This paper deals with ergodic theorems for particular time-inhomogeneous Markov processes, whose time-inhomogeneity is asymptotically periodic. Under a Lyapunov/minorization condition, it is shown that, for any measurable bounded function f, the time average $\frac{1}{t} \int_0^t f(X_s)ds$ converges in $\mathbb{L}^2$ towards a limiting distribution, starting from any initial distribution for the process $(X_t)_{t \geq 0}$ . This convergence can be improved to an almost sure convergence under an additional assumption on the initial measure. This result is then applied to show the existence of a quasi-ergodic distribution for processes absorbed by an asymptotically periodic moving boundary, satisfying a conditional Doeblin condition.

This work presents a moving mesh methodology based on the solution of a pseudo flow problem. The mesh motion is modeled as a pseudo Stokes problem solved by an explicit finite element projection method. The mesh quality requirements are satisfied by employing a null divergent velocity condition. This methodology is applied to triangular unstructured meshes and compared to well known approaches such as the ones based on diffusion and pseudo structural problems. One of the test cases is an airfoil with a fully meshed domain. A specific rotation velocity is imposed as the airfoil boundary condition. The other test is a set of two cylinders that move toward each other. A mesh quality criteria is employed to identify critically distorted elements and to evaluate the performance of each mesh motion approach. The results obtained for each test case show that the pseudo-flow methodology produces satisfactory meshes during the moving process.

This article describes a number of velocity-based moving mesh numerical methods for multidimensional nonlinear time-dependent partial differential equations (PDEs). It consists of a short historical review followed by a detailed description of a recently developed multidimensional moving mesh finite element method based on conservation. Finite element algorithms are derived for both mass-conserving and non mass-conserving problems, and results shown for a number of multidimensional nonlinear test problems, including the second order porous medium equation and the fourth order thin film equation as well as a two-phase problem. Further applications and extensions are referenced.

Les aciers TRIP présentent une ductilité élevée et une forte résistance. Cette propriété remarquable trouve son origine dans la transformation martensitique de l'austénite. Ce comportement est modélisé avec une approche par transition d'échelle en utilisant le cadre de la thermomicromécanique pour déterminer les forces motrices ainsi que les lois de germination et de croissance qui contrôlent l'évolution de la microstructure. La présence de frontières mobiles et la discontinuité du champ de déformation au passage de ces interfaces sont prises en compte dans l'écriture des équations de champs. On montre que le champ de contrainte interne associé à la transformation se compose d'une contribution à longue distance responsable de la sélection des variantes et d'une contribution locale liée à la morphologie des microdomaines et au mode de croissance. On utilise une approche cristallographique à variables internes pour déterminer ces contributions. L'utilisation d'un modèle auto-cohérent classique permet ensuite de décrire le comportement global. Les résultats numériques obtenus sont comparés avec les données expérimentales pour un alliage FeNi et un acier TRIP 0,2C-1,5Si-1,5Mn élaboré par l'IRSID.

Let X be a one-dimensional strong Markov process with continuous sample paths. Using Volterra-Stieltjes integral equation techniques we investigate Hölder continuity and differentiability of first passage time distributions of X with respect to continuous lower and upper moving boundaries. Under mild assumptions on the transition function of X we prove the existence of a continuous first passage time density to one-sided differentiable moving boundaries and derive a new integral equation for this density. We apply our results to Brownian motion and its nonrandom Markovian transforms, in particular to the Ornstein-Uhlenbeck process.