97 research outputs found
A new fictitious domain method: Optimal convergence without cut elements
AbstractWe present a method of the fictitious domain type for the Poisson–Dirichlet problem. The computational mesh is obtained from a background (typically uniform Cartesian) mesh by retaining only the elements intersecting the domain where the problem is posed. The resulting mesh does not thus fit the boundary of the problem domain. Several finite element methods (XFEM, CutFEM) adapted to such meshes have been recently proposed. The originality of the present article consists in avoiding integration over the elements cut by the boundary of the problem domain, while preserving the optimal convergence rates, as confirmed by both the theoretical estimates and the numerical results
Duality-based Asymptotic-Preserving method for highly anisotropic diffusion equations
The present paper introduces an efficient and accurate numerical scheme for
the solution of a highly anisotropic elliptic equation, the anisotropy
direction being given by a variable vector field. This scheme is based on an
asymptotic preserving reformulation of the original system, permitting an
accurate resolution independently of the anisotropy strength and without the
need of a mesh adapted to this anisotropy. The counterpart of this original
procedure is the larger system size, enlarged by adding auxiliary variables and
Lagrange multipliers. This Asymptotic-Preserving method generalizes the method
investigated in a previous paper [arXiv:0903.4984v2] to the case of an
arbitrary anisotropy direction field
Spectral methods for kinetic theory models of viscoelastic fluids
This work is dedicated to the construction of numerical techniques for the models of viscoelastic fluids that result from polymer kinetic theory. Our main contributions are as follows: Inspired by the interpretation of the Oldroyd B model of dilute polymer solutions as a suspension of Hookean dumbbells in a Newtonian solvent, we have constructed new numerical methods for this model that respect some important properties of the underlying differential equations, namely the positive definiteness of the conformation tensor and an energy estimate. These methods have been implemented on the basis of a spectral discretization for simple Couette and Poiseuille planar flows as well as flow past a cylinder in a channel. Numerical experiments confirm the enhanced stability of our approach. Spectral methods have been designed and implemented for the simulation of mesoscopic models of polymeric liquids that do not possess closed-form constitutive equations. The methods are based on the Fokker-Planck equations rather than on the equivalent stochastic differential equations. We have considered the FENE dumbbell model of dilute polymer solutions and the Ă–ttinger reptation model of concentrated polymer solutions. The comparison with stochastic simulation techniques has been performed in the cases of both homogeneous flows and the flow past a cylinder in a channel. Our method turned out to be more efficient in most cases
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