Mathematical Models and Methods in Applied Sciences

In this work we address a problem governed by linear parabolic partial differential equations set in two adjoining domains, coupled by nonlinear interface conditions of Neumann type. In particular, we address the existence and uniqueness of strong solutions by applying the strong maximum principle, the Schauder fixed point theorem and the fundamental solutions of linear parabolic partial differential equations.

In the first part of this work, we consider the properties of a linear parabolic partial differential equation set on a single domain with a nonlinear boundary condition. After having addressed the well-posedness and some comparison results for the problem on one domain, in the second part of this work we address the case of coupled problems on adjoining domains. In both cases, we complete the understanding of the behavior of the solution of the problems at hand by means of numerical simulations.

The theoretical results obtained here are applied to study the behavior of a biological model for the transfer of chemicals through thin biological membranes. This model represents the dynamics of the concentration u of a chemical solution separated from the exterior by a semi-permeable membrane.

The analysis of the two-domain problem that we carry out could also be used to investigate the convergence property of iterative substructuring methods applied to the approximation of multidomain problems with nonlinear coupling of Neumann type.

Acoustic waves travelling in axisymmetric pipes with visco-thermal losses at the wall obey a Webster–Lokshin model. Their simulation may be achieved by concatenating scattering matrices of elementary transfer functions associated with nearly constant parameters (e.g. curvature). These functions are computed analytically and involve diffusive pseudo-differential operators, for which we have representation formula and input-output realizations, yielding direct numerical approximations of finite order. The method is based on some involved complex analysis.

In this paper the vanishing Debye length limit (space charge neutral limit) of bipolar time-dependent drift-diffusion models for semiconductors with p-n-junctions (i.e. with a fixed bipolar background charge) is studied in the multi-dimensional case. For generally smooth sign-changing doping profiles with good boundary conditions, the quasineutral limit (zero-Debye-length limit) is justified rigorously in the Sobolev's norm uniformly in time. The proof is based on the elaborate energy method and the relative entropy functional method which yields the uniform estimates with respect to the scaled Debye length.

The paper deals with a phase transition model applied to a two-phase system. There is a wide literature on the study of phase transition processes in case that no voids nor overlapping can occur between the two phases. The main novelty of our approach is the possibility of having voids during the phase change. This aspect is described in the model by the mass balance equation whose effects are included by means of the pressure of the system in the dynamical relations. The state variables are the absolute temperature (whose evolution is ruled by the entropy balance equation), the strain tensor (satisfying a quasi-static macroscopic equation of motion), and the volume fractions of the two phases (whose evolutions are described by a vectorial equation coming from the principle of virtual power and related to the microscopic motions). Well-posedness of the initial-boundary value problem associated to the PDEs system resulting from this model is proved.

Various types of oil traps have been found to be associated with salt domes in subsurface geology. In this paper the diapiric rise of light salt layers through a denser overburden — the surrounding rocks — is modeled assuming that, in a geological time scale, salt and rocks layers behave like Newtonian fluids. A Lagrangian approach is adopted to track the interface between layers, within the framework of a finite element space discretization. An accurate description of large deformations due to salt movement is achieved using a grid adaptation technique based on geometrical refinement. Different geological cases have been simulated in order to describe the behavior of rocks and estimate the effect on diapiric growth of buoyancy force, differential loading, gravitational gliding and thin-skinned regional extension. Our computational model accounts also for sedimentation and compaction of the overburden.

This paper is concerned with a finite element method (FEM) for multiscale problems in linear elasticity. We propose a method which discretizes the physical problem directly by a macroscopic FEM, coupled with a microscopic FEM resolving the micro scale on small cells or patches. The assembly process of the unknown macroscopic model is done without iterative cycles. The method allows to recover the macroscopic properties of the material in an efficient and cheap way. The microscale behavior can be reconstructed from the known micro and macro solutions. We give a fully discrete convergence analysis for the proposed method which takes into account the discretization errors at both micro and macro levels. In the case of a periodic elastic tensor, we give a priori error estimates for the displacement and for the macro and micro strains and stresses as well as an error estimate for the numerical homogenized tensor.