277 research outputs found
A quasi-isometric embedding theorem for groups
We show that every group of at most exponential growth with respect to
some left invariant metric admits a bi-Lipschitz embedding into a finitely
generated group such that is amenable (respectively, solvable,
satisfies a non-trivial identity, elementary amenable, of finite decomposition
complexity, etc.) whenever is. We also discuss some applications to
compression functions of Lipschitz embeddings into uniformly convex Banach
spaces, F{\o}lner functions, and elementary classes of amenable groups
Error analysis of a space-time finite element method for solving PDEs on evolving surfaces
In this paper we present an error analysis of an Eulerian finite element
method for solving parabolic partial differential equations posed on evolving
hypersurfaces in , . The method employs discontinuous
piecewise linear in time -- continuous piecewise linear in space finite
elements and is based on a space-time weak formulation of a surface PDE
problem. Trial and test surface finite element spaces consist of traces of
standard volumetric elements on a space-time manifold resulting from the
evolution of a surface. We prove first order convergence in space and time of
the method in an energy norm and second order convergence in a weaker norm.
Furthermore, we derive regularity results for solutions of parabolic PDEs on an
evolving surface, which we need in a duality argument used in the proof of the
second order convergence estimate
An Eulerian Finite Element Method for PDEs in time-dependent domains
The paper introduces a new finite element numerical method for the solution
of partial differential equations on evolving domains. The approach uses a
completely Eulerian description of the domain motion. The physical domain is
embedded in a triangulated computational domain and can overlap the
time-independent background mesh in an arbitrary way. The numerical method is
based on finite difference discretizations of time derivatives and a standard
geometrically unfitted finite element method with an additional stabilization
term in the spatial domain. The performance and analysis of the method rely on
the fundamental extension result in Sobolev spaces for functions defined on
bounded domains. This paper includes a complete stability and error analysis,
which accounts for discretization errors resulting from finite difference and
finite element approximations as well as for geometric errors coming from a
possible approximate recovery of the physical domain. Several numerical
examples illustrate the theory and demonstrate the practical efficiency of the
method.Comment: 27 pages, 3 figures, 8 table
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