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Inner products in covolume and mimetic methods.(English)Zbl 1155.65103

A class of compatible spatial discretizations for solving partial differential equations is presented. A discrete exact framework is developed to classify these methods which include the mimetic and the covolume methods as well as certain low-order finite element methods. This construction ensures discrete analogs of the differential operators that satisfy the identities and theorems of vector calculus, in particular a Helmholtz decomposition theorem. It is shown that these methods differ only in their choice of discrete inner products. Finally, certain uniqueness results for the covolume inner product are shown.

MSC:

65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
65N06 Finite difference methods for boundary value problems involving PDEs
35J25 Boundary value problems for second-order elliptic equations

Cite

References:

[1]D.N. Arnold, Differential complexes and numerical stability, in Proceedings of the International Congress of Mathematicians, Vol. I, Higher Ed. Press, Beijing (2002) 137-157. ·Zbl 1023.65113
[2]M. Berndt, K. Lipnikov, D. Moulton and M. Shashkov, Convergence of mimetic finite difference discretizations of the diffusion equation. East-West J. Numer. Math9 (2001) 253-316. Zbl1014.65114 ·Zbl 1014.65114
[3]P. Bochev and J.M. Hyman, Principles of mimetic discretizations of differential operators, in Compatible Spatial Discretizations, D. Arnold, P. Bochev, R. Lehoucq, R. Nicolaides and M. Shashkov Eds., IMA Volumes in Mathematics and its Applications142, Springer, New York (2006). ·Zbl 1110.65103
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[6]A. Hirani, Discrete Exterior Calculus. Ph.D. thesis, California Institute of Technology, USA (2003).
[7]J.M. Hyman and M. Shashkov, The adjoint operators for the natural discretizations for the divergence, gradient, and curl on logically rectangular grids. IMACS J. Appl. Num. Math.25 (1997) 1-30. Zbl1005.65024 ·Zbl 1005.65024 ·doi:10.1016/S0168-9274(97)00097-4
[8]J.M. Hyman and M. Shashkov, Natural discretizations for the divergence, gradient, and curl on logically rectangular grids. Comput. Math. Appl.33 (1997) 81-104. ·Zbl 0868.65006 ·doi:10.1016/S0898-1221(97)00009-6
[9]J.M. Hyman and M. Shashkov, Mimetic discretizations for Maxwell’s equations. J. Comp. Phys.151 (1999) 881-909. ·Zbl 0956.78015 ·doi:10.1006/jcph.1999.6225
[10]J.M. Hyman and M. Shashkov, The orthogonal decomposition theorems for mimetic finite difference methods. SIAM J. Numer. Anal.36 (1999) 788-818. ·Zbl 0972.65077 ·doi:10.1137/S0036142996314044
[11]J.C. Nedelec, Mixed finite elements in \Bbb R 3 . Numer. Math.35 (1980) 315-341. ·Zbl 0419.65069 ·doi:10.1007/BF01396415
[12]J.C. Nedelec, A new family of mixed finite elements in \Bbb R 3 . Numer. Math.50 (1986) 57-81. ·Zbl 0625.65107 ·doi:10.1007/BF01389668
[13]R.A. Nicolaides, Direct discretization of planar div-curl problems. SIAM J. Numer. Anal.29 (1992) 32-56. Zbl0745.65063 ·Zbl 0745.65063 ·doi:10.1137/0729003
[14]R. Nicolaides and K. Trapp, Covolume discretizations of differential forms, in Compatible Spatial Discretizations, D. Arnold, P. Bochev, R. Lehoucq, R. Nicolaides and M. Shashkov Eds., IMA Volumes in Mathematics and its Applications142, Springer, New York (2006). ·Zbl 1110.65024
[15]R.A. Nicolaides and D.Q. Wang, Convergence analysis of a covolume scheme for Maxwell’s equations in three dimensions. Math. Comp.67 (1998) 947-963. ·Zbl 0907.65116 ·doi:10.1090/S0025-5718-98-00971-5
[16]R.A. Nicolaides and X. Wu, Covolume solutions of three-dimensional div-curl equations. SIAM J. Numer. Anal.34 (1997) 2195-2203. ·Zbl 0889.35006 ·doi:10.1137/S0036142994277286
[17]P.A. Raviart and J.M. Thomas, A mixed finite elemnt method for second order elliptic problems, in Springer Lecture Notes in Mathematics606, Springer-Verlag (1977) 292-315. ·Zbl 0362.65089
[18]K. Trapp, A Class of Compatible Discretizations with Applications to Div-Curl Systems. Ph.D. thesis, Carnegie Mellon University, USA (2004).
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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