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Mimetic scalar products of discrete differential forms.(English)Zbl 1352.65417

Summary: We propose a strategy for the systematic construction of the mimetic inner products on cochain spaces for the numerical approximation of partial differential equations on unstructured polygonal and polyhedral meshes. The mimetic inner products are locally built in a recursive way on each \(k\)-cell and, then, globally assembled. This strategy is similar to the implementation of the finite element methods. The effectiveness of this approach is documented by deriving mimetic discretizations and testing their behavior on a set of problems related to the Maxwell equations.

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65N06 Finite difference methods for boundary value problems involving PDEs

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Triangle

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[1]Aarnes, J. E.; Krogstad, S.; Lie, K.-A., Multiscale mixed/mimetic methods on corner-point grids, Comput. Geosci., 12, 3, 297-315 (2007) ·Zbl 1259.76065
[2]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. Equ., 23, 1, 145-195 (2007) ·Zbl 1111.65101
[3]Antonietti, P. F.; Beirão da Veiga, L.; Verani, M., Hierarchical a posteriori error estimators for the mimetic discretization of elliptic problems, SIAM J. Numer. Anal., 51, 654-675 (2013) ·Zbl 1267.65159
[4]Antonietti, P. F.; Beirão da Veiga, L.; Verani, M., A mimetic discretization of elliptic obstacle problems, Math. Comput., 82, 1379-1400 (2013) ·Zbl 1271.65136
[5]Arnold, D. N.; Falk, R. S.; Winther, R., Finite element exterior calculus, homological techniques, and applications, Acta Numer., 15, 1-155 (2006) ·Zbl 1185.65204
[6]Arnold, D. N.; Falk, R. S.; Winther, R., Finite element exterior calculus: from Hodge theory to numerical stability, Bull. Am. Math. Soc. (N.S.), 47, 2, 281-354 (2010) ·Zbl 1207.65134
[7]Batista, E. D.; Castillo, J. E., Mimetic schemes on non-uniform structured meshes, Electron. Trans. Numer. Anal., 34, 152-162 (2009) ·Zbl 1188.65112
[8]Beirão da Veiga, L., A residual based error estimator for the mimetic finite difference method, Numer. Math., 108, 3, 387-406 (2008) ·Zbl 1144.65067
[9]Beirão da Veiga, L., A mimetic discretization method for linear elasticity, Math. Model. Numer. Anal., 44, 2, 231-250 (2010) ·Zbl 1258.74206
[10]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, 01, 199-214 (2013) ·Zbl 1416.65433
[11]Beirão da Veiga, L.; Droniou, J.; Manzini, G., A unified approach to handle convection term in finite volumes and mimetic discretization methods for elliptic problems, IMA J. Numer. Anal., 31, 4, 1357-1401 (2011) ·Zbl 1263.65102
[12]Beirão da Veiga, L.; Gyrya, V.; Lipnikov, K.; Manzini, G., Mimetic finite difference method for the Stokes problem on polygonal meshes, J. Comput. Phys., 228, 19, 7215-7232 (2009) ·Zbl 1172.76032
[13]Beirão da Veiga, L.; Lipnikov, K., A mimetic discretization of the Stokes problem with selected edge bubbles, SIAM J. Sci. Comput., 32, 2, 875-893 (2010) ·Zbl 1352.76021
[14]Beirão da Veiga, L.; Lipnikov, K.; Manzini, G., Convergence analysis of the high-order mimetic finite difference method, Numer. Math., 113, 3, 325-356 (2009) ·Zbl 1183.65132
[15]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
[16]Beirão da Veiga, L.; Lipnikov, K.; Manzini, G., Error analysis for a mimetic discretization of the steady Stokes problem on polyhedral meshes, SIAM J. Numer. Anal., 48, 1419-1443 (2011) ·Zbl 1352.76022
