Domain decomposition for an asymptotic geological fault modeling

We present in this paper a domain decomposition method to treat faults in geological basin modeling. The particularity of this model is that the faults whose widths are very small in comparison with the basin size, are not characterized as subdomains any more but as interfaces between sedimentary blocks. The originality of this work lies in the formulation of this new fault model and in the definition and the computation of the interface conditions between the subdomains. To cite this article: E. Flauraud et al., C. R. Mecanique 331 (2003).     

A Nonlinear Drift - Diffusion System with Electric Convection Arising in Electrophoretic and Semiconductor Modeling

A multi-dimensional transient drift-diffusion model for (at most) three charged particles, consisting of the continuity equations for the concentrations of the species and the Poisson equation for the electric potential, is considered. The diffusion terms depend on the concentrations. Such a system arises in electrophoretic modeling of three species (neutrally, positively and negatively charged) and in semiconductor theory for two species (positively charged holes and negatively charged electrons). Diffusion terms of degenerate type are also possible in semiconductor modeling. For the initial boundary value problem with mixed Dirichlet - Neumann boundary conditions and general reaction rates, a global existence result is proved. Uniqueness of solutions follows in the Dirichlet boundary case if the diffusion terms are uniformly parabolic or if the initial and boundary densities are strictly positive. Finally, we prove that solutions exist which are positive uniformly in time and globally bounded if the reaction rates satisfy appropriate growth conditions.     

Cyclic reduction and FACR methods for piecewise hermite bicubic orthogonal spline collocation

Cyclic reduction and Fourier analysis-cyclic reduction (FACR) methods are presented for the solution of the linear systems which arise when orthogonal spline collocation with piecewise Hermite bicubics is applied to boundary value problems for certain separable partial differential equations on a rectangle. On anN×N uniform partition, the cyclic reduction and Fourier analysis-cyclic reduction methods requireO(N ²log₂ N) andO(N ²log₂log₂ N) arithmetic operations, respectively.     

Differential Conditions for Constrained Nonlinear Programming via Pareto Optimization

We deal with the differential conditions for local optimality. The conditions that we derive for inequality constrained problems do not require constraint qualifications and are the broadest conditions based on only first-order and second-order derivatives. A similar result is proved for equality constrained problems, although the necessary conditions require the regularity of the equality constraints.     

On the properties of solutions to the Goursat-Darboux problem with boundary and distributed controls

We consider a control system described by the Goursat-Darboux equation. The system is controlled by distributed and boundary controls. The controls are subject to the constraints given as multivalued mappings with closed, possibly nonconvex, values depending on the phase variable. Alongside the initial constraints, we consider the convexified constraints and the constraints whose values are the extreme points of the convexified constraints. We study the questions of existence of solutions and establish connections between the solutions under various constraints.     

Fixed-sign solutions for a system of singular focal boundary value problems

We consider a system of focal boundary value problems where the nonlinearities may be singular in the independent variable and may also be singular in the dependent arguments. Using Schauder fixed point theorem, we establish criteria such that the system of boundary value problems has at least one fixed-sign solution.     

On the persistence of elliptic lower-dimensional tori in Hamiltonian systems under the first Melnikov condition and Rüssmann’s non-degeneracy condition

In this paper we formulate a theorem on the persistence of elliptic lower-dimensional invariant tori for nearly integrable analytic Hamiltonian systems under the first Melnikov condition and Rüssmann’s non-degeneracy condition, and give the measure estimates of parameters for the non-resonance conditions under Rüssmann’s non-degeneracy condition, which is essential for the proof of our result.     

On regularity of a weak solution to the Navier–Stokes equation with generalized impermeability boundary conditions

We consider the 3D Navier–Stokes equation with generalized impermeability boundary conditions. As auxiliary results, we prove the local in time existence of a strong solution (‘strong’ in a limited sense) and a theorem on structure. Then, taking advantage of the boundary conditions, we formulate sufficient conditions for regularity up to the boundary of a weak solution by means of requirements on one of the eigenvalues of the rate of deformation tensor. Finally, we apply these general results to the case of an axially symmetric flow with zero angular velocity.     

Further generalization of the Dubovicki-Milutin theorem

Combining the results of Walczak (Ref. 1) and Lasiecka (Refs. 2 and 3), we extend the well-known Dubovicki-Milutin theorem in two directions: (i) We allow for treatment of several equality constraints in optimization problems; and (ii) We assume less regularity of optimization problems by requiring the existence of external cones instead of tangent cones. An example of an optimization problem is also considered.     

On the convergence rate and optimization of a numerical method with splitting of boundary conditions for the stokes system in a spherical layer in the axisymmetric case: Modification for thick layers

The convergence rate of a fast-converging second-order accurate iterative method with splitting of boundary conditions constructed by the authors for solving an axisymmetric Dirichlet boundary value problem for the Stokes system in a spherical gap is studied numerically. For R/r exceeding about 30, where r and R are the radii of the inner and outer boundary spheres, it is established that the convergence rate of the method is lower (and considerably lower for large R/r) than the convergence rate of its differential version. For this reason, a really simpler, more slowly converging modification of the original method is constructed on the differential level and a finite-element implementation of this modification is built. Numerical experiments have revealed that this modification has the same convergence rate as its differential counterpart for R/r of up to 5 × 10³. When the multigrid method is used to solve the split and auxiliary boundary value problems arising at iterations, the modification is more efficient than the original method starting from R/r ～ 30 and is considerably more efficient for large values of R/r. It is also established that the convergence rates of both methods depend little on the stretching coefficient η of circularly rectangular mesh cells in a range of η that is well sufficient for effective use of the multigrid method for arbitrary values of R/r smaller than ～ 5 × 10³.