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243
An Adaptive Finite Element Method for the Incompressible NavierStokes Equations on Timedependent Domains
, 1995
"... Contents 1 Introduction and Notations 1 2 Moving Boundary Problems 9 2.1 Flow in a Channel with a Moving Indentation : : : : : : : : : 9 2.2 Flow in a Water Pump : : : : : : : : : : : : : : : : : : : : : : 14 2.3 Time Discretization : : : : : : : : : : : : : : : : : : : : : : : : 21 2.3.1 Investiga ..."
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Cited by 23 (4 self)
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Contents 1 Introduction and Notations 1 2 Moving Boundary Problems 9 2.1 Flow in a Channel with a Moving Indentation : : : : : : : : : 9 2.2 Flow in a Water Pump : : : : : : : : : : : : : : : : : : : : : : 14 2.3 Time Discretization : : : : : : : : : : : : : : : : : : : : : : : : 21 2.3.1 Investigation of the continuous problem : : : : : : : : : 21 2.3.2 Semidiscretization : : : : : : : : : : : : : : : : : : : : 25 2.3.3 Full discretization : : : : : : : : : : : : : : : : : : : : : 29 3 Adaptive Finite Elements 31 3.1 Adaptive Algorithm : : : : : : : : : : : : : : : : : : : : : : : : 32 3.2 A residualbased error estimator : : : : : : : : : : : : : : : : : 37 3.3 Multigrid method on locally refined meshes : : : : : : : : : : : 40 4 Error estimators for the Stokes Equations 49 4.1 Discretization of the
Optimal a priori error estimates for the hpversion of the local discontinuous Galerkin method for convectiondiffusion problems
 Math. Comp
"... Abstract. We study the convergence properties of the hpversion of the local discontinuous Galerkin finite element method for convectiondiffusion problems; we consider a model problem in a onedimensional space domain. We allow arbitrary meshes and polynomial degree distributions and obtain upper b ..."
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Cited by 22 (7 self)
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Abstract. We study the convergence properties of the hpversion of the local discontinuous Galerkin finite element method for convectiondiffusion problems; we consider a model problem in a onedimensional space domain. We allow arbitrary meshes and polynomial degree distributions and obtain upper bounds for the energy norm of the error which are explicit in the meshwidth h, in the polynomial degree p, and in the regularity of the exact solution. We identify a special numerical flux for which the estimates are optimal in both h and p. The theoretical results are confirmed in a series of numerical examples. 1.
Fast deterministic pricing of options on Lévy driven assets
 M2AN Math. Model. Numer. Anal
, 2002
"... A partial integrodifferential equation (PIDE) ∂tu + A[u] = 0 for European contracts on assets with general jumpdiffusion price process of Lévy type is derived. The PIDE is localized to bounded domains and the error due to this localization is estimated. The localized PIDE is discretized by the θ ..."
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Cited by 21 (3 self)
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A partial integrodifferential equation (PIDE) ∂tu + A[u] = 0 for European contracts on assets with general jumpdiffusion price process of Lévy type is derived. The PIDE is localized to bounded domains and the error due to this localization is estimated. The localized PIDE is discretized by the θscheme in time and a wavelet Galerkin method with N degrees of freedom in space. The full Galerkin matrix for A can be replaced with a sparse matrix in the wavelet basis, and the linear systems for each time step are solved approximatively with GMRES in linear complexity. The total work of the algorithm for M time steps is bounded by O(MN(ln N) 2) operations and O(N ln(N)) memory. The deterministic algorithm gives optimal convergence rates (up to logarithmic terms) for the computed solution in the same complexity as finite difference approximations of the standard BlackScholes equation. Computational examples for various Lévy price processes (VG, CGMY) are presented. 1
A parallel method for time discretization of parabolic equations based on Laplace transformation and quadrature
, 2003
"... ..."
A Phase Field Model for Continuous Clustering on Vector Fields
 IEEE Transactions on Visualization and Computer Graphics
"... A new method for the simplification of flow fields is presented. It is based on continuous clustering. A wellknown physical clustering model, the Cahn Hilliard model which describes phase separation, is modified to reflect the properties of the data to be visualized. Clusters are defined implicitly ..."
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Cited by 17 (3 self)
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A new method for the simplification of flow fields is presented. It is based on continuous clustering. A wellknown physical clustering model, the Cahn Hilliard model which describes phase separation, is modified to reflect the properties of the data to be visualized. Clusters are defined implicitly as connected components of the positivity set of a density function. An evolution equation for this function is obtained as a suitable gradient flow of an underlying anisotropic energy functional. Here, time serves as the scale parameter. The evolution is characterized by a successive coarsening of patterns — the actual clustering — during which the underlying simulation data specifies preferable pattern boundaries. We introduce specific physical quantities in the simulation to control the shape, orientation and distribution of the clusters, as a function of the underlying flow field. In addition the model is expanded involving elastic effects. Thereby in early stages of the evolution shear layer type representation of the flow field can be generated, whereas for later stages the distribution of clusters can be influenced. Furthermore, we incorporate upwind ideas to give the clusters an oriented drop–shaped appearance. Here we discuss the applicability of this new type of approach mainly for flow fields, where the cluster energy penalizes cross streamline boundaries. However, the method also carries provisions for other fields as well. The clusters can be displayed directly as a flow texture. Alternatively, the clusters can be visualized by iconic representations, which are positioned by using a skeletonization algorithm. 1
Studies of the Accuracy of Time Integration Methods for ReactionDiffusion Equations
 J. COMPUT. PHYS
, 2005
"... In this study we present numerical experiments of time integration methods applied to systems of reactiondi#usion equations. Our main interest is in evaluating the relative accuracy and asymptotic order of accuracy of the methods on problems which exhibit an approximate balance between the compe ..."
