Degenerate parabolic variational inequalities with convection are solved bymeans of a combined relaxation method and method of characteristics. Themathematical problem is motivated by Richard's equation, modelling theunsaturated – saturated flow in porous media. By means of the relaxationmethod we control the degeneracy. The dominance of the convection iscontrolled by the method of characteristics.

In this paper, the main purpose is to reveal what kind of qualitative dynamicalchanges a continuous age-structured model may undergo as continuous reproduction is replaced withan annual birth pulse. Using the discrete dynamical system determined by the stroboscopic map we obtain an exact periodic solution of system with density-dependent fertility and obtain the threshold conditions for its stability. We also present formal proofs of the supercritical flip bifurcation at the bifurcation as well as extensive analysis of dynamics in unstableparameter regions. Above this threshold, there is a characteristic sequence of bifurcations, leading to chaotic dynamics, which implies that the dynamical behavior of the single species model with birth pulses are very complex, including small-amplitude annual oscillations, large-amplitude multi-annual cycles, and chaos. This suggests that birth pulse, in effect, provides a natural period or cyclicity that allowsfor a period-doubling route to chaos. Finally, we discuss the effects of generation delay on stability of positiveequilibrium (or positive periodic solution), and show that generation delay is found to act both as a destabilizing and a stabilizing effect.

The paper deals with the numerical resolution of the convection-diffusion system which arises when modeling combustion for turbulent flow. The considered model is of compressible turbulent reacting type where the turbulence-chemistry interactions are governed by additional balance equations. The system of PDE's, that governs such a model, turns out to be in non-conservation form and usual numerical approaches grossly fail in the capture of viscous shock layers. Put in other words, classical finite volume methods induce large errors when approximated the convection-diffusion extracted system. To solve this difficulty, recent works propose a nonlinear projection scheme based on cancellation phenomenon of relevant dissipation rates of entropy. Unfortunately, such a property never holds in the present framework. The nonlinear projection procedures are thus extended.

We study in this paper some numerical schemes for hyperbolic systemswith unilateral constraint. In particular, we deal with the scalar case, the isentropicgas dynamics system and the full-gas dynamics system.We prove the convergence of the scheme to an entropy solutionof the isentropicgas dynamics with unilateral constraint on the density and mass loss.We also study the non-trivial steady states of the system.

In this paper we propose a finite element method for the approximation ofsecond order elliptic problems on composite grids. The method isbased on continuous piecewise polynomial approximation on eachgrid and weak enforcement of the proper continuity at anartificial interface defined by edges (or faces) of one the grids.We prove optimal order a priori and energy type a posteriori error estimates in 2 and 3 space dimensions,and present some numerical examples.

In this paper, we study the exterior boundary value problems of the Darwinmodel to the Maxwell's equations. The variational formulation is establishedand the existence and uniqueness is proved. We use the infinite element methodto solve the problem, only a small amount of computational work is needed.Numerical examples are given as well as a proof of convergence.

We study the gradient flow for the total variation functional, which arises in image processing and geometric applications. We propose a variational inequality weak formulation for the gradient flow,and establish well-posedness of the problem by the energy method. The main idea of our approach is to exploit the relationship betweenthe regularized gradient flow (characterized by a small positive parameterε, see (1.7)) and the minimal surface flow [21]and the prescribed mean curvature flow [16].Since our approach is constructive and variational, finite element methods can be naturally applied to approximate weak solutions of the limiting gradient flow problem. We propose a fully discrete finite element method and establish convergence tothe regularized gradient flow problem as h,k → 0, and to the total variation gradient flow problem as h,k,ε → 0in general cases.Provided that the regularized gradient flow problem possessesstrong solutions, which is proved possible if the datum functionsare regular enough, we establish practical a priori error estimates for the fully discrete finite element solution, in particular, by focusing on the dependence of the error bounds on the regularization parameter ε. Optimal order error bounds are derived for the numerical solution under the meshrelation k = O(h2). In particular, it is shown thatall error bounds depend on $\frac{1}{\varepsilon}$ onlyin some lower polynomial order for small ε.