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267
Additive Schwarz algorithms for parabolic convectiondiffusion equations
 Numer. Math
, 1991
"... In this paper, we consider the solution of linear systems of algebraic equations that arise from parabolic finite element problems. We introduce three additive Schwarz type domain decomposition methods for general, not necessarily selfadjoint, linear, second order, parabolic partial differential equ ..."
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Cited by 41 (6 self)
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In this paper, we consider the solution of linear systems of algebraic equations that arise from parabolic finite element problems. We introduce three additive Schwarz type domain decomposition methods for general, not necessarily selfadjoint, linear, second order, parabolic partial differential equations and also study the convergence rates of these algorithms. The resulting preconditioned linear system of equations is solved by the generalized minimal residual method. Numerical results are also reported. Key words Schwarz’s alternating method, domain decomposition, parabolic convectiondiffusion equation, finite elements. AMS(MOS) subject classifications. 65N30, 65F10 1
Asynchronous Variational Integrators
 ARCH. RATIONAL MECH. ANAL.
, 2003
"... We describe a new class of asynchronous variational integrators (AVI) for nonlinear elastodynamics. The AVIs are distinguished by the following attributes: (i) The algorithms permit the selection of independent time steps in each element, and the local time steps need not bear an integral relation t ..."
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Cited by 40 (9 self)
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We describe a new class of asynchronous variational integrators (AVI) for nonlinear elastodynamics. The AVIs are distinguished by the following attributes: (i) The algorithms permit the selection of independent time steps in each element, and the local time steps need not bear an integral relation to each other; (ii) the algorithms derive from a spacetime form of a discrete version of Hamilton’s variational principle. As a consequence of this variational structure, the algorithms conserve local momenta and a local discrete multisymplectic structure exactly. To guide the development of the discretizations, a spacetime multisymplectic formulation of elastodynamics is presented. The variational principle used incorporates both configuration and spacetime reference variations. This allows a unified treatment of all the conservation properties of the system. A discrete version of reference configuration is also considered, providing a natural definition of a discrete energy. The possibilities for discrete energy conservation are evaluated. Numerical tests reveal that, even when local energy balance is not enforced exactly, the global and local energy behavior of the AVIs is quite remarkable, a property which can probably be traced to the symplectic nature of the algorithm.
A Posteriori Error Estimates for Elliptic Problems in Two and Three Space Dimensions
 SIAM J. Numer. Anal
, 1993
"... . Let u 2 H be the exact solution of a given selfadjoint elliptic boundary value problem, which is approximated by some ~ u 2 S, S being a suitable finite element space. Efficient and reliable a posteriori estimates of the error jj u \Gamma ~ u jj, measuring the (local) quality of ~ u, play a cruc ..."
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Cited by 38 (5 self)
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. Let u 2 H be the exact solution of a given selfadjoint elliptic boundary value problem, which is approximated by some ~ u 2 S, S being a suitable finite element space. Efficient and reliable a posteriori estimates of the error jj u \Gamma ~ u jj, measuring the (local) quality of ~ u, play a crucial role in termination criteria and in the adaptive refinement of the underlying mesh. A wellknown class of error estimates can be derived systematically by localizing the discretized defect problem using domain decomposition techniques. In the present paper, we provide a guideline for the theoretical analysis of such error estimates. We further clarify the relation to other concepts. Our analysis leads to new error estimates, which are specially suited to three space dimensions. The theoretical results are illustrated by numerical computations. Key words: adaptive finite element methods, aposteriori error estimates AMS (MOS) subject classifications: 65N30, 65N50, 65N55, 35J25 1 submi...
Large Dense Numerical Linear Algebra in 1993: The Parallel Computing Influence
 International Journal Supercomputer Applications
, 1994
"... This paper surveys the current state of applications of large dense numerical linear algebra, and the influence of parallel computing. Furthermore, we attempt to crystalize many important ideas that we feel have been sometimes been misunderstood in the rush to write fast programs. 1 Introduction Th ..."
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Cited by 35 (2 self)
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This paper surveys the current state of applications of large dense numerical linear algebra, and the influence of parallel computing. Furthermore, we attempt to crystalize many important ideas that we feel have been sometimes been misunderstood in the rush to write fast programs. 1 Introduction This paper represents my continuing efforts to track the status of large dense linear algebra problems. The goal is to shatter the barriers that separate the various interested communities while commenting on the influence of parallel computing. A secondary goal is to crystalize the most important ideas that have all too often been obscured by the details of machines and algorithms. Parallel supercomputing is in the spotlight. In the race towards the proliferation of papers on person X's experiences with machine Y (and why his algorithm runs faster than person Z's), sometimes we have lost sight of the applications for which these algorithms are meant to be useful. This paper concentrates on la...
