We consider a special configuration of vorticity that consists of a pair ofexternally tangent circular vortex sheets, each having a circularly symmetric core of bounded vorticity concentric to the sheet, and each core precisely balancing the vorticity mass of the sheet. This configuration is a stationary weak solution of the 2D incompressible Euler equations. We propose to perform numerical experiments to verify that certain approximations of this flow configuration converge to a non-stationary weak solution. Preliminary simulations presented here suggest this isindeed the case. We establish a convergence theorem for the vortex blob method that applies to this problem. This theorem and the preliminary calculations we carried out support the existence of two distinct weak solutions with the same initial data.


Singularly perturbed reaction-diffusionproblems exhibit in general solutions with anisotropic features,e.g. strong boundary and/or interior layers.This anisotropy is reflected in a discretization by using mesheswith anisotropic elements. The quality of the numerical solutionrests on the robustness of the a posteriori error estimator withrespect to both, the perturbation parameters of the problemand the anisotropy of the mesh. The equilibrated residual method has been shown to provide one of the most reliable error estimates for the reaction-diffusion problem. Its modification suggested byAinsworth and Babuška has been proved to be robust for the case of singular perturbation. In the present work we investigate the modified method on anisotropic meshes. The method in the form of Ainsworth and Babuška is shown here to fail on anisotropic meshes. We suggest a new modification based on the stretching ratios of the mesh elements. The resulting error estimator is equivalent tothe equilibrated residual method in the case of isotropic meshesand is proved to be robust on anisotropic meshes as well. Among others, the equilibrated residual method involves the solution of an infinite dimensional local problem on each element. In practical computations an approximate solution to this local problem was successfully computed. Nevertheless, up to now no rigorous analysis has been done showing the appropriateness of any computable approximation. This demands special attention since an improper approximate solution to the local problem can be fatal for the robustness of the whole method. In the present work we provide one of the desired approximations. We prove that the method is not affected by the approximate solution of the local problem.

For a plate subject to stress boundary condition, the deformationdetermined by the Reissner–Mindlin plate bending model could bebending dominated, transverse shear dominated, or neither(intermediate), depending on the load. We show that theReissner–Mindlin model has a wider range of applicability thanthe Kirchhoff–Love model, but it does not always converge to theelasticity theory. In the case of bending domination, both the twomodels are accurate. In the case of transverse shear domination,the Reissner–Mindlin model is accurate but the Kirchhoff–Lovemodel totally fails. In the intermediate case, while theKirchhoff–Love model fails, the Reissner–Mindlin solution alsohas a relative error comparing to the elasticity solution, whichdoes not decrease when the plate thickness tends to zero. We alsoshow that under the conventional definition of the resultantloading functional, the well known shear correction factor 5/6in the Reissner–Mindlin model should be replaced by 1.Otherwise, the range of applicability of the Reissner–Mindlinmodel is not wider than that of Kirchhoff–Love's.

In this paper, we study the dynamic frictional contact of a viscoelastic beam with a deformableobstacle. The beam is assumed to be situated horizontally and to move, in both horizontal andtangential directions, by the effect of applied forces. The left end of the beam is clampedand the right one is free. Its horizontal displacement is constrained because of the presenceof a deformable obstacle, the so-called foundation, which is modelled by a normal compliance contact condition.The effect of the friction is included in the vertical motion ofthe free end, by using Tresca's law or Coulomb's law. In both cases, the variationalformulation leads to a nonlinear variational equation for the horizontal displacement coupledwith a nonlinear variational inequality for the vertical displacement. We recall an existenceand uniqueness result. Then, by using the finite element method to approximate the spatial variableand an Euler scheme to discretize the time derivatives, a numerical scheme is proposed. Error estimates on the approximative solutions are derived. Numerical results demonstrate the application of the proposed algorithm.

We consider the use of finite volume methods for the approximation of aparabolic variational inequality arising in financial mathematics.We show, under some regularityconditions, the convergence of the upwind implicit finite volume schemeto a weak solution of the variational inequality in a bounded domain.Some results, obtained in comparison with other methodson two dimensional cases, show that finite volume schemes can be accurate and efficient.

In this work, we propose a general framework for the construction of pressure law for phase transition. These equations of state are particularly suitable for a use in a relaxation finite volume scheme. The approach is based on a constrained convex optimizationproblem on the mixture entropy. It is valid for both miscible and immiscible mixtures. We also propose a rough pressure law for modelling a super-critical fluid.

In this work, the quasistatic thermoviscoelastic thermistor problem isconsidered. The thermistor model describes the combination of the effects due tothe heat, electrical current conduction and Joule's heat generation. The variationalformulation leads to a coupled system of nonlinear variational equations for whichthe existence of a weak solution is recalled.Then, a fully discrete algorithm is introduced based on the finite elementmethod to approximate the spatial variable and an Euler scheme to discretizethe time derivatives. Error estimates are derived and, under suitableregularity assumptions, the linear convergence of the scheme is deduced.Finally, some numerical simulations are performed in order to show the behaviourof the algorithm.

We consider the lowest-order Raviart–Thomas mixed finite elementmethod for second-order elliptic problems on simplicial meshes intwo and three space dimensions. This method produces saddle-pointproblems for scalar and flux unknowns. We show how to easily andlocally eliminate the flux unknowns, which implies the equivalencebetween this method and a particular multi-point finite volumescheme, without any approximate numerical integration. The matrixof the final linear system is sparse, positive definite for alarge class of problems, but in general nonsymmetric. We next showthat these ideas also apply to mixed and upwind-mixed finiteelement discretizations of nonlinear parabolicconvection–diffusion–reaction problems. Besides the theoreticalrelationship between the two methods, the results allow forimportant computational savings in the mixed finite elementmethod, which we finally illustrate on a set of numericalexperiments.

In this paper we propose a solution of the Lambertian shape-from-shading (SFS) problem by designing a new mathematical framework based on the notion of viscosity solution. The power of our approach is twofolds: (1) it defines a notion of weak solutions (in the viscosity sense) which does notnecessarily require boundary data. Moreover, it allows to characterize theviscosity solutions by their “minimums”; and (2) it unifies the works of [Rouy and Tourin, SIAM J. Numer. Anal.29 (1992) 867–884], [Lions et al., Numer. Math.64 (1993) 323–353], [Falcone and Sagona, Lect. Notes Math.1310 (1997) 596–603],[Prados et al., Proc. 7th Eur. Conf. Computer Vision2351 (2002) 790–804; Prados and Faugeras, IEEE Comput. Soc. Press2 (2003) 826–831], based on the notion of viscosity solutions and the work of [Dupuis and Oliensis, Ann. Appl. Probab.4 (1994) 287–346] dealing with classical solutions.

In this paper we prove the discrete compactness property for a discontinuous Galerkin approximation of Maxwell's systemon quite general tetrahedral meshes. As a consequence, a discrete Friedrichs inequality is obtainedand the convergence of the discrete eigenvalues to the continuous ones is deducedusing the theory of collectively compact operators.Some numerical experiments confirm the theoretical predictions.