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A Massively Parallel Adaptive Finite Element Method with Dynamic Load Balancing
 Appl. Numer. Math
, 1993
"... We construct massively parallel adaptive finite element methods for the solution of hyperbolic conservation laws. Spatial discretization is performed by a discontinuous Galerkin finite element method using a basis of piecewise Legendre polynomials. Temporal discretization utilizes a RungeKutta meth ..."
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Cited by 109 (14 self)
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We construct massively parallel adaptive finite element methods for the solution of hyperbolic conservation laws. Spatial discretization is performed by a discontinuous Galerkin finite element method using a basis of piecewise Legendre polynomials. Temporal discretization utilizes a RungeKutta method. Dissipative fluxes and projection limiting prevent oscillations near solution discontinuities. The resulting method is of high order and may be parallelized efficiently on MIMD computers. We demonstrate parallel efficiency through computations on a 1024processor nCUBE/2 hypercube. We present results using adaptiverefinement to reduce the computational cost of the method, and tiling, a dynamic, elementbased data migration system that maintains global load balance of the adaptive method by overlapping neighborhoods of processors that each perform local balancing. 1. Introduction We are studying massively parallel adaptive finite element methods for solving systems ofdimensional hyper...
Edge stabilization for Galerkin approximations of convectiondiffusionreaction problems
 Comp. Methods Appl. Mech. Engrg
"... Abstract. We analyze a nonlinear shockcapturing scheme for H 1conforming, piecewiseaffine finite element approximations of linear elliptic problems. The meshes are assumed to satisfy two standard conditions: a local quasiuniformity property and the Xu–Zikatanov condition ensuring that the stiffne ..."
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Cited by 77 (20 self)
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Abstract. We analyze a nonlinear shockcapturing scheme for H 1conforming, piecewiseaffine finite element approximations of linear elliptic problems. The meshes are assumed to satisfy two standard conditions: a local quasiuniformity property and the Xu–Zikatanov condition ensuring that the stiffness matrix associated with the Poisson equation is an Mmatrix. A discrete maximum principle is rigorously established in any space dimension for convectiondiffusionreaction problems. We prove that the shockcapturing finite element solution converges to that without shockcapturing if the cell Péclet numbers are sufficiently small. Moreover, in the diffusiondominated regime, the difference between the two finite element solutions superconverges with respect to the actual approximation error. Numerical experiments on test problems with stiff layers confirm the sharpness of the a priori error estimates. 1.
Fast Nonsymmetric Iterations and Preconditioning for NavierStokes Equations
 SIAM J. Sci. Comput
, 1994
"... Discretization and linearization of the steadystate NavierStokes equations gives rise to a nonsymmetric indefinite linear system of equations. In this paper, we introduce preconditioning techniques for such systems with the property that the eigenvalues of the preconditioned matrices are bounded i ..."
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Cited by 74 (10 self)
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Discretization and linearization of the steadystate NavierStokes equations gives rise to a nonsymmetric indefinite linear system of equations. In this paper, we introduce preconditioning techniques for such systems with the property that the eigenvalues of the preconditioned matrices are bounded independently of the mesh size used in the discretization. We confirm and supplement these analytic results with a series of numerical experiments indicating that Krylov subspace iterative methods for nonsymmetric systems display rates of convergence that are independent of the mesh parameter. In addition, we show that preconditioning costs can be kept small by using iterative methods for some intermediate steps performed by the preconditioner. * This work was supported by the U. S. Army Research Office under grant DAAL0392G0016 and the U. S. National Science Foundation under grant ASC8958544 at the University of Maryland, and the Science and Engineering Research Council of Great Britain V...
An augmented Lagrangianbased approach to the Oseen problem
 SIAM J. SCI. COMPUT
, 2006
"... 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 mult ..."
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Cited by 68 (19 self)
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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.
Mantle Convection with a Brittle Lithosphere: Thoughts on the Global Tectonic Styles of the Earth and Venus
, 1998
"... . Plates are an integral part of the convection system in the fluid mantle, but plate boundaries are the product of brittle faulting and plate motions are strongly influenced by the existence of such faults. The conditions for plate tectonics are studied by considering brittle behaviour, using B ..."
