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1,524
An optimal order error analysis of the onedimensional quasicontinuum approximation
 SIAM J. Numer. Anal
"... Abstract. We derive a model problem for quasicontinuum approximations that allows a simple, yet insightful, analysis of the optimalorder convergence rate in the continuum limit for both the energybased quasicontinuum approximation and the quasinonlocal quasicontinuum approximation. For simplicity ..."
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Cited by 61 (19 self)
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Abstract. We derive a model problem for quasicontinuum approximations that allows a simple, yet insightful, analysis of the optimalorder convergence rate in the continuum limit for both the energybased quasicontinuum approximation and the quasinonlocal quasicontinuum approximation. For simplicity, the analysis is restricted to the case of secondneighbor interactions and is linearized about a uniformly stretched reference lattice. The optimalorder error estimates for the quasinonlocal quasicontinuum approximation are given for all strains up to the continuum limit strain for fracture. The analysis is based on an explicit treatment of the coupling error at the atomistictocontinuum interface, combined with an analysis of the error due to the atomistic and continuum schemes using the stability of the quasicontinuum approximation.
On the Approximation Power of Bivariate Splines
 Advances in Comp. Math. 9
, 1996
"... . We show how to construct stable quasiinterpolation schemes in the bivariate spline spaces S r d (4) with d 3r+2 which achieve optimal approximation order. In addition to treating the usual max norm, we also give results in the Lp norms, and show that the methods also approximate derivatives to ..."
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Cited by 59 (36 self)
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. We show how to construct stable quasiinterpolation schemes in the bivariate spline spaces S r d (4) with d 3r+2 which achieve optimal approximation order. In addition to treating the usual max norm, we also give results in the Lp norms, and show that the methods also approximate derivatives to optimal order. We pay special attention to the approximation constants, and show that they depend only on the the smallest angle in the underlying triangulation and the nature of the boundary of the domain. AMS(MOS) Subject Classifications: 41A15, 41A63, 41A25, 65D10 Keywords and phrases: Bivariate Splines, Approximation Order by Splines, Stable Approximation Schemes, Super Splines. x1. Introduction Let\Omega be a bounded polygonal domain in IR 2 . Given a finite triangulation 4 of \Omega\Gamma we are interested in spaces of splines of smoothness r and degree d of the form S r d (4) := fs 2 C r(\Omega\Gamma : sj T 2 P d ; for all T 2 4g; where P d denotes the space of polynomials of...
A compiler for variational forms
 ACM Trans. Math. Software
"... As a key step towards a complete automation of the finite element method, we present a new algorithm for automatic and efficient evaluation of multilinear variational forms. The algorithm has been implemented in the form of a compiler, the FEniCS Form Compiler FFC. We present benchmark results for a ..."
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Cited by 58 (21 self)
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As a key step towards a complete automation of the finite element method, we present a new algorithm for automatic and efficient evaluation of multilinear variational forms. The algorithm has been implemented in the form of a compiler, the FEniCS Form Compiler FFC. We present benchmark results for a series of standard variational forms, including the incompressible Navier– Stokes equations and linear elasticity. The speedup compared to the standard quadraturebased approach is impressive; in some cases the speedup is as large as a factor 1000.
A twolevel additive Schwarz preconditioner for nonconforming plate elements
 Numer. Math
, 1994
"... Abstract. Twolevel additive Schwarz preconditioners are developed for the nonconforming P1 finite element approximation of scalar secondorder symmetric positive definite elliptic boundary value problems, the Morley finite element approximation of the biharmonic equation, and the divergencefree no ..."
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Cited by 58 (5 self)
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Abstract. Twolevel additive Schwarz preconditioners are developed for the nonconforming P1 finite element approximation of scalar secondorder symmetric positive definite elliptic boundary value problems, the Morley finite element approximation of the biharmonic equation, and the divergencefree nonconforming P1 finite element approximation of the stationary Stokes equations. The condition numbers of the preconditioned systems are shown to be bounded independent of mesh sizes and the number of subdomains in the case of generous overlap. 1.
Finite element heterogeneous multiscale methods with near optimal . . .
"... This paper is concerned with a numerical method for multiscale elliptic problems. Using the framework of the Heterogeneous Multiscale Methods (HMM), we propose a micromacro approache which combines finite element method (FEM) for the macroscopic solver and the pseudospectral method for the micro s ..."
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Cited by 58 (24 self)
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This paper is concerned with a numerical method for multiscale elliptic problems. Using the framework of the Heterogeneous Multiscale Methods (HMM), we propose a micromacro approache which combines finite element method (FEM) for the macroscopic solver and the pseudospectral method for the micro solver. Unlike the micromacro methods based on standard FEM proposed so far in HMM we obtain, for periodic homogenization problems, a method that has almostlinear complexity in the number of degrees of freedom of the discretization of the macro (slow) variable.
