January  2012, 11(1): 147-172. doi: 10.3934/cpaa.2012.11.147

Approximation of nonlinear parabolic equations using a family of conformal and non-conformal schemes

1. 

Université Paris-Est Marne-La-Vallée, 5 bd Descartes, Champs-sur-Marne, 77454 Marne-la-Vallée Cedex 2

2. 

Slovak University of Technology, Slovak Republic, Slovak Republic

Received  December 2009 Revised  September 2010 Published  September 2011

We consider a family of space discretisations for the approximation of nonlinear parabolic equations, such as the regularised mean curvature flow level set equation, using semi-implicit or fully implicit time schemes. The approximate solution provided by such a scheme is shown to converge thanks to compactness and monotony arguments. Numerical examples show the accuracy of the method.
Citation: Robert Eymard, Angela Handlovičová, Karol Mikula. Approximation of nonlinear parabolic equations using a family of conformal and non-conformal schemes. Communications on Pure & Applied Analysis, 2012, 11 (1) : 147-172. doi: 10.3934/cpaa.2012.11.147
References:
[1]

Ivar Aavatsmark, T. Barkve, O. Boe and T. Mannseth, Discretization on non-orthogonal, quadrilateral grids for inhomogeneous, anisotropic media,, J. Comput. Phys., 127 (1996), 2. doi: 10.1006/jcph.1996.0154. Google Scholar

[2]

Leo Agelas and Roland Masson, Convergence of the finite volume MPFA O scheme for heterogeneous anisotropic diffusion problems on general meshes,, C. R., 346 (2008), 1007. doi: 10.1016/j.crma.2008.07.015. Google Scholar

[3]

Y. Chen, Y. Giga and S. Goto, Uniqueness and existence of viscosity solutions of generalized mean curvature flow equation,, J. Differential Geometry, 33 (1991), 749. Google Scholar

[4]

S. Corsaro, K. Mikula, A. Sarti and F. Sgallari, Semi-implicit co-volume method in 3D image segmentation,, SIAM Journal on Scientific Computing, 28 (2006), 2248. doi: 10.1137/060651203. Google Scholar

[5]

L. C. Evans and J. Spruck, Motion of level sets by mean curvature I,, J. Differential Geometry, 33 (1991), 635. Google Scholar

[6]

R. Eymard, T. Gallouët and R. Herbin, Finite volume methods,, Techniques of Scientific Computing, (2000), 713. Google Scholar

[7]

R. Eymard, T. Gallouët and R. Herbin, Discretisation of heterogeneous and anisotropic diffusion problems on general nonconforming meshes. SUSHI: a scheme using stabilization and hybrid interfaces,, IMA Journal of Numerical Analysis, (2009). doi: 10.1093/imanum/drn084. Google Scholar

[8]

R. Eymard, A. Handlovičová and K. Mikula, Study of a finite volume scheme for the regularized mean curvature flow level set equation,, IMA Journal of Numerical Analysis, (2010). doi: 10.1093/imanum/drq025. Google Scholar

[9]

R. Eymard, R. Herbin and J. C. Latché, Convergence analysis of a colocated finite volume scheme for the incompressible Navier-Stokes equations on general 2D or 3D meshes,, SIAM Journal on Numerical Analysis, 45 (2007), 1. doi: 10.1137/040613081. Google Scholar

[10]

P. Frolkovič and K. Mikula, Flux based levelset method: a finite volume method for evolving interfaces,, Applied Numerical Mathematics, 57 (2007), 436. doi: 10.1016/j.apnum.2006.06.002. Google Scholar

[11]

A. Handlovičová, K. Mikula and F. Sgallari, Semi-implicit complementary volume scheme for solving level set like equations in image processing and curve evolution,, Numer. Math., 93 (2003), 675. doi: 10.1007/s002110100374. Google Scholar

[12]

J. Leray and J.-L. Lions, Quelques résultats de Višik sur les problèmes elliptiques nonlinéaires par les méthodes de Minty-Browder,, Bull. Soc. Math. France, 93 (1965), 97. Google Scholar

[13]

K. Lipnikov, M. Shashkov and I. Yotov, Local flux mimetic finite difference methods,, Numer. Math., 112 (2009), 115. doi: 10.1007/s00211-008-0203-5. Google Scholar

[14]

