Raffinement de la borne spectrale d'un faisceau de matrices
Rate of convergence of finite-difference approximations for degenerate linear parabolic equations with and coefficients.
Realization of Dirichlet conditions in RKPM
Recent developments in wavelet methods for the solution of PDE's
After reviewing some of the properties of wavelet bases, and in particular the property of characterisation of function spaces via wavelet coefficients, we describe two new approaches to, respectively, stabilisation of numerically unstable PDE's and to non linear (adaptive) solution of PDE's, which are made possible by these properties.
Recent results in the approximation of free boundaries
Reconstruction algorithms for an inverse medium problem
In this paper, we consider a two-dimensional inverse medium problem from noisy observation data. We propose effective reconstruction algorithms to detect the number, the location and the size of the piecewise constant medium within a body, and then we try to recover the unknown shape of inhomogeneous media. This problem is nonlinear and ill-posed, thus we should consider stable and elegant approaches in order to improve the corresponding approximation. We give several examples to show the viability...
Reconstruction of the right-hand side of the elliptical equation from observation data obtained at the boundary
Reconstruction of thickness variation of a dielectric coating through the generalized impedance boundary conditions
We deal with an inverse scattering problem whose aim is to determine the thickness variation of a dielectric thin coating located on a conducting structure of unknown shape. The inverse scattering problem is solved through the application of the Generalized Impedance Boundary Conditions (GIBCs) which contain the thickness, curvature as well as material properties of the coating and they have been obtained in the previous work [B. Aslanyürek, H. Haddar and H.Şahintürk, Wave Motion 48 (2011) 681–700]...
Reduced continuity finite element methods for first order scalar hyperbolic equations
Reducing the bandwidth in solving linear algebraic systems arising in the finite element method
The matrix of the system of linear algebraic equations, arising in the application of the finite element method to one-dimensional problems, is a bandmatrix. In approximations of high order, the band is very wide but the elements situated far from the diagonal of the matrix are negligibly small as compared with the diagonal elements. The aim of the paper is to show on a model problem that in practice it is possible to work with a matrix of the system the bandwidth of which is reduced. A simple...
Reduktionsverfahren für Differenzengleichungen bei Randwertaufgaben II. -Solving Difference Equations for Boundary Value Problems by Reduction Methods II.
Reduktionsverfahren für Differenzengleichungen bei Randwertaufgaben I.
Regularization and stabilization of discrete saddle-point variational problems.
Regularization of nonlinear ill-posed problems by exponential integrators
The numerical solution of ill-posed problems requires suitable regularization techniques. One possible option is to consider time integration methods to solve the Showalter differential equation numerically. The stopping time of the numerical integrator corresponds to the regularization parameter. A number of well-known regularization methods such as the Landweber iteration or the Levenberg-Marquardt method can be interpreted as variants of the Euler method for solving the Showalter differential...
Reissner-Mindlin model for plates of variable thickness. Solution by mixed-interpolated elements
Hard clamped and hard simply supported elastic plate is considered. The mixed finite element analysis combined with some interpolation, proposed by Brezzi, Fortin and Stenberg, is extended to the case of variable thickness and anisotropic material.
Relaxation of an optimal design problem in fracture mechanic: the anti-plane case
In the framework of the linear fracture theory, a commonly-used tool to describe the smooth evolution of a crack embedded in a bounded domain Ω is the so-called energy release rate defined as the variation of the mechanical energy with respect to the crack dimension. Precisely, the well-known Griffith's criterion postulates the evolution of the crack if this rate reaches a critical value. In this work, in the anti-plane scalar case, we consider the shape design problem which consists in optimizing...
Relèvement polynômial de traces et applications
Reliable solutions of problems in the deformation theory of plasticity with respect to uncertain material function
Maximization problems are formulated for a class of quasistatic problems in the deformation theory of plasticity with respect to an uncertainty in the material function. Approximate problems are introduced on the basis of cubic Hermite splines and finite elements. The solvability of both continuous and approximate problems is proved and some convergence analysis presented.
Remarks on balanced norm error estimates for systems of reaction-diffusion equations
Error estimates of finite element methods for reaction-diffusion problems are often realized in the related energy norm. In the singularly perturbed case, however, this norm is not adequate. A different scaling of the seminorm leads to a balanced norm which reflects the layer behavior correctly. We discuss the difficulties which arise for systems of reaction-diffusion problems.
Remarks on the Ciarlet-Raviart mixed finite element.