Adaptive numerical treatment of elliptic systems on manifolds
 Advances in Computational Mathematics, 15(1):139
, 2001
"... ABSTRACT. Adaptive multilevel finite element methods are developed and analyzed for certain elliptic systems arising in geometric analysis and general relativity. This class of nonlinear elliptic systems of tensor equations on manifolds is first reviewed, and then adaptive multilevel finite element ..."
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Cited by 57 (26 self)
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ABSTRACT. Adaptive multilevel finite element methods are developed and analyzed for certain elliptic systems arising in geometric analysis and general relativity. This class of nonlinear elliptic systems of tensor equations on manifolds is first reviewed, and then adaptive multilevel finite element methods for approximating solutions to this class of problems are considered in some detail. Two a posteriori error indicators are derived, based on local residuals and on global linearized adjoint or dual problems. The design of Manifold Code (MC) is then discussed; MC is an adaptive multilevel finite element software package for 2 and 3manifolds developed over several years at Caltech and UC San Diego. It employs a posteriori error estimation, adaptive simplex subdivision, unstructured algebraic multilevel methods, global inexact Newton methods, and numerical continuation methods for the numerical solution of nonlinear covariant elliptic systems on 2 and 3manifolds. Some of the more interesting features of MC are described in detail, including some new ideas for topology and geometry representation in simplex meshes, and an unusual partition of unitybased method for exploiting parallel computers. A short example is then given which involves the Hamiltonian and momentum constraints in the Einstein equations, a representative nonlinear 4component covariant elliptic system on a Riemannian 3manifold which arises in general relativity. A number of operator properties and solvability results recently established are first summarized, making possible two quasioptimal a priori error estimates for Galerkin approximations which are then derived. These two results complete the theoretical framework for effective use of adaptive multilevel finite element methods. A sample calculation using the MC software is then presented.
Mesh Parameterization: Theory and Practice
 SIGGRAPH ASIA 2008 COURSE NOTES
, 2008
"... Mesh parameterization is a powerful geometry processing tool with numerous computer graphics applications, from texture mapping to animation transfer. This course outlines its mathematical foundations, describes recent methods for parameterizing meshes over various domains, discusses emerging tools ..."
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Cited by 56 (5 self)
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Mesh parameterization is a powerful geometry processing tool with numerous computer graphics applications, from texture mapping to animation transfer. This course outlines its mathematical foundations, describes recent methods for parameterizing meshes over various domains, discusses emerging tools like global parameterization and intersurface mapping, and demonstrates a variety of parameterization applications.
Adaptive Isogeometric Analysis by Local hRefinement with TSplines
"... Isogeometric analysis based on NURBS (NonUniform Rational BSplines) as basis functions preserves the exact geometry but suffers from the drawback of a rectangular grid of control points in the parameter space, which renders a purely local refinement impossible. This paper demonstrates how this dif ..."
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Cited by 53 (5 self)
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Isogeometric analysis based on NURBS (NonUniform Rational BSplines) as basis functions preserves the exact geometry but suffers from the drawback of a rectangular grid of control points in the parameter space, which renders a purely local refinement impossible. This paper demonstrates how this difficulty can be overcome by using Tsplines instead. Tsplines allow the introduction of socalled Tjunctions, which are related to hanging nodes in the standard FEM. Obeying a few straightforward rules, rectangular patches in the parameter space of the Tsplines can be subdivided and thus a local refinement becomes feasible while still preserving the exact geometry. Furthermore, it is shown how stateoftheart a posteriori error estimation techniques can be combined with refinement by TSplines. Numerical examples underline the potential of isogeometric analysis with Tsplines and give hints for further developments. Key words: adaptivity, a posteriori error estimation, isogeometric analysis, NURBS, CAD,
Energy norm a posteriori error estimation for discontinuous Galerkin methods
 COMPUT. METHODS APPL. MECH. ENGRG
, 2001
"... In this note we present a residualbased a posteriori error estimate of a natural mesh dependent energy norm of the error in a family of discontinuous Galerkin approximations of elliptic problems. The theory is developed for an elliptic model problem in two and three spatial dimensions and genera ..."
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Cited by 50 (2 self)
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In this note we present a residualbased a posteriori error estimate of a natural mesh dependent energy norm of the error in a family of discontinuous Galerkin approximations of elliptic problems. The theory is developed for an elliptic model problem in two and three spatial dimensions and general nonconvex polygonal domains are allowed. We also present some illustrating numerical examples.
Variational Formulation for the Stationary Fractional Advection Dispersion Equation
"... In this paper a theoretical framework for the Galerkin finite element approximation to the steady state fractional advection dispersion equation is presented. Appropriate fractional derivative spaces are defined and shown to be equivalent to the usual fractional dimension Sobolev spaces H^s. Existen ..."
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Cited by 49 (1 self)
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In this paper a theoretical framework for the Galerkin finite element approximation to the steady state fractional advection dispersion equation is presented. Appropriate fractional derivative spaces are defined and shown to be equivalent to the usual fractional dimension Sobolev spaces H^s. Existence and uniqueness results are proven, and error estimates for the Galerkin approximation derived. Numerical results are included which confirm the theoretical estimates.