[17]Beirão da Veiga, L.; Manzini, G., An a posteriori error estimator for the mimetic finite difference approximation of elliptic problems, Int. J. Numer. Methods Eng., 76, 11, 1696-1723 (2008) ·Zbl 1195.65146
[18]Beirão da Veiga, L.; Manzini, G., A higher-order formulation of the mimetic finite difference method, SIAM J. Sci. Comput., 31, 1, 732-760 (2008) ·Zbl 1185.65201
[19]Beirão da Veiga, L.; Mora, D., A mimetic discretization of the Reissner-Mindlin plate bending problem, Numer. Math., 117, 3, 425-462 (2011) ·Zbl 1428.74132
[20]Bertolazzi, E.; Manzini, G., A second-order maximum principle preserving finite volume method for steady convection-diffusion problems, SIAM J. Numer. Anal., 43, 5, 2172-2199 (2005) ·Zbl 1145.65326
[21]Bertolazzi, E.; Manzini, G., On vertex reconstructions for cell-centered finite volume approximations of 2-D anisotropic diffusion problems, Math. Models Methods Appl. Sci., 17, 1, 1-32 (2007) ·Zbl 1119.65115
[22]Bochev, P.; Hyman, J. M., Principle of mimetic discretizations of differential operators, (Arnold, D.; Bochev, P.; Lehoucq, R.; Nicolaides, R.; Shashkov, M., Compatible Discretizations. Proceedings of IMA Hot Topics Workshop on Compatible Discretizations. Compatible Discretizations. Proceedings of IMA Hot Topics Workshop on Compatible Discretizations, The IMA Volumes in Mathematics and its Applications Series, vol. 142 (2006), Springer-Verlag), 89-120 ·Zbl 1110.65103
[23]Boffi, D.; Fernandes, P.; Gastaldi, L.; Perugia, I., Computational models of electromagnetic resonators: analysis of edge element approximation, SIAM J. Numer. Anal., 36, 1264-1290 (1999) ·Zbl 1025.78014
[24]Bossavit, A., Mixed finite elements and the complex of Whitney forms, (The Mathematics of Finite Elements and Applications, VI. The Mathematics of Finite Elements and Applications, VI, Uxbridge, 1987 (1988), Academic Press: Academic Press London), 137-144 ·Zbl 0692.65053
[25]Bossavit, A., Whitney forms: a class of finite elements for three dimensional computation in electromagnetism, IEE Proc., 135, 493-500 (1988)
[26]Bossavit, A., Differential forms and the computation of fields and forces in electromagnetism, Eur. J. Mech. B, Fluids, 10, 5, 474-488 (1991) ·Zbl 0741.76088
[27]Bossavit, A., Mixed methods and the marriage between “mixed” finite elements and boundary elements, Numer. Methods Partial Differ. Equ., 7, 4, 347-362 (1991) ·Zbl 0744.65086
[28]Bossavit, A., Computational Electromagnetism (1998), Academic Press: Academic Press San Diego, CA ·Zbl 0945.78001
[29]Brezzi, F.; Buffa, A., Innovative mimetic discretizations for electromagnetic problems, J. Comput. Appl. Mech., 234, 6, 1980-1989 (2010) ·Zbl 1191.78056
[30]Brezzi, F.; Fortin, M., Mixed and Hybrid Finite Element Methods (1991), Springer-Verlag ·Zbl 0788.73002
[31]Brezzi, F.; Lipnikov, K.; Shashkov, M., Convergence of the mimetic finite difference method for diffusion problems on polyhedral meshes, SIAM J. Numer. Anal., 43, 5, 1872-1896 (2005) ·Zbl 1108.65102
[32]Brezzi, F.; Lipnikov, K.; Simoncini, V., A family of mimetic finite difference methods on polygonal and polyhedral meshes, Math. Models Methods Appl. Sci., 15, 10, 1533-1551 (2005) ·Zbl 1083.65099
[33]Buffa, A., Remarks on the discretization of some noncoercive operator with applications to heterogeneous Maxwell equations, SIAM J. Numer. Anal., 43, 1, 1-18 (2006) ·Zbl 1128.78010
[34]Buffa, A.; Christiansen, S. H., A dual finite element complex on the barycentric refinement, Math. Comput., 76, 260, 1743-1769 (2007) ·Zbl 1130.65108
[35]Campbell, J.; Shashkov, M., A tensor artificial viscosity using a mimetic finite difference algorithm, J. Comput. Phys., 172, 739-765 (2001) ·Zbl 1002.76082
