RAIRO Model Research Articles

A Superconvergent $$C^0$$ C 0 Discontinuous Galerkin Method for Kirchhoff Plates: Error Estimates, Hybridization and Postprocessing

This paper is devoted to a superconvergent $$C^0$$C0 discontinuous Galerkin (SCDG) method for Kirchhoff plates. First of all, following the ideas in Huang et al. (Comput Methods Appl Mech Eng 199(23---24):1446---1454, 2010) but with the normal bending moment and twisting moment as new numerical traces, we propose a modified framework of CDG methods for Kirchhoff plates. Then by a technical choice of the numerical traces, we obtain our SCDG method. Observing that the famous Hellan---Herrmann---Johnson (HHJ) method is a special case of the SCDG method, we are motivated to borrow some techniques for analyzing the HHJ method to derive optimal and superconvergent error estimates for the SCDG method. Under some assumption on the stabilization parameters, we consider the hybridization of the SCDG method. Furthermore, we construct a superconvergent discrete deflection by postprocessing the solution of the method using similar technique in Stenberg (RAIRO Model Math Anal Numer 25(1):151---167, 1991). Some numerical results are performed to demonstrate the theoretical results for the SCDG method.

Local Existence of Weak Solutions to Kinetic Models of Granular Media

We prove in any dimension \({d \geqq 1}\) a local in time existence of weak solutions to the Cauchy problem for the kinetic equation of granular media, $$\partial_t f+v\cdot \nabla_x f = {div}_v[f(\nabla W *_v f)]$$ when the initial data are nonnegative, integrable and bounded functions with compact support in velocity, and the interaction potential \({W}\) is a \({C^2({\mathbb{R}}^d)}\) radially symmetric convex function. Our proof is constructive and relies on a splitting argument in position and velocity, where the spatially homogeneous equation is interpreted as the gradient flow of a convex interaction energy with respect to the quadratic Wasserstein distance. Our result generalizes the local existence result obtained by Benedetto et al. (RAIRO Model Math Anal Numer 31(5):615–641, 1997) on the one-dimensional model of this equation for a cubic power-law interaction potential.

Homogenization of a class of Neumann problems in perforated domains

We consider elliptic problems in periodically perforated domains in R N , N 3, with nonhomogeneous Neumann conditions on the boundary of the holes. The aim is to give the asymptotic behavior of the solutions as the period e goes to zero. Two geometries are considered. In the first one, all the holes are small, i.e., their size is of order of er(e) with r(e) → 0. The second geometry is more general, there are small holes as before but also holes of size of the order of e (the last ones corresponding to the classical homogenization situation). Our study is performed by the periodic unfolding method from C. R. Acad. Sci. Paris Ser. I 335 (2002), 99-104, adapted to the case of holes of size er(e )( seeJ. Math. Pures Appl. 89 (2008), 248- 277). The use of this method allows us to study second-order operators with highly oscillating coefficients and so, to generalize here the results of RAIRO Model. Math. Anal. Numer. 4(22) (1988), 561-608. In both cases, if r(e) = exp(N/N − 1), an additional term appears in the right-hand side of the limit equation.

On the Mathematical Analysis and Numerical Approximation of a System of Nonlinear Parabolic PDEs

In this paper we consider a boundary value problem for a system of 2 nonlinear parabolic PDEs e.g. arising in the context of flow and transport in porous media. The flow model is based on tho nonlinear Richard’s equation problem and is combined with the transport equation through saturation and Darcy’s velocity (discharge) terms. The convective terms are approximated by means of the method of characteristics initiated by P. Pironneau [Num. Math. 38 (1982), 871–885] and R. Douglas and T. F. Russel [SIAM J. Num. Anal. 19 (1982), 309–332]. The nonlinear terms in Richard’s equation are approximated by means of a relaxation scheme applied by W. Jäger and J. Kačur [RAIRO Model. Math. Anal. Num. 29 (1995), 605–627] and J. Kačur [IMA J. Num. Anal. 19 (1999), 119–154; SIAM J. Num. Anal. 39 (1999), c 290–316]. The convergence of the approximation method is proved.

Uniform stability analysis of Austin, Manteuffel and McCormick finite elements and fast and robust iterative methods for the Stokes‐like equations

AbstractIn this paper, we discuss uniformly stable discretizations and fast solvers for the steady and semi‐discrete time‐dependent Stokes equations, which will be called the Stokes‐like equations. We study two pairs of conforming finite elements that are uniformly accurate with respect to all relevant parameters in the equations, one of which is the Scott–Vogelius (RAIRO Modél. Math. Anal. Numér. 1985; 19:111–143) element and the other of which is the finite element pair by Austin et al. (Numer. Linear Algebra Appl. 2004; 11:115–140). We prove that the finite element pair by Austin, Manteuffel and McCormick is uniformly accurate and we design and analyze the fast and robust solution techniques for the linear algebraic systems arising from the discretizations of the Stokes‐like equations as well. Our method is based on the augmented Lagrangian Uzawa iterative methods (Augmented Lagrangian Methods. North‐Holland: Amsterdam, 1983) and the multigrid methods designed by Austin et al. (Numer. Linear Algebra Appl. 2004; 11:115–140) for the linear algebraic equations from the discretizations of the differential operator ρ2I−κ2Δ−µ2∇div. We prove our methods are robust with respect to all relevant parameters that appear in the equations and also the mesh size, thereby, achieve the fast and robust methods for the solutions to the Stokes‐like equations. In particular, the convergence analysis for the multigrid methods designed by Austin et al., has been posed as an open question (Numer. Linear Algebra Appl. 2004; 11:115–140), which is resolved in this paper. Copyright © 2009 John Wiley & Sons, Ltd.

A priori and a posteriori error estimations for the dual mixed finite element method of the Navier‐Stokes problem

AbstractThis article is concerned with a dual mixed formulation of the Navier‐Stokes system in a polygonal domain of the plane with Dirichlet boundary conditions and its numerical approximation. The gradient tensor, a quantity of practical interest, is introduced as a new unknown. The problem is then approximated by a mixed finite element method. Quasi‐optimal a priori error estimates are obtained. These a priori error estimates, an abstract nonlinear theory (similar to (Verfürth, RAIRO Model Math Anal Numer 32 (1998), 817–842)) and a posteriori estimates for the Stokes system from (Farhloul et al., Numer Funct Anal Optim 27 (2006), 831–846) lead to an a posteriori error estimate for the Navier‐Stokes system. © 2008 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2009

Coefficients of the Singularities for Elliptic Boundary Value Problems on Domains with Conical Points. III: Finite Element Methods on Polygonal Domains

In the two first parts of this work [RAIRO Model. Math. Anal. Numer., 24 (1990), pp. 27e52], [RAIRO Model. Math. Anal. Numer., 24 (1990), pp. 343–367] formulas giving the coefficients arising in the singular expansion of the solutions of elliptic boundary value problems on nonsmooth domains are investigated. Now, for the case of homogeneous strongly elliptic operators with constant coefficients on polygonal domains, the solution of such problems by the finite element method is considered. In order to approximate the solution or the coefficients, different methods are used based on the expressions of the coefficients that were obtained in the first two parts; the dual singular function method is also generalized.