Results 1  10
of
211
Estimating differential quantities using polynomial fitting of osculating jets
"... This paper addresses the pointwise estimation of differential properties of a smooth manifold S —a curve in the plane or a surface in 3D — assuming a point cloud sampled over S is provided. The method consists of fitting the local representation of the manifold using a jet, and either interpolation ..."
Abstract

Cited by 87 (2 self)
 Add to MetaCart
This paper addresses the pointwise estimation of differential properties of a smooth manifold S —a curve in the plane or a surface in 3D — assuming a point cloud sampled over S is provided. The method consists of fitting the local representation of the manifold using a jet, and either interpolation or approximation. A jet is a truncated Taylor expansion, and the incentive for using jets is that they encode all local geometric quantities —such as normal, curvatures, extrema of curvature. On the way to using jets, the question of estimating differential properties is recasted into the more general framework of multivariate interpolation / approximation, a wellstudied problem in numerical analysis. On a theoretical perspective, we prove several convergence results when the samples get denser. For curves and surfaces, these results involve asymptotic estimates with convergence rates depending upon the degree of the jet used. For the particular case of curves, an error bound is also derived. To the best of our knowledge, these results are among the first ones providing accurate estimates for differential quantities of order three and more. On the algorithmic side, we solve the interpolation/approximation problem using Vandermonde systems. Experimental results for surfaces of R 3 are reported. These experiments illustrate the asymptotic convergence results, but also the robustness of the methods on general Computer Graphics models.
An augmented Lagrangianbased approach to the Oseen problem
 SIAM J. Sci. Comput
, 2006
"... Abstract. We describe an effective solver for the discrete Oseen problem based on an augmented Lagrangian formulation of the corresponding saddle point system. The proposed method is a block triangular preconditioner used with a Krylov subspace iteration like BiCGStab. The crucial ingredient is a no ..."
Abstract

Cited by 52 (23 self)
 Add to MetaCart
Abstract. We describe an effective solver for the discrete Oseen problem based on an augmented Lagrangian formulation of the corresponding saddle point system. The proposed method is a block triangular preconditioner used with a Krylov subspace iteration like BiCGStab. The crucial ingredient is a novel multigrid approach for the (1,1) block, which extends a technique introduced by Schöberl for elasticity problems to nonsymmetric problems. Our analysis indicates that this approach results in fast convergence, independent of the mesh size and largely insensitive to the viscosity. We present experimental evidence for both isoP2P0 and isoP2P1 finite elements in support of our conclusions. We also show results of a comparison with two stateoftheart preconditioners, showing the competitiveness of our approach. Key words. Navier–Stokes equations, finite element, iterative methods, multigrid, preconditioning AMS subject classifications. 65F10, 65N22, 65F50 DOI. 10.1137/050646421 1. Introduction. We consider the numerical solution of the steady Navier– Stokes equations governing the flow of a Newtonian, incompressible viscous fluid. Let Ω ⊂ R d (d =2,3) be a bounded, connected domain with a piecewise smooth
Numerical mathematics
, 2000
"... Abstract. In this paper we introduce some basic differential models for the description of blood flow in the circulatory system. We comment on their mathematical properties, their meaningfulness and their limitation to yield realistic and accurate numerical simulations, and their contribution for a ..."
Abstract

Cited by 40 (5 self)
 Add to MetaCart
Abstract. In this paper we introduce some basic differential models for the description of blood flow in the circulatory system. We comment on their mathematical properties, their meaningfulness and their limitation to yield realistic and accurate numerical simulations, and their contribution for a better understanding of cardiovascular physiopathology. Mathematics Subject Classification (2000). 92C50,96C10,76Z05,74F10,65N30,65M60. Keywords. Cardiovascular mathematics; mathematical modeling; fluid dynamics; Navier– Stokes equations; numerical approximation; finite element method; differential equations. 1.
On the computation of crystalline microstructure
 Acta Numerica
, 1996
"... Microstructure is a feature of crystals with multiple symmetryrelated energyminimizing states. Continuum models have been developed explaining microstructure as the mixture of these symmetryrelated states on a fine scale to minimize energy. This article is a review of numerical methods and the num ..."
Abstract

Cited by 39 (16 self)
 Add to MetaCart
Microstructure is a feature of crystals with multiple symmetryrelated energyminimizing states. Continuum models have been developed explaining microstructure as the mixture of these symmetryrelated states on a fine scale to minimize energy. This article is a review of numerical methods and the numerical analysis for the computation of crystalline microstructure.
Iterative Methods for Problems in Computational Fluid Dynamics
 ITERATIVE METHODS IN SCIENTIFIC COMPUTING
, 1996
"... We discuss iterative methods for solving the algebraic systems of equations arising from linearization and discretization of primitive variable formulations of the incompressible NavierStokes equations. Implicit discretization in time leads to a coupled but linear system of partial differential equ ..."
Abstract