[36]Cangiani, A.; Gardini, F.; Manzini, G., Convergence of the mimetic finite difference method for eigenvalue problems in mixed form, Comput. Methods Appl. Mech. Eng., 200, 9-12, 1150-1160 (2011) ·Zbl 1225.65106
[37]Cangiani, A.; Manzini, G., Flux reconstruction and pressure post-processing in mimetic finite difference methods, Comput. Methods Appl. Mech. Eng., 197, 9-12, 933-945 (2008) ·Zbl 1169.76404
[38]Cangiani, A.; Manzini, G.; Russo, A., Convergence analysis of the mimetic finite difference method for elliptic problems, SIAM J. Numer. Anal., 47, 4, 2612-2637 (2009) ·Zbl 1204.65128
[39]Castillo, J.; Hyman, J.; Shashkov, M.; Steinberg, S., The sensitivity and accuracy of fourth order finite-difference schemes on nonuniform grids in one dimension, Comput. Math. Appl., 30, 8, 41-55 (1995) ·Zbl 0836.65025
[40]Castillo, J.; Hyman, J.; Shashkov, M.; Steinberg, S., Fourth- and sixth-order conservative finite-difference approximations of the divergence and gradient, Appl. Numer. Math., 37, 171-187 (2001) ·Zbl 0978.65011
[41]Castillo, J. E.; Grone, R. D., A matrix analysis approach to higher-order approximations for divergence and gradients satisfying a global conservation law, SIAM J. Matrix Anal. Appl., 25, 1, 128-142 (2003) ·Zbl 1061.65018
[42]Castillo, J. E.; Hyman, J. M.; Shashkov, M. J.; Steinberg, S.; Ilin, A. V.; Scott, L. R., High-order mimetic finite difference methods on nonuniform grids, Houston J. Math., Special Issue, 347-361 (1995)
[43]Ciarlet, P. G., The Finite Element Method for Elliptic Problems (1978), North-Holland: North-Holland Amsterdam ·Zbl 0383.65058
[45]Coudière, Y.; Manzini, G., The discrete duality finite volume method for convection-diffusion problems, SIAM J. Numer. Anal., 47, 6, 4163-4192 (2010) ·Zbl 1210.65183
[47]Dodziuk, J., Finite-difference approach to the Hodge theory of harmonic forms, Am. J. Math., 98, 1, 79-104 (1976) ·Zbl 0324.58001
[48]Domelevo, K.; Omnes, P., A finite volume method for the Laplace equation on almost arbitrary two-dimensional grids, M2AN Math. Model. Numer. Anal., 39, 6, 1203-1249 (2005) ·Zbl 1086.65108
[49]Eymard, R.; Henry, G.; Herbin, R.; Hubert, F.; Klöfkorn, R.; Manzini, G., 3D benchmark on discretization schemes for anisotropic diffusion problems on general grids, (Fořt, J.; Fürst, J.; Halama, J.; Herbin, R.; Hubert, F., FVCA6: Finite Volume for Complex Applications VI. FVCA6: Finite Volume for Complex Applications VI, Prague, Czech Republic, March 2011. FVCA6: Finite Volume for Complex Applications VI. FVCA6: Finite Volume for Complex Applications VI, Prague, Czech Republic, March 2011, Springer Proceedings in Mathematics (2011), Springer) ·Zbl 1246.76053
[50]Gallo, C.; Manzini, G., 2-D numerical modeling of bioremediation in heterogeneous saturated soils, Transp. Porous Media, 31, 1, 67-88 (1998)
[51]Gallo, C.; Manzini, G., A mixed finite element/finite volume approach for solving biodegradation transport in groundwater, Int. J. Numer. Methods Fluids, 26, 5, 533-556 (1998) ·Zbl 0927.76048
[52]Gross, P. W.; Kotiuga, P. R., Data structures for geometric and topological aspects of finite element algorithms, (Teixeira, F. L., Geometric Methods in Computational Electromagnetics, PIER 32 (2001), EMW Publishing: EMW Publishing Cambridge, MA), 151-169
[53]Gross, P. W.; Kotiuga, P. R., Finite element-based algorithms to make cuts for magnetic scalar potentials: topological constraints and computational complexity, (Teixeira, F. L., Geometric Methods in Computational Electromagnetics, PIER 32 (2001), EMW Publishing: EMW Publishing Cambridge, MA), 207-245