S. Nemadjieu and M Rumpf, Finite volume schemes on simplices,, Personal communication, (2009). Google Scholar

[15]

A. M. Oberman, A convergent monotone difference scheme for motion of level sets by mean curvature,, Numer. Math., 99 (2004), 365. doi: 10.1007/s00211-004-0566-1. Google Scholar

[16]

N. J. Walkington, Algorithms for computing motion by mean curvature,, SIAM J. Numer. Anal., 33 (1996), 2215. doi: 10.1137/S0036142994262068. Google Scholar

show all references

References:
[1]

Ivar Aavatsmark, T. Barkve, O. Boe and T. Mannseth, Discretization on non-orthogonal, quadrilateral grids for inhomogeneous, anisotropic media,, J. Comput. Phys., 127 (1996), 2. doi: 10.1006/jcph.1996.0154. Google Scholar

[2]

Leo Agelas and Roland Masson, Convergence of the finite volume MPFA O scheme for heterogeneous anisotropic diffusion problems on general meshes,, C. R., 346 (2008), 1007. doi: 10.1016/j.crma.2008.07.015. Google Scholar

[3]

Y. Chen, Y. Giga and S. Goto, Uniqueness and existence of viscosity solutions of generalized mean curvature flow equation,, J. Differential Geometry, 33 (1991), 749. Google Scholar

[4]

S. Corsaro, K. Mikula, A. Sarti and F. Sgallari, Semi-implicit co-volume method in 3D image segmentation,, SIAM Journal on Scientific Computing, 28 (2006), 2248. doi: 10.1137/060651203. Google Scholar

[5]

L. C. Evans and J. Spruck, Motion of level sets by mean curvature I,, J. Differential Geometry, 33 (1991), 635. Google Scholar

[6]

R. Eymard, T. Gallouët and R. Herbin, Finite volume methods,, Techniques of Scientific Computing, (2000), 713. Google Scholar

[7]

R. Eymard, T. Gallouët and R. Herbin, Discretisation of heterogeneous and anisotropic diffusion problems on general nonconforming meshes. SUSHI: a scheme using stabilization and hybrid interfaces,, IMA Journal of Numerical Analysis, (2009). doi: 10.1093/imanum/drn084. Google Scholar

[8]

R. Eymard, A. Handlovičová and K. Mikula, Study of a finite volume scheme for the regularized mean curvature flow level set equation,, IMA Journal of Numerical Analysis, (2010). doi: 10.1093/imanum/drq025. Google Scholar

[9]

R. Eymard, R. Herbin and J. C. Latché, Convergence analysis of a colocated finite volume scheme for the incompressible Navier-Stokes equations on general 2D or 3D meshes,, SIAM Journal on Numerical Analysis, 45 (2007), 1. doi: 10.1137/040613081. Google Scholar

[10]

P. Frolkovič and K. Mikula, Flux based levelset method: a finite volume method for evolving interfaces,, Applied Numerical Mathematics, 57 (2007), 436. doi: 10.1016/j.apnum.2006.06.002. Google Scholar

[11]

A. Handlovičová, K. Mikula and F. Sgallari, Semi-implicit complementary volume scheme for solving level set like equations in image processing and curve evolution,, Numer. Math., 93 (2003), 675. doi: 10.1007/s002110100374. Google Scholar

[12]

J. Leray and J.-L. Lions, Quelques résultats de Višik sur les problèmes elliptiques nonlinéaires par les méthodes de Minty-Browder,, Bull. Soc. Math. France, 93 (1965), 97. Google Scholar

[13]

K. Lipnikov, M. Shashkov and I. Yotov, Local flux mimetic finite difference methods,, Numer. Math., 112 (2009), 115. doi: 10.1007/s00211-008-0203-5. Google Scholar

[14]

S. Nemadjieu and M Rumpf, Finite volume schemes on simplices,, Personal communication, (2009). Google Scholar

[15]

A. M. Oberman, A convergent monotone difference scheme for motion of level sets by mean curvature,, Numer. Math., 99 (2004), 365. doi: 10.1007/s00211-004-0566-1. Google Scholar

[16]

N. J. Walkington, Algorithms for computing motion by mean curvature,, SIAM J. Numer. Anal., 33 (1996), 2215. doi: 10.1137/S0036142994262068. Google Scholar

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