4,382 research outputs found
A Day in Linguistic History
The day begins like any other day. A few students straggle toward the Union in search of coffee and eggs. Delivery trucks come on the campus. At the entrance, the sign still reads SUSAN DOE UNIVERSITY FOUNDED 1894 NO SOLICITORS. This is it, the University, known affectionately by students and faculty as Sue Doe U
LĀ² -estimates for the evolving surface finite element method
In this paper we consider the evolving surface ļ¬nite element method
for the advection and diļ¬usion of a conserved scalar quantity on a moving
surface. In an earlier paper using a suitable variational formulation in time
dependent Sobolev space we proposed and analysed a ļ¬nite element method
using surface ļ¬nite elements on evolving triangulated surfaces. An optimal
order HĀ¹ -error bound was proved for linear ļ¬nite elements. In this work we
prove the optimal error bound in LĀ² (Ī(t)) uniformly in time
On Approximations of the Curve Shortening Flow and of the Mean Curvature Flow based on the DeTurck trick
In this paper we discuss novel numerical schemes for the computation of the
curve shortening and mean curvature flows that are based on special
reparametrizations. The main idea is to use special solutions to the harmonic
map heat flow in order to reparametrize the equations of motion. This idea is
widely known from the Ricci flow as the DeTurck trick. By introducing a
variable time scale for the harmonic map heat flow, we obtain families of
numerical schemes for the reparametrized flows. For the curve shortening flow
this family unveils a surprising geometric connection between the numerical
schemes in [5] and [9]. For the mean curvature flow we obtain families of
schemes with good mesh properties similar to those in [3]. We prove error
estimates for the semi-discrete scheme of the curve shortening flow. The
behaviour of the fully-discrete schemes with respect to the redistribution of
mesh points is studied in numerical experiments. We also discuss possible
generalizations of our ideas to other extrinsic flows
Second order splitting of a class of fourth order PDEs with point constraints
We formulate a well-posedness and approximation theory for a class of generalised saddle point problems with a specific form of constraints. In this way we develop an approach to a class of fourth order elliptic partial differential equations with point constraints using the idea of splitting into coupled second order equations. An approach is formulated using a penalty method to impose the constraints. Our main motivation is to treat certain fourth order equations involving the biharmonic operator and point Dirichlet constraints for example arising in the modelling of biomembranes on curved and flat surfaces but the approach may be applied more generally. The theory for well-posedness and approximation is presented in an abstract setting. Several examples are described together with some numerical experiments
A fully discrete evolving surface finite element method
In this paper we consider a time discrete evolving surface finite element method for the advection and diffusion of a conserved scalar quantity on a moving surface. In earlier papers using a suitable variational formulation in time dependent Sobolev space we proposed and analyzed a finite element method using surface finite elements on evolving triangulated surfaces [IMA J. Numer Anal., 25 (2007), pp. 385--407; Math. Comp., to appear]. Optimal order L2(Ī(t)) and H1(Ī(t)) error bounds were proved for linear finite elements. In this work we prove optimal order error bounds for a backward Euler time discretization
Analysis and computations for a model of quasi-static deformation of a thinning sheet arising in superplastic forming
We consider a mathematical model for the quasi-static deformation of a thinning sheet. The model couples a first-order equation for the thickness of the sheet to a prescribed curvature equation for the displacement of the sheet. We prove a local in time existence and uniqueness theorem for this system when the sheet can be written as a graph. A contact problem is formulated for a sheet constrained to be above a mould. Finally we present some computational results
An abstract framework for parabolic PDEs on evolving spaces
We present an abstract framework for treating the theory of well-posedness of
solutions to abstract parabolic partial differential equations on evolving
Hilbert spaces. This theory is applicable to variational formulations of PDEs
on evolving spatial domains including moving hypersurfaces. We formulate an
appropriate time derivative on evolving spaces called the material derivative
and define a weak material derivative in analogy with the usual time derivative
in fixed domain problems; our setting is abstract and not restricted to
evolving domains or surfaces. Then we show well-posedness to a certain class of
parabolic PDEs under some assumptions on the parabolic operator and the data.Comment: 38 pages. Minor typos correcte
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