[54]Gyrya, V.; Lipnikov, K., High-order mimetic finite difference method for diffusion problems on polygonal meshes, J. Comput. Phys., 227, 8841-8854 (2008) ·Zbl 1152.65101
[55]Hiptmair, R., Canonical construction of finite elements, Math. Comput., 68, 228, 1325-1346 (1999) ·Zbl 0938.65132
[56]Hiptmair, R., Finite elements in computational electromagnetism, Acta Numer., 11, 237-339 (2002) ·Zbl 1123.78320
[57]Hyman, J. M.; Scovel, J. C., Deriving mimetic difference approximations to differential operators using algebraic topology (1988), Los Alamos National Laboratory: Los Alamos National Laboratory Los Alamos, NM
[58]Hyman, J. M.; Shashkov, M. J., Adjoint operators for the natural discretizations of the divergence, gradient and curl on logically rectangular grids, Appl. Numer. Math., 25, 4, 413-442 (1997) ·Zbl 1005.65024
[59]Hyman, J. M.; Shashkov, M. J., Natural discretizations for the divergence, gradient, and curl on logically rectangular grids, Comput. Math. Appl., 33, 4, 81-104 (1997) ·Zbl 0868.65006
[60]Hyman, J. M.; Shashkov, M. J., Mimetic finite difference methods for Maxwellʼs equations and the equations of magnetic diffusion, (Teixeira, F. L., Geometric Methods in Computational Electromagnetics, PIER 32 (2001), EMW Publishing: EMW Publishing Cambridge, MA), 89-121
[61]Hyman, J. M.; Shashkov, M. J.; Steinberg, S., The numerical solution of diffusion problems in strongly heterogeneous non-isotropic materials, J. Comput. Phys., 132, 1, 130-148 (1997) ·Zbl 0881.65093
[62]Hyman, J. M.; Shashkov, M. J.; Steinberg, S., The effect of inner products for discrete vector fields on the accuracy of mimetic finite difference methods, Comput. Math. Appl., 42, 12, 1527-1547 (2001) ·Zbl 0998.65107
[63]Lipnikov, K.; Manzini, G.; Brezzi, F.; Buffa, A., The mimetic finite difference method for 3D magnetostatics fields problems, J. Comput. Phys., 230, 2, 305-328 (2011) ·Zbl 1207.78041
[64]Lipnikov, K.; Manzini, G.; Svyatskiy, D., Analysis of the monotonicity conditions in the mimetic finite difference method for elliptic problems, J. Comput. Phys., 230, 7, 2620-2642 (2011) ·Zbl 1218.65117
[65]Lipnikov, K.; Morel, J.; Shashkov, M., Mimetic finite difference methods for diffusion equations on non-orthogonal non-conformal meshes, J. Comput. Phys., 199 (2004) ·Zbl 1057.65071
[66]Lipnikov, K.; Moulton, J. D.; Svyatskiy, D., A multilevel multiscale mimetic \((M^3)\) method for two-phase flows in porous media, J. Comput. Phys., 227, 6727-6753 (2008) ·Zbl 1338.76096
[67]Manzini, G.; Russo, A., A finite volume method for advection-diffusion problems in convection-dominated regimes, Comput. Methods Appl. Mech. Eng., 197, 13-16, 1242-1261 (2008) ·Zbl 1159.76356
[68]Margolin, L.; Shashkov, M.; Smolarkiewicz, P., A discrete operator calculus for finite difference approximations, Comput. Methods Appl. Mech. Eng., 187, 365-383 (2000) ·Zbl 0978.76063
[69]Mattiussi, C., An analysis of finite volume, finite element, and finite difference methods using some concepts from algebraic topology, J. Comput. Phys., 133, 2, 289-309 (1997) ·Zbl 0878.65091
[70]Mattiussi, C., The finite volume, finite difference and finite elements methods as numerical methods for physical field problems (2000), Online publication available at:
[71]Morel, J.; Roberts, R.; Shashkov, M., A local support-operators diffusion discretization scheme for quadrilateral \(r - z\) meshes, J. Comput. Phys., 144, 17-51 (1998) ·Zbl 1395.76052
[72]Munkres, J., Elements of Algebraic Topology (January 1984), Prentice Hall ·Zbl 0673.55001
