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Displaying 61 –
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114
We introduce and analyze a numerical strategy
to approximate effective coefficients in stochastic homogenization of discrete elliptic
equations. In particular, we consider the simplest case possible: An elliptic equation on
the d-dimensional lattice
with independent and identically distributed conductivities on the associated edges.
Recent results by Otto and the author quantify the error made by approximating
the homogenized coefficient by the averaged energy of a regularized
corrector (with...
In this paper, we present numerical methods for the determination of solitons, that consist in spatially localized stationary states of nonlinear scalar equations or coupled systems arising in nonlinear optics. We first use the well-known shooting method in order to find excited states (characterized by the number of nodes) for the classical nonlinear Schrödinger equation. Asymptotics can then be derived in the limits of either large are large nonlinear exponents . In a second part, we compute...
In this paper, we present numerical methods
for the determination of solitons, that consist in spatially localized
stationary states of nonlinear scalar equations or coupled systems
arising in nonlinear optics.
We first use the well-known shooting method in order to find
excited states (characterized by the number k of nodes) for the
classical nonlinear Schrödinger equation. Asymptotics can then
be derived in the limits of either large k are large nonlinear
exponents σ.
In a second part, we compute...
These notes give a state of the art of numerical homogenization methods for linear
elliptic equations. The guideline of these notes is analysis. Most of the numerical
homogenization methods can be seen as (more or less different) discretizations of the same
family of continuous approximate problems, which H-converges to the homogenized problem.
Likewise numerical correctors may also be interpreted as approximations of Tartar’s
correctors. Hence the...
We study existence and some properties of solutions of the nonlinear elliptic equation N(x,a(u))Lu = f in unbounded domains. The above method is not a variational problem. Our techniques involve fixed point arguments and Galerkin method.
We study the 3-D elasticity problem in the case of a
non-symmetric heterogeneous rod. The asymptotic expansion of the solution is
constructed. The coercitivity of the homogenized equation is proved. Estimates
are derived for the difference between the truncated series and the exact solution.
An axisymmetric second order elliptic problem with mixed boundarz conditions is considered. A part of the boundary has to be found so as to minimize one of four types of cost functionals. The numerical realization is presented in detail. The convergence of piecewise linear approximations is proved. Several numerical examples are given.
The shape of the meridian curve of an elastic body is optimized within a class of Lipschitz functions. Only axisymmetric mixed boundary value problems are considered. Four different cost functionals are used and approximate piecewise linear solutions defined on the basis of a finite element technique. Some convergence and existence results are derived by means of the theory of the appropriate weighted Sobolev spaces.
Currently displaying 61 –
80 of
114