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Mimetic discretization of two-dimensional magnetic diffusion equations.(English)Zbl 1349.78095

Summary: In case of non-constant resistivity, cylindrical coordinates, and highly distorted polygonal meshes, a consistent discretization of the magnetic diffusion equations requires new discretization tools based on a discrete vector and tensor calculus. We developed a new discretization method using the mimetic finite difference framework. It is second-order accurate on arbitrary polygonal meshes and a consistent calculation of the Joule heating is intrinsic within it. The second-order convergence rates in \(L^2\) and \(L^1\) norms were verified with numerical experiments.

MSC:

78M20 Finite difference methods applied to problems in optics and electromagnetic theory
65M06 Finite difference methods for initial value and initial-boundary value problems involving PDEs

Software:

AMG1R5

Cite

References:

[1]Beirao da Veiga, L.; Lipnikov, K.; Manzini, G., Arbitrary-order nodal mimetic discretizations of elliptic problems on polygonal meshes, SIAM J. Numer. Anal., 49, 1737-1760 (2011) ·Zbl 1242.65215
[2]Bochev, P. B.; Robinson, A. C., Matching algorithms with physics: exact sequences of finite element spaces, (Collected Lectures of the Preservation of Stability Under Discretization (2002), SIAM) ·Zbl 1494.78020
[3]Brezzi, F.; Buffa, A.; Lipnikov, K., Mimetic finite differences for elliptic problems, ESAIM: Math. Model. Numer. Anal., 43, 277-295 (2009) ·Zbl 1177.65164
[4]Brezzi, F.; Fortin, M., Mixed and Hybrid Finite Element Methods (1991), Springer-Verlag: Springer-Verlag New York ·Zbl 0788.73002
[5]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
[6]Brezzi, F.; Lipnikov, K.; Shashkov, M.; Simoncini, V., A new discretization methodology for diffusion problems on generalized polyhedral meshes, Comput. Methods Appl. Mech. Eng., 196, 3682-3692 (2007) ·Zbl 1173.76370
[7]Brezzi, F.; Lipnikov, K.; Simoncini, V., A family of mimetic finite difference methods on polygonal and polyhedral meshes, Math. Model. Methods Appl. Sci., 15, 10, 1533-1551 (2005) ·Zbl 1083.65099
[8]Droniou, J.; Eymard, R.; Gallouet, T.; Herbin, R., A unified approach to mimetic finite difference, hybrid finite volume and mixed finite volume method, Math. Model. Methods Appl. Sci., 20, 2, 1-31 (2010)
[9]Hyman, J.; Shashkov, M., The approximation of boundary conditions for mimetic finite difference methods, Comput. Math. Appl., 36, 79-99 (1998) ·Zbl 0932.65111
[10]Hyman, J.; Shashkov, M., The orthogonal decomposition theorems for mimetic finite difference methods, SIAM J. Numer. Anal., 36, 3, 788-818 (1999) ·Zbl 0972.65077
[11]Hyman, J.; Shashkov, M.; Steinberg, S., The numerical solution of diffusion problems in strongly heterogeneous non-isotropic materials, J. Comput. Phys., 132, 130-148 (1997) ·Zbl 0881.65093
[13]Lipnikov, K.; Manzini, G.; Svyatskiy, D., Analysis of the monotonicity conditions in the mimetic finite difference method for elliptic problems, J. Comput. Phys., 230, 2620-2642 (2011) ·Zbl 1218.65117
[15]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
[17]Olson, P., Summation by parts, projections, and stability. I, Math. Comput., 64, 211, 1035-1065 (1995) ·Zbl 0828.65111
[18]Rieben, R. N.; White, D. A., Verification of high-order mixed finite element solution of transient magnetic field problems, IEEE Trans. Magn., 42, 1, 25-39 (2006)
[19]Roberts, J. E.; Thomas, J.-M., Mixed and hybrid methods, (Ciarlet, P. G.; Lions, J. L., Handbook of Numerical Analysis, Finite Element Methods, vol. II (1991), Elsevier/North Holland: Elsevier/North Holland Amsterdam) ·Zbl 0875.65090
[20]Samarskii, A. A.; Tishkin, V. F.; Favorskii, A. P.; Yu, Shashkov M., Operator-difference schemes, Differentsialnye Uravneniya, 17, 7, 1317-1327 (1981), (in Russian) ·Zbl 0485.65060
[21]Shashkov, M., Conservative Finite-Difference Methods on General Grids (1996), CRC Press: CRC Press Boca Raton ·Zbl 0844.65067
[22]Stüben, K., Algebraic multigrid (AMG): experiences and comparisons, Appl. Math. Comput., 13, 419-452 (1983) ·Zbl 0533.65064
[23]Hyman, J.; Shashkov, M., Mimetic discretizations for Maxwell’s equations and the equations of magnetic diffusion, Prog. Electromagn. Res., 32, 89-121 (2001)
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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