52 research outputs found
A convexity-preserving and perimeter-decreasing parametric finite element method for the area-preserving curve shortening flow
We propose and analyze a semi-discrete parametric finite element scheme for
solving the area-preserving curve shortening flow. The scheme is based on
Dziuk's approach (SIAM J. Numer. Anal. 36(6): 1808-1830, 1999) for the
anisotropic curve shortening flow. We prove that the scheme preserves two
fundamental geometric structures of the flow with an initially convex curve:
(i) the convexity-preserving property, and (ii) the perimeter-decreasing
property. To the best of our knowledge, the convexity-preserving property of
numerical schemes which approximate the flow is rigorously proved for the first
time. Furthermore, the error estimate of the semi-discrete scheme is
established, and numerical results are provided to demonstrate the
structure-preserving properties as well as the accuracy of the scheme.Comment: 24 pages, 2 figure
A high-order discontinuous Galerkin method for the poro-elasto-acoustic problem on polygonal and polyhedral grids
The aim of this work is to introduce and analyze a finite element
discontinuous Galerkin method on polygonal meshes for the numerical
discretization of acoustic waves propagation through poroelastic materials.
Wave propagation is modeled by the acoustics equations in the acoustic domain
and the low-frequency Biot's equations in the poroelastic one. The coupling is
introduced by considering (physically consistent) interface conditions, imposed
on the interface between the domains, modeling both open and sealed pores.
Existence and uniqueness is proven for the strong formulation based on
employing the semigroup theory. For the space discretization we introduce and
analyze a high-order discontinuous Galerkin method on polygonal and polyhedral
meshes, which is then coupled with Newmark- time integration schemes. A
stability analysis both for the continuous problem and the semi-discrete one is
presented and error estimates for the energy norm are derived for the
semidiscrete problem. A wide set of numerical results obtained on test cases
with manufactured solutions are presented in order to validate the error
analysis. Examples of physical interest are also presented to test the
capability of the proposed methods in practical cases.Comment: The proof of the well-posedness contains an error. This has an impact
on the whole paper. We need time to fix the issu
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Discrete Differential Geometry
This is the collection of extended abstracts for the 26 lectures and the open problems session at the second Oberwolfach workshop on Discrete Differential Geometry
Nonlocal Flow of Convex Plane Curves and Isoperimetric Inequalities
In the first part of the paper we survey some nonlocal flows of convex plane
curves ever studied so far and discuss properties of the flows related to
enclosed area and length, especially the isoperimetric ratio and the
isoperimetric difference. We also study a new nonlocal flow of convex plane
curves and discuss its evolution behavior. In the second part of the paper we
discuss necessary and sufficient conditions (in terms of the (mixed)
isoperimetric ratio or (mixed) isoperimetric difference) for two convex closed
curves to be homothetic or parallel.Comment: 23 page
High-order Discontinuous Galerkin methods for the elastodynamics equation on polygonal and polyhedral meshes
We propose and analyze a high-order Discontinuous Galerkin Finite Element Method for the approximate solution of wave propagation problems modeled by the elastodynamics equations on computational meshes made by polygonal and polyhedral elements. We analyze the well posedness of the resulting formulation, prove hp-version error a-priori estimates, and present a dispersion analysis, showing that polygonal meshes behave as classical simplicial/quadrilateral grids in terms of dispersion properties. The theoretical estimates are confirmed through various two-dimensional numerical verifications
Solution-adaptive Cartesian cell approach for viscous and inviscid flows
Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/77313/1/AIAA-13171-269.pd
A Posteriori Error Estimates for Surface Finite Element Methods
Problems involving the solution of partial differential equations over surfaces appear in many engineering and scientific applications. Some of those applications include crystal growth, fluid mechanics and computer graphics. Many times analytic solutions to such problems are not available. Numerical algorithms, such as Finite Element Methods, are used in practice to find approximate solutions in those cases.
In this work we present L2 and pointwise a posteriori error estimates for Adaptive Surface Finite Elements solving the Laplace-Beltrami equation −△Γ u = f . The two sources of errors for Surface Finite Elements are a Galerkin error, and a geometric error that comes from replacing the original surface by a computational mesh. A posteriori error estimates on flat domains only have a Galerkin component. We use residual type error estimators to measure the Galerkin error. The geometric component of our error estimate becomes zero if we consider flat domains, but otherwise has the same order as the residual one. This is different from the available energy norm based error estimates on surfaces, where the importance of the geometric components diminishes asymptotically as the mesh is refined. We use our results to implement an Adaptive Surface Finite Element Method.
An important tool for proving a posteriori error bounds for non smooth functions is the Scott-Zhang interpolant. A refined version of a standard Scott-Zhang interpolation bound is also proved during our analysis. This local version only requires the interpolated function to be in a Sobolev space defined over an element T instead of an element patch containing T.
In the last section we extend our elliptic results to get estimates for the surface heat equation ut − △Γ u = f using the elliptic reconstruction technique
Moduli space of simple polygons
In this work we study the space S(n) of positively oriented, simple n-gons
with labeled vertices up to oriented similarity and M(n) the moduli space of
simple n-gons as a quotient of S(n). We give a local description of S(n) and
M(n) around the regular n-gon and describe completely the two spaces when n = 4
using a topological Morse function. We also provide an asymptotic description
of the compactifications of S(4) and M(4) up to a complicated set at their
boundaries.Comment: 53 pages, 18 figure
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