Robust Numerical Methods for Contingent Claims under Jump Diffusion Processes
 IMA Journal of Numerical Analysis
, 2003
"... An implicit method is developed for the numerical solution of option pricing models where it is assumed that the underlying process is a jump diffusion. This method can be applied to a variety of contingent claim valuations, including American options, various kinds of exotic options, and models wit ..."
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Cited by 34 (13 self)
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An implicit method is developed for the numerical solution of option pricing models where it is assumed that the underlying process is a jump diffusion. This method can be applied to a variety of contingent claim valuations, including American options, various kinds of exotic options, and models with uncertain volatility or transaction costs. Proofs of timestepping stability and convergence of a fixed point iteration scheme are presented. For typical model parameters, it is shown that the fixed point iteration reduces the error by two orders of magnitude at each iteration. The correlation integral is computed using a fast Fourier transform (FFT) method. Techniques are developed for avoiding wraparound effects. Numerical tests of convergence for a variety of options are presented.
What Is a Good Linear Finite Element?  Interpolation, Conditioning, Anisotropy, and Quality Measures
 In Proc. of the 11th International Meshing Roundtable
, 2002
"... When a mesh of simplicial elements (triangles or tetrahedra) is used to form a piecewise linear approximation of a function, the accuracy of the approximation depends on the sizes and shapes of the elements. In finite element methods, the conditioning of the stiffness matrices also depends on the si ..."
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Cited by 33 (0 self)
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When a mesh of simplicial elements (triangles or tetrahedra) is used to form a piecewise linear approximation of a function, the accuracy of the approximation depends on the sizes and shapes of the elements. In finite element methods, the conditioning of the stiffness matrices also depends on the sizes and shapes of the elements. This article explains the mathematical connections between mesh geometry, interpolation errors, discretization errors, and stiffness matrix conditioning. These relationships are expressed by error bounds and element quality measures that determine the fitness of a triangle or tetrahedron for interpolation or for achieving low condition numbers. Unfortunately, the quality measures for these purposes do not fully agree with each other; for instance, small angles are bad for matrix conditioning but not for interpolation or discretization. The upper and lower bounds on interpolation error and element stiffness matrix conditioning given here are tighter than those usually seen in the literature, so the quality measures are likely to be unusually precise indicators of element fitness. Bounds are included for anisotropic cases, wherein long, thin elements perform better than equilateral ones. Surprisingly, there are circumstances wherein interpolation, conditioning, and discretization error are each best served by elements of different aspect ratios or orientations.
Variational time integrators
 Int. J. Numer. Methods Eng
"... The purpose of this paper is to review and further develop the subject of variational integration algorithms as it applies to mechanical systems of engineering interest. In particular, the conservation properties of both synchronous and asynchronous variational integrators (AVIs) are discussed in de ..."
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Cited by 31 (10 self)
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The purpose of this paper is to review and further develop the subject of variational integration algorithms as it applies to mechanical systems of engineering interest. In particular, the conservation properties of both synchronous and asynchronous variational integrators (AVIs) are discussed in detail. We present selected numerical examples which demonstrate the excellent accuracy, conservation and convergence characteristics of AVIs. In these tests, AVIs are found to result in substantial speedups, at equal accuracy, relative to explicit Newmark. A mathematical proof of convergence of the AVIs is also presented in this paper. Finally, we develop the subject of horizontal variations and configurational forces in discrete dynamics. This theory leads to exact pathindependent characterizations of the configurational forces acting on discrete systems. Notable examples are the configurational forces acting on material nodes in a finite element discretisation; and the Jintegral at the tip of a crack in
A monotone finite element scheme for convectiondiffusion equations
 Math. Comp
, 1999
"... Abstract. A simple technique is given in this paper for the construction and analysis of a class of finite element discretizations for convectiondiffusion problems in any spatial dimension by properly averaging the PDE coefficients on element edges. The resulting finite element stiffness matrix is ..."
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Cited by 30 (3 self)
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Abstract. A simple technique is given in this paper for the construction and analysis of a class of finite element discretizations for convectiondiffusion problems in any spatial dimension by properly averaging the PDE coefficients on element edges. The resulting finite element stiffness matrix is an Mmatrix under some mild assumption for the underlying (generally unstructured) finite element grids. As a consequence the proposed edgeaveraged finite element scheme is particularly interesting for the discretization of convection dominated problems. This scheme admits a simple variational formulation, it is easy to analyze, and it is also suitable for problems with a relatively smooth flux variable. Some simple numerical examples are given to demonstrate its effectiveness for convection dominated problems. 1.
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 ..."
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Cited by 28 (5 self)
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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...
The Sounds of Physical Shapes
 Presence
, 1996
"... We propose a general framework for the simulation of sounds produced by colliding ..."
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Cited by 27 (9 self)
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We propose a general framework for the simulation of sounds produced by colliding