340 research outputs found
Correctors for the Neumann problem in thin domains with locally periodic oscillatory structure
In this paper we are concerned with convergence of solutions of the Poisson
equation with Neumann boundary conditions in a two-dimensional thin domain
exhibiting highly oscillatory behavior in part of its boundary. We deal with
the resonant case in which the height, amplitude and period of the oscillations
are all of the same order which is given by a small parameter .
Applying an appropriate corrector approach we get strong convergence when we
replace the original solutions by a kind of first-order expansion through the
Multiple-Scale Method.Comment: to appear in Quarterly of Applied Mathematic
Error estimates for a Neumann problem in highly oscillating thin domains
In this work we analyze convergence of solutions for the Laplace operator
with Neumann boundary conditions in a two-dimensional highly oscillating domain
which degenerates into a segment (thin domains) of the real line. We consider
the case where the height of the thin domain, amplitude and period of the
oscillations are all of the same order, given by a small parameter .
We investigate strong convergence properties of the solutions using an
appropriate corrector approach. We also give error estimates when we replace
the original solutions for the second-order expansion through the
Multiple-Scale Method
Multilevel quadrature for elliptic problems on random domains by the coupling of FEM and BEM
Elliptic boundary value problems which are posed on a random domain can be
mapped to a fixed, nominal domain. The randomness is thus transferred to the
diffusion matrix and the loading. While this domain mapping method is quite
efficient for theory and practice, since only a single domain discretisation is
needed, it also requires the knowledge of the domain mapping.
However, in certain applications, the random domain is only described by its
random boundary, while the quantity of interest is defined on a fixed,
deterministic subdomain. In this setting, it thus becomes necessary to compute
a random domain mapping on the whole domain, such that the domain mapping is
the identity on the fixed subdomain and maps the boundary of the chosen fixed,
nominal domain on to the random boundary.
To overcome the necessity of computing such a mapping, we therefore couple
the finite element method on the fixed subdomain with the boundary element
method on the random boundary. We verify the required regularity of the
solution with respect to the random domain mapping for the use of multilevel
quadrature, derive the coupling formulation, and show by numerical results that
the approach is feasible
Multiscale methods for problems with complex geometry
We propose a multiscale method for elliptic problems on complex domains, e.g.
domains with cracks or complicated boundary. For local singularities this paper
also offers a discrete alternative to enrichment techniques such as XFEM. We
construct corrected coarse test and trail spaces which takes the fine scale
features of the computational domain into account. The corrections only need to
be computed in regions surrounding fine scale geometric features. We achieve
linear convergence rate in energy norm for the multiscale solution. Moreover,
the conditioning of the resulting matrices is not affected by the way the
domain boundary cuts the coarse elements in the background mesh. The analytical
findings are verified in a series of numerical experiments
Kinetic Brownian motion on Riemannian manifolds
International audienceWe consider in this work a one parameter family of hypoelliptic diffusion processes on the unit tangent bundle T 1 M of a Riemannian manifold (M, g), collectively called kinetic Brownian motions, that are random perturbations of the geodesic flow, with a parameter σ quantifying the size of the noise. Projection on M of these processes provides random C 1 paths in M. We show, both qualitively and quantitatively, that the laws of these M-valued paths provide an interpolation between geodesic and Brownian motions. This qualitative description of kinetic Brownian motion as the parameter σ varies is complemented by a thourough study of its long time asymptotic behaviour on rotationally invariant manifolds, when σ is fixed, as we are able to give a complete description of its Poisson boundary in geometric terms
Multiple Scale Systems-Modeling, Analysis and Numerics
[no abstract available
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