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A new nonlinear Galerkin method based on nite element discretization is presented in this paper for a class of second order nonlinear parabolic equations. The new scheme is based on two di erent nite element spaces de ned respectively on one coarse grid with grid size H and one ne grid with grid size h H. Nonlinearity and time dependence are both treated on the coarse space and only a xed stationary equation needs to be solved on the ne space at each time. With linear nite element discretizations, it is proved that the di erence between the new nonlinear Galerkin solution and the standard Galerkin solution in H 1 () norm is of the order of H 3 .

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Abstract. When two miscible fluids, such as glycerol (glycerin) and water, are brought in contact, they immediately diffuse in each other. However if the diffusion is sufficiently slow, large concentration gradients exist during some time. They can lead to the appearance of an "effective interfacial tension". To study these phenomena we use the mathematical model consisting of the diffusion equation with convective terms and of the Navier-Stokes equations with the Korteweg stress. We prove the global existence and uniqueness of the solution for the associated initial-boundary value problem in a twodimensional bounded domain. We study the longtime behavior of the solution and show that it converges to the uniform composition distribution with zero velocity field. We also present numerical simulations of miscible drops and show how transient interfacial phenomena can change their shape.Mathematics Subject Classification. 35K50, 76D05.

International audienceSpectral properties of some integro-differential operators on R1 are studied. Characterization of the principal eigenvalue is obtained in terms of the positive eigenfunction. These results are used to prove local and global stability of travelling waves and to find their speed

The paper is concerned with the existence of pulses for monotone reaction-diffusion systems of two equations. For a general class of systems we prove that pulses exist if and only if the wave solutions propagate at positive speed. This result is applied to investigate the existence of pulses for the system of competition of species.