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A unified analysis of elliptic problems with various boundary conditions and their approximation. (English) Zbl 07217139
Summary: We design an abstract setting for the approximation in Banach spaces of operators acting in duality. A typical example are the gradient and divergence operators in Lebesgue-Sobolev spaces on a bounded domain. We apply this abstract setting to the numerical approximation of Leray-Lions type problems, which include in particular linear diffusion. The main interest of the abstract setting is to provide a unified convergence analysis that simultaneously covers (i) all usual boundary conditions, (ii) several approximation methods. The considered approximations can be conforming (that is, the approximation functions can belong to the energy space relative to the problem) or not, and include classical as well as recent numerical schemes. Convergence results and error estimates are given. We finally briefly show how the abstract setting can also be applied to some models such as flows in fractured medium, elasticity equations and diffusion equations on manifolds.
MSC:
65J05 General theory of numerical analysis in abstract spaces
65N99 Numerical methods for partial differential equations, boundary value problems
47A58 Linear operator approximation theory
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[1] Andreianov, B.; Boyer, F.; Hubert, F., Besov regularity and new error estimates for finite volume approximations of the \(p\)-Laplacian, Numer. Math. 100 (2005), 565-592
[2] Andreianov, B.; Boyer, F.; Hubert, F., On the finite-volume approximation of regular solutions of the \(p\)-Laplacian, IMA J. Numer. Anal. 26 (2006), 472-502
[3] Andreianov, B.; Boyer, F.; Hubert, F., Discrete Besov framework for finite volume approximation of the \(p\)-Laplacian on non-uniform Cartesian grids, ESAIM Proc. 18 (2007), 1-10
[4] Andreianov, B.; Boyer, F.; Hubert, F., Discrete duality finite volume schemes for Leray-Lions-type elliptic problems on general 2D meshes, Numer. Methods Partial Differ. Equations 23 (2007), 145-195
[5] Antonietti, P. F.; Bigoni, N.; Verani, M., Mimetic finite difference approximation of quasilinear elliptic problems, Calcolo 52 (2015), 45-67
[6] Barrett, J. W.; Liu., W. B., Finite element approximation of the \(p\)-Laplacian, Math. Comput. 61 (1993), 523-537
[7] Barrett, J. W.; Liu, W. B., Finite element approximation of the parabolic \(p\)-Laplacian, SIAM J. Numer. Anal. 31 (1994), 413-428
[8] Barrett, J. W.; Liu, W. B., Quasi-norm error bounds for the finite element approximation of a non-Newtonian flow, Numer. Math. 68 (1994), 437-456
[9] Beurling, A.; Livingston, A. E., A theorem on duality mappings in Banach spaces, Ark. Mat. 4 (1962), 405-411
[10] Brenner, K.; Groza, M.; Guichard, C.; Lebeau, G.; Masson, R., Gradient discretization of hybrid dimensional Darcy flows in fractured porous media, Numer. Math. 134 (2016), 569-609
[11] Brezis, H., Functional Analysis, Sobolev Spaces and Partial Differential Equations, Universitext, Springer, New York (2011)
[12] Browder, F. E., On a theorem of Beurling and Livingston, Can. J. Math. 17 (1965), 367-372
[13] Browder, F. E.; Figueiredo, D. G. de, \(J\)-monotone nonlinear operators in Banach spaces, Djairo G. de Figueiredo. Selected Papers D. G. Costa Springer, Cham (2013), 1-9
[14] Burman, E.; Ern, A., Discontinuous Galerkin approximation with discrete variational principle for the nonlinear Laplacian, C. R. Math. Acad. Sci. Paris 346 (2008), 1013-1016
[15] P. G. Ciarlet; P. Ciarlet, Jr., Another approach to linearized elasticity and a new proof of Korn’s inequality, Math. Models Methods Appl. Sci. 15 (2005), 259-271
[16] Deimling, K., Nonlinear Functional Analysis, Springer, Berlin (1985)
[17] Pietro, D. A. Di; Droniou, J., A hybrid high-order method for Leray-Lions elliptic equations on general meshes, Math. Comput. 86 (2017), 2159-2191
[18] Droniou, J., Finite volume schemes for fully non-linear elliptic equations in divergence form, ESAIM Math. Model. Numer. Anal. 40 (2006), 1069-1100
[19] Droniou, J.; Eymard, R.; Gallouët, T.; Guichard, C.; Herbin, R., The Gradient Discretisation Method, Mathematics & Applications 82, Springer, Cham (2018)
[20] Eymard, R.; Gallouët, T.; Herbin, R., Cell centred discretisation of non linear elliptic problems on general multidimensional polyhedral grids, J. Numer. Math. 17 (2009), 173-193
[21] Eymard, R.; Guichard, C., Discontinuous Galerkin gradient discretisations for the approximation of second-order differential operators in divergence form, Comput. Appl. Math. 37 (2018), 4023-4054
[22] Glazyrina, L. L.; Pavlova, M. F., On an approximate solution method for the problem of surface and groundwater combined movement with exact approximation on the section line, Uch. Zap. Kazan. Univ., Ser. Fiz.-Mat. Nauki 158 (2016), 482-499 Russian
[23] Glowinski, R.; Rappaz, J., Approximation of a nonlinear elliptic problem arising in a non-Newtonian fluid flow model in glaciology, M2AN Math. Model. Numer. Anal. 37 (2003), 175-186
[24] Kato, T., Introduction to the theory of operators in Banach spaces, Perturbation Theory for Linear Operators Classics in Mathematics, Springer, Berlin (1995), 126-188
[25] Leray, J.; Lions, J.-L., Quelques résultats de Višik sur les problèmes elliptiques non linéaires par les méthodes de Minty-Browder, Bull. Soc. Math. Fr. 93 (1965), 97-107 French
[26] Lindenstrauss, J., On nonseparable reflexive Banach spaces, Bull. Am. Math. Soc. 72 (1966), 967-970
[27] Liu, W. B.; Barrett, J. W., A further remark on the regularity of the solutions of the \(p\)-Laplacian and its applications to their finite element approximation, Nonlinear Anal., Theory Methods Appl. 21 (1993), 379-387
[28] Liu, W. B.; Barrett, J. W., A remark on the regularity of the solutions of the \(p\)-Laplacian and its application to their finite element approximation, J. Math. Anal. Appl. 178 (1993), 470-487
[29] Minty, G. J., On a “monotonicity” method for the solution of nonlinear equations in Banach spaces, Proc. Natl. Acad. Sci. USA 50 (1963), 1038-1041
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