D magnetostatics fields problems. J. Comput. Phys. 230 (2011) 305–328. [CrossRef] [MathSciNet] [Google Scholar]
- K. Lipnikov, G. Manzini and M. Shashkov, Mimetic finite difference method. J. Comput. Phys. B 257 (2014) 1163–1227. [Google Scholar]
- K. Lipnikov, G. Manzini and D. Svyatskiy, Analysis of the monotonicity conditions in the mimetic finite difference method for elliptic problems. J. Comput. Phys. 230 (2011) 2620–2642. [Google Scholar]
- K.N. Lipnikov, J.D. Moulton, G. Manzini and M.J. Shashkov, The mimetic finite difference method for elliptic and parabolic problems with a staggered discretization of diffusion coefficient. Technical Report LA-UR-15-23755, Los Alamos National Laboratory, 2015. To appear in J. Comput. Phys. (2015). [Google Scholar]
- K. Lipnikov, J.D. Moulton and D. Svyatskiy, A Multilevel Multiscale Mimetic (M3) method for two-phase flows in porous media. J. Comp. Phys. 227 (2008) 6727–6753. [Google Scholar]
- K. Lipnikov, M. Shashkov and I. Yotov, Local flux mimetic finite difference methods. Numer. Math. 112 (2009) 115–152. [CrossRef] [MathSciNet] [Google Scholar]
- G. Manzini, A. Russo and N. Sukumar, New perspectives on polygonal and polyhedral finite element methods. Math. Models Methods Appl. Sci. 24 (2014) 1665–1699. [CrossRef] [MathSciNet] [Google Scholar]
- L. Margolin, M. Shashkov and P. Smolarkiewicz, A discrete operator calculus for finite difference approximations. Comput. Methods Appl. Mech. Engrg. 187 (2000) 365–383. [CrossRef] [MathSciNet] [Google Scholar]
- A. Palha, P.P. Rebelo, R. Hiemstra, J. Kreeft and M. Gerritsma, Physics-compatible discretization techniques on single and dual grids, with application to the Poisson equation of volume forms. J. Comput. Phys. B 257 (2014) 1394–1422. [CrossRef] [Google Scholar]
- N. Sukumar and E. Malsch, Recent advances in the construction of polygonal finite element interpolants. Arch. Comput. Methods Engrg. 13 (2006) 129–163. [CrossRef] [MathSciNet] [Google Scholar]
- N. Sukumar and A. Tabarraei, Conforming polygonal finite elements. Int. J. Numer. Meth. Engrg. 61 (2004) 2045–2066. [Google Scholar]
- E. Wachspress, A rational Finite Element Basis. Academic Press (1975). [Google Scholar]
- J. Wang and X. Ye, A weak Galerkin finite element method for second-order elliptic problems. J. Comput. Appl. Math. 241 (2013) 103–115. [CrossRef] [MathSciNet] [Google Scholar]
- J. Wang and X. Ye, A weak Galerkin mixed finite element method for second-order elliptic problems. Math. Comput. 83 (2014) 2101–2126. [Google Scholar]
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