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223
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 22 (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
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
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 19 (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.
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 14 (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.
Finite Volume Element Approximations of Nonlocal in Time Onedimensional Flows in Porous Media
 Computing
, 1999
"... Various finite volume element schemes for parabolic integrodifferential equations in 1D are derived and studied. These types of equations arise in modeling reactive flows or material with memory effects. Our main goal is to develop a general framework for obtaining finite volume element approximat ..."
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Cited by 14 (9 self)
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Various finite volume element schemes for parabolic integrodifferential equations in 1D are derived and studied. These types of equations arise in modeling reactive flows or material with memory effects. Our main goal is to develop a general framework for obtaining finite volume element approximations and to study the error analysis. We consider the lowestorder (linear and Lsplines) finite volume elements, although higherorder volume elements can be considered as well under this framework. It is proved that finite volume element approximations are convergent with optimal order in H 1 norms, suboptimal order in the L 2 norm and superconvergent order in a discrete H 1 norm.
A MemoryEfficient Finite Element Method for Systems of ReactionDiffusion Equations with NonSmooth Forcing
, 2003
"... The release of calcium ions in a human heart cell is modeled by a system of reactiondi #usion equations, which describe the interaction of the chemical species and the e#ects of various cell processes on them. The release is modeled by a forcing term in the calcium equation that involves a superposi ..."
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Cited by 13 (10 self)
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The release of calcium ions in a human heart cell is modeled by a system of reactiondi #usion equations, which describe the interaction of the chemical species and the e#ects of various cell processes on them. The release is modeled by a forcing term in the calcium equation that involves a superposition of many Dirac delta functions in space; such a nonsmooth righthand side leads to divergence for many numerical methods. The calcium ions enter the cell at a large number of regularly spaced points throughout the cell; to resolve those points adequately for a cell with realistic threedimensional dimensions, an extremely fine spatial mesh is needed. A finite element method is developed that addresses the two crucial issues for this and similar applications: Convergence of the method is demonstrated in extension of the classical theory that does not apply to nonsmooth forcing functions like the Dirac delta function; and the memory usage of the method is optimal and thus allows for extremely fine threedimensional meshes with many millions of degrees of freedom, already on a serial computer. Additionally, a coarsegrained parallel implementation of the algorithm allows for the solution on meshes with yet finer resolution than possible in serial.
SPACETIME ADAPTIVE WAVELET METHODS FOR PARABOLIC EVOLUTION PROBLEMS
"... Abstract. With respect to spacetime tensorproduct wavelet bases, parabolic initial boundary value problems are equivalently formulated as biinfinite matrix problems. Adaptive wavelet methods are shown to yield sequences of approximate solutions which converge at the optimal rate. In case the spat ..."
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Cited by 13 (3 self)
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Abstract. With respect to spacetime tensorproduct wavelet bases, parabolic initial boundary value problems are equivalently formulated as biinfinite matrix problems. Adaptive wavelet methods are shown to yield sequences of approximate solutions which converge at the optimal rate. In case the spatial domain is of product type, the use of spatial tensor product wavelet bases is proved to overcome the socalled curse of dimensionality, i.e., the reduction of the convergence rate with increasing spatial dimension. 1.