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Cited by 65 (9 self)
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. Plates are an integral part of the convection system in the fluid mantle, but plate boundaries are the product of brittle faulting and plate motions are strongly influenced by the existence of such faults. The conditions for plate tectonics are studied by considering brittle behaviour, using Byerlee's law to limit the maximum stress in the lithosphere, in a mantle convection model with temperature dependent viscosity. When the yield stress is high, convection is confined below a thick, stagnant lithosphere. At low yield stress, brittle deformation mobilizes the lithosphere which becomes a part of the overall circulation; surface deformation occurs in localized regions close to upwelling and downwellings in the system. At intermediate levels of the yield stress, there is a cycling between these two states: thick lithosphere episodically mobilizes and collapses into the interior before reforming. The mobilelid regime resembles convection of a fluid with temperature depende...
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 56 (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.
Mixed finite element methods for viscoelastic flow analysis: A review
 J. NonNewtonian Fluid Mech
, 1998
"... The progress made during the past decade in the application of mixed finite element methods to solve viscoelastic flow problems using differential constitutive equations is reviewed. The algorithmic developments are discussed in detail. Starting with the classical mixed formulation, the elastic visc ..."
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Cited by 54 (3 self)
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The progress made during the past decade in the application of mixed finite element methods to solve viscoelastic flow problems using differential constitutive equations is reviewed. The algorithmic developments are discussed in detail. Starting with the classical mixed formulation, the elastic viscous stress splitting (EVSS) method as well as the related discrete EVSS and the socalled EVSSG method are discussed among others. Furthermore, stabilization techniques such as the streamline upwind PetrovGalerkin (SUPG) and the discontinuous Galerkin (DG) are reviewed. The performance of the numerical schemes for both smooth and nonsmooth benchmark problems is discussed. Finally, the capabilities of viscoelastic flow solvers to predict experimental observations are reviewed. 1
Bubble Functions Prompt Unusual Stabilized Finite Element Methods
"... A second order linear scalar differential equation including a zeroth order term is approximated using first the standard Galerkin method enriched with bubble functions. Static condensation of the bubbles suggest an unusual stabilized finite element method for which we establish a convergence study ..."
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Cited by 50 (11 self)
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A second order linear scalar differential equation including a zeroth order term is approximated using first the standard Galerkin method enriched with bubble functions. Static condensation of the bubbles suggest an unusual stabilized finite element method for which we establish a convergence study and obtain successful numerical simulations. The method is generalized to allow for a convection operator in the equation. This work may be employed as a starting point for simulation of nonlinear equations governing turbulence phenomena, flows with chemical reactions, and other important problems. Submitted to: Computer Methods in Applied Mechanics and Engineering Preprint June 1994 i L.P.Franca and C.Farhat Preprint, June 1994 1 1. INTRODUCTION We have pointed out in a recent communication [9] that for a certain model problem, bubble functions added to the usual finite element polynomials seem to subtract stability from the formulation. This finding contrasts our experience with other...
The Discontinuous Enrichment Method
, 2000
"... We propose a finite element based discretization method in which the standard polynomial field is enriched within each element by a nonconforming field that is added to it. The enrichment contains freespace solutions of the homogeneous differential equation that are not represented by the underlyin ..."
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Cited by 49 (6 self)
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We propose a finite element based discretization method in which the standard polynomial field is enriched within each element by a nonconforming field that is added to it. The enrichment contains freespace solutions of the homogeneous differential equation that are not represented by the underlying polynomial field. Continuity of the enrichment across element interfaces is enforced weakly by Lagrange multipliers. In this manner, we expect to attain high coarsemesh accuracy without significant degradation of conditioning, assuring good performance of the computation at any mesh resolution. Examples of application to acoustics and advectiondiffusion are presented. Key words: Finite elements; discontinuous enrichment; acoustics; advectiondiffusion 1 Introduction The standard finite element method is based on continuous, piecewise polynomial, Galerkin approximation. This approach is optimal for the Laplace operator in the sense that it minimizes the error in the energy normthe H ...
A Review of A Posteriori Error Estimation
 and Adaptive MeshRefinement Techniques, Wiley & Teubner
, 1996
"... linear parabolic equations ..."
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