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Cited by 17 (3 self)
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In this study we present numerical experiments of time integration methods applied to systems of reactiondi#usion equations. Our main interest is in evaluating the relative accuracy and asymptotic order of accuracy of the methods on problems which exhibit an approximate balance between the competing component time scales. Nearly balanced
A posteriori analysis of the finite element discretization of some parabolic equations
 MR2136996 RECONSTRUCTION FOR DISCRETE PARABOLIC PROBLEMS 1657
, 2005
"... Abstract. We are interested in the discretization of parabolic equations, either linear or semilinear, by an implicit Euler scheme with respect to the time variable and finite elements with respect to the space variables. The main result of this paper consists of building error indicators with respe ..."
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Cited by 17 (4 self)
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Abstract. We are interested in the discretization of parabolic equations, either linear or semilinear, by an implicit Euler scheme with respect to the time variable and finite elements with respect to the space variables. The main result of this paper consists of building error indicators with respect to both time and space approximations and proving their equivalence with the error, in order to work with adaptive time steps and finite element meshes. Résumé. Nous considérons la discrétisation d’équations paraboliques, soit linéaires soit semilinéaires, par un schéma d’Euler implicite en temps et par éléments finis en espace. L’idée de cet article est de construire des indicateurs d’erreur liés à l’approximation en temps et en espace et de prouver leur équivalence avec l’erreur, dans le but de travailler avec des pas de temps adaptatifs et des maillages d’éléments finis adaptés à la solution. 1.
A posteriori error estimates for the Crank– Nicolson method for parabolic equations
 Math. Comp
"... Abstract. We derive optimal order a posteriori error estimates for time discretizations by both the Crank–Nicolson and the Crank–Nicolson–Galerkin methods for linear and nonlinear parabolic equations. We examine both smooth and rough initial data. Our basic tool for deriving a posteriori estimates a ..."
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Cited by 16 (5 self)
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Abstract. We derive optimal order a posteriori error estimates for time discretizations by both the Crank–Nicolson and the Crank–Nicolson–Galerkin methods for linear and nonlinear parabolic equations. We examine both smooth and rough initial data. Our basic tool for deriving a posteriori estimates are secondorder Crank–Nicolson reconstructions of the piecewise linear approximate solutions. These functions satisfy two fundamental properties: (i) they are explicitly computable and thus their difference to the numerical solution is controlled a posteriori, and (ii) they lead to optimal order residuals as well as to appropriate pointwise representations of the error equation of the same form as the underlying evolution equation. The resulting estimators are shown to be of optimal order by deriving upper and lower bounds for them depending only on the discretization parameters and the data of our problem. As a consequence we provide alternative proofs for known a priori rates of convergence for the Crank–Nicolson method. 1.
Discontinuous Galerkin Methods for Incompressible and Nearly Incompressible Elasticity by Nitsche's Method
 COMPUTER METHODS IN APPLIED MECHANICS AND ENGINEERING 2002
, 2000
"... We propose and analyze a discontinuous finite element method for nearly incompressible linear elasticity, on triangular or tetrahedral meshes. We show optimal error estimates that are uniform with respect to Poisson's ratio. The method is thus locking free. We also introduce an equivalent mixed form ..."
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Cited by 15 (2 self)
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We propose and analyze a discontinuous finite element method for nearly incompressible linear elasticity, on triangular or tetrahedral meshes. We show optimal error estimates that are uniform with respect to Poisson's ratio. The method is thus locking free. We also introduce an equivalent mixed formulation, allowing for completely incompressible elasticity problems. Numerical results are presented.
Optimal error estimates of finite difference methods for the GrossPitaevskii equation with angular momentum rotation
 Math. Comp
"... Abstract. We analyze finite difference methods for the GrossPitaevskii equation with an angular momentum rotation term in two and three dimensions and obtain the optimal convergence rate, for the conservative CrankNicolson finite difference (CNFD) method and semiimplicit finite difference (SIFD) ..."
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Cited by 15 (10 self)
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Abstract. We analyze finite difference methods for the GrossPitaevskii equation with an angular momentum rotation term in two and three dimensions and obtain the optimal convergence rate, for the conservative CrankNicolson finite difference (CNFD) method and semiimplicit finite difference (SIFD) method, at the order of O(h 2 + τ 2)inthel 2norm and discrete H 1norm with time step τ and mesh size h. Besides the standard techniques of the energy method, the key technique in the analysis for the SIFD method is to use the mathematical induction, and resp., for the CNFD method is to obtain apriori bound of the numerical solution in the l ∞norm by using the inverse inequality and the l 2norm error estimate. In addition, for the SIFD method, we also derive error bounds on the errors between the mass and energy in the discretized level and their corresponding continuous counterparts, respectively, which are at the same order of the convergence rate as that of the numerical solution itself. Finally, numerical results are reported to confirm our error estimates of the numerical methods. 1.