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, L.—SPRUCK, J.: Motion of level sets by mean curvature. I ,J. Differ.Geom. 33 (1991), 635–681. [8] EYMARD, R.—HANDLOVIČOVÁ, A. H.—MIKULA, K.: Study of a finite volume scheme for regularized mean curvature flow level set equation , J. Numer. Anal. 31 (2011), no. 3, 813–846. [9] EYMARD, R.—GALLOUËT, T.—HERBIN, R.: Discretization of heterogeneous and anisotropic diffusion problems on general nonconforming meshes. SUSHI: A scheme using stabilization and hybrid interfaces , IMA J. Numer. Anal. 30 (2010), no. 4, 1009–1043 [10] LINDEBERG, T.: Scale-space theory

equations by the ‘superimplicit’ Stone method [1]. Section 4 gives the results of numerical experiments which illustrate both ap- proximation and iterative properties of algorithms proposed. 2. CONSTRUCTION OF EXPONENTIAL FINITE VOLUME APPROXIMATIONS In this section, we successively consider the domain discretization, the approxi- mation of surface and volume integrals in the balance relations, allowance for the boundary conditions, the principles of the construction of local matrices and assem- bling of the global matrices as well as convergence of grid solutions. 2

J. Numer. Math., Vol. 12, No. 4, pp. 255–284 (2004) c© VSP 2004 Some refined finite volume element methods for the Stokes and Navier–Stokes systems with corner singularities K. DJADEL∗ and S. NICAISE† Received July 11, 2003 Received in revised form June 10, 2004 Abstract — It is well known that the solution of the Stokes or Navier–Stokes system in a non convex polygonal domain of R2 has a singular behaviour near non convex corners. Consequently we investi- gate different refined (non conforming) finite volume-element methods to approximate the solution of such

H1- norm to the exact solution, with a rate of convergence of order hs (where h is the size of the mesh). Keywords: convection–diffusion equations, Finite Volume, convergence rate, interpolation 1. INTRODUCTION 1.1. The problem Let W be a polygonal open subset of RN (N = 2 or 3). We study a Ž nite volume discretization of ½ ¡ Du+ div(vu) + bu = f + div(G) in W u = 0 on ¶W (1.1) where v 2 (C(W))N , b 2 L¥(W) is nonnegative, f 2 L2(W) and G 2 (Hs(W))N with s 2 [0;1] (if s = 0, H0(W) is to be understood as L2(W)). The solution to (1.1) is taken in a weak sense as in

Environment, vol.60, pp.137-149. [20] Kamiński, M. and Ossowski, R. L. (2009): The Stochastic perturbation - based Finite Volume Method for the flow problems. - Journal of Technical Physics, vol.50, No.1, pp.297-315. [21] Kamiński M. (2001): Stochastic problem of viscous incompressible fluid flow with heat transfer. - Zeitschrift für Angewandte Mathematik und Mechanik, vol.81, No.12, pp.827-837. [22] Cueto - Felgueroso L. and Peraire J. (2008): A time - adaptive Finite Volume Method for the Cahn-Hilliard and Kuramoto - Sivashinsky equations. - Journal of Computational

References Bank, R.E. and Rose, D.J. (1987). Some error estimates for the box methods, SIAM Journal on Numerical Analysis 24 (4): 777–787. Cai, J.X. and Miao, J. (2012). New explicit multisymplectic scheme for the complex modified Korteweg–de Vries equation, Chinese Physics Letters 29 (3): 030201. Cai, Z.Q. (1991). On the finite volume element method, Numerische Mathematik 58 (7): 713–735. Costa, R., Machado, G.J. and Clain, S. (2015). A sixth-order finite volume method for the 1D biharmonic operator: Application to intramedullary nail simulation

,4] for details): c Ihj j Ihc c j Xh (2.5) c1 c 2 c 2 c2 c 2 c Xh c2 c1 0 c 2 c Ihc (2.6) K Ihc dx K c dx c Xh for any K Th (2.7) e Ihc ds e c ds c Xh for any side e of K Th (2.8) Ihc L¥ e c L¥ e c Xh for any side e of K Th (2.9) c Ihc Lp K ChK c W 1p K c Xh 1 p ¥ (2.10) Two-scale Ž nite volume element method 123 3. A TWO-SCALE FINITE VOLUME ELEMENT METHOD WITH OVERSAMPLING This section is devoted to the formulation of two-scale Ž nite volume element method for solving (2.1) and its necessary venue for the analysis. In Subsection 3.1 we brie y review a homogenization

1 Introduction In this work, we consider the discretization by the cell-centered finite volume method of the following convection-diffusion problem: (1.1) { - Δ ⁢ u + div ⁡ ( 𝒗 ⁢ u ) + b ⁢ u = f in ⁢ Ω , u = 0 on ⁢ ∂ ⁡ Ω , $\left\{\begin{aligned} \displaystyle-\Delta u+\operatorname{div}(\boldsymbol{v% }u)+bu&\displaystyle=f&&\displaystyle\quad\text{in }\Omega,\\ \displaystyle u&\displaystyle=0&&\displaystyle\quad\text{on }\partial\Omega,% \end{aligned}\right.$ where Ω is an open bounded polygonal subset of ℝ d ${\mathbb{R}^{d}}$ , d ≥ 2 ${d\geq 2}$ , 𝒗

COMPUTATIONAL METHODS IN APPLIED MATHEMATICS, Vol.6(2006), No.3, pp.306–325 c© 2006 Institute of Mathematics of the National Academy of Sciences of Belarus ON FINITE VOLUME DISCRETIZATION OF THE THREE-DIMENSIONAL BIOT POROELASTICITY SYSTEM IN MULTILAYER DOMAINS A.NAUMOVICH1 Abstract — In this paper we propose a finite volume discretization for the three- dimensional Biot poroelasticity system in multilayer domains. For stability reasons, staggered grids are used. The discretization takes into account discontinuity of the coefficients across the interfaces between

References [1] B. Andreianov, F. Boyer, and F. Hubert. Discrete duality finite volume schemes for leray-lions-type elliptic problems on general 2d meshes. 23(1):145–195. [2] Y. Boubendir and S. Tlupova. Stokesdarcy boundary integral solutions using preconditioners. 228(23):8627–8641. [3] F. Boyer, F. Hubert, and S. Krell. Non-overlapping schwarz algorithm for solving 2d m-DDFV schemes. 30(4):Pp 1062–1100. [4] M. Cai, M. Mu, and J. Xu. Preconditioning techniques for a mixed stokes/darcy model in porous media applications. 233(2):346–355. [5] C. Cancs, C. Chainais