[73]Nicolaides, R. A.; Trapp, K. A., Covolume discretizations of differential forms, (Arnold, D.; Bochev, P.; Lehoucq, R.; Nicolaides, R.; Shashkov, M., Compatible Discretizations. Proceedings of IMA Hot Topics Workshop on Compatible Discretizations. Compatible Discretizations. Proceedings of IMA Hot Topics Workshop on Compatible Discretizations, The IMA Volumes in Mathematics and its Applications Series, vol. 142 (2006), Springer-Verlag), 161-172 ·Zbl 1110.65024
[74]Nicolaides, R. A.; Wang, D.-Q., Convergence analysis of a covolume scheme for Maxwellʼs equations in three dimensions, Math. Comput., 67, 223, 947-963 (1998) ·Zbl 0907.65116
[75]Nicolaides, R. A., Direct discretization of planar div-curl problems, SIAM J. Numer. Anal., 29, 1, 32-56 (1992) ·Zbl 0745.65063
[76]Nicolaides, R. A.; Wu, X., Covolume solutions of three-dimensional div-curl equations, SIAM J. Numer. Anal., 34, 6, 2195-2203 (1997) ·Zbl 0889.35006
[77]Perot, J. B.; Subramanian, V., Discrete calculus methods for diffusion, J. Comput. Phys., 224, 1, 59-81 (2007) ·Zbl 1120.65325
[78]Perot, J. B.; Vidovic, D.; Wesseling, P., Mimetic reconstructions of vectors, (Arnold, D.; Bochev, P.; Lehoucq, R.; Nicolaides, R.; Shashkov, M., Compatible Discretizations. Proceedings of IMA Hot Topics Workshop on Compatible Discretizations. Compatible Discretizations. Proceedings of IMA Hot Topics Workshop on Compatible Discretizations, The IMA Volumes in Mathematics and its Applications Series, vol. 142 (2006), Springer-Verlag), 173-188 ·Zbl 1110.65108
[79]Robidoux, N., Numerical solution of the steady diffusion equation with discontinuous coefficients (May 2002), University of New Mexico: University of New Mexico Albuquerque, NM, PhD thesis
[80]Shashkov, M.; Steinberg, S., Support-operator finite-difference algorithms for general elliptic problems, J. Comput. Phys., 118, 1, 131-151 (1995) ·Zbl 0824.65101
[81]Shashkov, M.; Steinberg, S., Solving diffusion equations with rough coefficients in rough grids, J. Comput. Phys., 129, 2, 383-405 (1996) ·Zbl 0874.65062
[82]Shewchuk, J. R., Triangle: Engineering a 2D quality mesh generator and Delaunay triangulator, (Lin, Ming C.; Manocha, Dinesh, Applied Computational Geometry: Towards Geometric Engineering. Applied Computational Geometry: Towards Geometric Engineering, Lecture Notes in Computer Science, vol. 1148 (May 1996), Springer-Verlag: Springer-Verlag Berlin), 203-222
[83]Shewchuk, J. R., Delaunay refinement algorithms for triangular mesh generation, Comput. Geom., 22, 1-3, 21-74 (May 2002) ·Zbl 1016.68139
[84]Si, H., A quality tetrahedral mesh generator and a 3D Delaunay triangulator, Weierstrass Institute for Applied Analysis and Stochastics (WIAS)
[85]Tonti, E., A mathematical model for physical theories. II, Atti Accad. Naz. Lincei, Rend. Cl. Sci. Fis. Mat. Nat., 8, 52, 350-356 (1972) ·Zbl 0256.70012
[86]Tonti, E., On the formal structure of the physical theories, (Quaderno dei Gruppi di Ricerca Matematica del CNR (1975), Consiglio Nazionale delle Ricerche, Istituto di Matematica del Politecnico di Milano), Available online at:
[87]Trapp, K. A., Inner products in covolume and mimetic methods, M2AN Math. Model. Numer. Anal., 42, 941-959 (2008) ·Zbl 1155.65103
[88]Whitney, H., Geometric Integration Theory (1957), Princeton University Press: Princeton University Press Princeton, NJ ·Zbl 0083.28204
[89]Yee, K. S., Numerical solution of initial boundary value problems involving Maxwellʼs equations in isotropic media, IEEE Trans. Antennas Propag., AP-14, 3, 302-307 (1966) ·Zbl 1155.78304
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
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