Cited by 28 (5 self)
 Add to MetaCart
We discuss iterative methods for solving the algebraic systems of equations arising from linearization and discretization of primitive variable formulations of the incompressible NavierStokes equations. Implicit discretization in time leads to a coupled but linear system of partial differential equations at each time step, and discretization in space then produces a series of linear algebraic systems. We give an overview of commonly used time and space discretization techniques, and we discuss a variety of algorithmic strategies for solving the resulting systems of equations. The emphasis is on preconditioning techniques, which can be combined with Krylov subspace iterative methods. In many cases the solution of subsidiary problems such as the discrete convectiondiffusion equation and the discrete Stokes equations plays a crucial role. We examine iterative techniques for these problems and show how they can be integrated into effective solution algorithms for the NavierStokes equa...
Overlapping Schwarz Methods For Maxwell's Equations In Three Dimensions
 Numer. Math
, 1997
"... . Twolevel overlapping Schwarz methods are considered for finite element problems of 3D Maxwell's equations. N'ed'elec elements built on tetrahedra and hexahedra are considered. Once the relative overlap is fixed, the condition number of the additive Schwarz method is bounded, independently of the ..."
Abstract

Cited by 25 (4 self)
 Add to MetaCart
. Twolevel overlapping Schwarz methods are considered for finite element problems of 3D Maxwell's equations. N'ed'elec elements built on tetrahedra and hexahedra are considered. Once the relative overlap is fixed, the condition number of the additive Schwarz method is bounded, independently of the diameter of the triangulation and the number of subregions. A similar result is obtained for a multiplicative method. These bounds are obtained for quasiuniform triangulations. In addition, for the Dirichlet problem, the convexity of the domain has to be assumed. Our work generalizes wellknown results for conforming finite elements for second order elliptic scalar equations. 1. Introduction. When timedependent Maxwell's equations are considered, the electric field u satisfies the following equation curlcurlu + " @ 2 u @t 2 + oe @u @t = \Gamma @J @t ; in \Omega\Gamma (1) where J(x; t) is the current density and ", , oe describe the electromagnetic properties of the medium. For their...
An Explicit Link between Gaussian Fields and . . .
 PREPRINTS IN MATHEMATICAL SCIENCES
, 2010
"... Continuously indexed Gaussian fields (GFs) is the most important ingredient in spatial statistical modelling and geostatistics. The specification through the covariance function gives an intuitive interpretation of its properties. On the computational side, GFs are hampered with the bign problem, ..."
Abstract

Cited by 21 (7 self)
 Add to MetaCart
Continuously indexed Gaussian fields (GFs) is the most important ingredient in spatial statistical modelling and geostatistics. The specification through the covariance function gives an intuitive interpretation of its properties. On the computational side, GFs are hampered with the bign problem, since the cost of factorising dense matrices is cubic in the dimension. Although the computational power today is alltimehigh, this fact seems still to be a computational bottleneck in applications. Along with GFs, there is the class of Gaussian Markov random fields (GMRFs) which are discretely indexed. The Markov property makes the involved precision matrix sparse which enables the use of numerical algorithms for sparse matrices, that for fields in R 2 only use the squareroot of the time required by general algorithms. The specification of a GMRF is through its full conditional distributions but its marginal properties are not transparent in such a parametrisation. In this paper, we show that using an approximate stochastic weak solution to (linear) stochastic partial differential equations (SPDEs), we can, for some GFs in the Matérn class, provide an explicit link, for any triangulation of R d, between GFs and GMRFs. The consequence is that we can take the best from the two worlds and do the modelling using GFs but do the computations using GMRFs. Perhaps more importantly,
Spectral Element Methods for Transitional Flows
 J. Sci. Comput
, 2002
"... We describe the development and implementation of an efficient spectral element code for simulating transitional flows in complex threedimensional domains. Critical to this effort is the use of geometrically nonconforming elements that allow localized refinement in regions of interest, coupled ..."
Abstract

Cited by 20 (7 self)
 Add to MetaCart
We describe the development and implementation of an efficient spectral element code for simulating transitional flows in complex threedimensional domains. Critical to this effort is the use of geometrically nonconforming elements that allow localized refinement in regions of interest, coupled with a stabilized highorder timesplit formulation of the semidiscrete NavierStokes equations. Simulations of transition in a model of an arteriovenous graft illustrate the potential of this approach in biomechanical applications.
On the approximation of the unsteady Navier–Stokes equations by finite element projection methods
 NUMER. MATH
, 1998
"... ..."
Finite element analysis of microstructure for the cubic to tetragonal transformation
 SIAM J. Numer. Anal
, 1998
"... Abstract. Martensitic crystals which can undergo a cubic to tetragonal phase transformation have a nonconvex energy density with three symmetryrelated, rotationally invariant energy wells. We give estimates for the numerical approximation of a firstorder laminate for such martensitic crystals. We ..."
Abstract

Cited by 19 (13 self)
 Add to MetaCart
Abstract. Martensitic crystals which can undergo a cubic to tetragonal phase transformation have a nonconvex energy density with three symmetryrelated, rotationally invariant energy wells. We give estimates for the numerical approximation of a firstorder laminate for such martensitic crystals. We give bounds for the L 2 convergence of directional derivatives in the “twin ” plane, for the L 2 convergence of the deformation, for the weak convergence of the deformation gradient, for the convergence of the microstructure, and for the convergence of nonlinear integrals of the deformation gradient.