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211
Optimization of the Hermitian and skewHermitian splitting iteration for saddlepoint problems
 BIT
"... Abstract. This paper is concerned with a generalization of the Hermitian and skewHermitian (HSS) splitting iteration for solving positive definite, nonHermitian linear systems. It is shown that the new scheme can outperform the standard HSS method in some situations and can be used as an effective ..."
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Cited by 19 (16 self)
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Abstract. This paper is concerned with a generalization of the Hermitian and skewHermitian (HSS) splitting iteration for solving positive definite, nonHermitian linear systems. It is shown that the new scheme can outperform the standard HSS method in some situations and can be used as an effective preconditioner for certain linear systems in saddle point form. Numerical experiments using discretizations of incompressible flow problems demonstrate the effectiveness of the generalized HSS preconditioner.
A locally conservative LDG method for the incompressible NavierStokes equations
 Math. Comp
"... Abstract. In this paper a new local discontinuous Galerkin method for the incompressible stationary NavierStokes equations is proposed and analyzed. Four important features render this method unique: its stability, its local conservativity, its highorder accuracy, and the exact satisfaction of the ..."
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Cited by 17 (10 self)
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Abstract. In this paper a new local discontinuous Galerkin method for the incompressible stationary NavierStokes equations is proposed and analyzed. Four important features render this method unique: its stability, its local conservativity, its highorder accuracy, and the exact satisfaction of the incompressibility constraint. Although the method uses completely discontinuous approximations, a globally divergencefree approximate velocity in H(div; Ω) is obtained by simple, elementbyelement postprocessing. Optimal error estimates are proven and an iterative procedure used to compute the approximate solution is shown to converge. This procedure is nothing but a discrete version of the classical fixed point iteration used to obtain existence and uniqueness of solutions to the incompressible NavierStokes equations by solving a sequence of Oseen problems. Numerical results are shown which verify the theoretical rates of convergence. They also confirm the independence of the number of fixed point iterations with respect to the discretization parameters. Finally, they show that the method works well for a wide range of Reynolds numbers. 1.
Nonconforming finite element approximation of crystalline microstructure
 Math. Comp
, 1998
"... Abstract. We consider a class of nonconforming finite element approximations of a simply laminated microstructure which minimizes the nonconvex variational problem for the deformation of martensitic crystals which can undergo either an orthorhombic to monoclinic (double well) or a cubic to tetragona ..."
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Cited by 17 (9 self)
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Abstract. We consider a class of nonconforming finite element approximations of a simply laminated microstructure which minimizes the nonconvex variational problem for the deformation of martensitic crystals which can undergo either an orthorhombic to monoclinic (double well) or a cubic to tetragonal (triple well) transformation. We first establish a series of error bounds in terms of elastic energies for the L 2 approximation of derivatives of the deformation in the direction tangential to parallel layers of the laminate, for the L 2 approximation of the deformation, for the weak approximation of the deformation gradient, for the approximation of volume fractions of deformation gradients, and for the approximation of nonlinear integrals of the deformation gradient. We then use these bounds to give corresponding convergence rates for quasioptimal finite element approximations. 1.
Reduced Basis Approximation and A Posteriori Error Estimation for the TimeDependent Viscous Burgers Equation
 CALCOLO
, 2008
"... In this paper we present rigorous a posteriori L 2 error bounds for reduced basis approximations of the unsteady viscous Burgers equation in one space dimension. The key new ingredient is accurate solution–dependent (Online) calculation of the exponential–in–time stability factor by the Successive C ..."
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Cited by 15 (4 self)
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In this paper we present rigorous a posteriori L 2 error bounds for reduced basis approximations of the unsteady viscous Burgers equation in one space dimension. The key new ingredient is accurate solution–dependent (Online) calculation of the exponential–in–time stability factor by the Successive Constraint Method. Numerical results indicate that the a posteriori error bounds are practicable for reasonably large times — many convective scales — and reasonably large Reynolds numbers — O(100) or larger.
The simply laminated microstructure in martensitic crystals that undergo a cubic to orthorhombic phase transformation
, 1999
"... Abstract. We study simply laminated microstructures of a martensitic crystal capable of undergoing a cubic to orthorhombic transformation of type P (432) → P (222)′. The free energy density modeling such a crystal is minimized on six energy wells that are pairwise rankone connected. We consider th ..."
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Cited by 14 (9 self)
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Abstract. We study simply laminated microstructures of a martensitic crystal capable of undergoing a cubic to orthorhombic transformation of type P (432) → P (222)′. The free energy density modeling such a crystal is minimized on six energy wells that are pairwise rankone connected. We consider the energy minimization problem with Dirichlet boundary data compatible with an arbitrary but fixed simple laminate. We first show that for all but a few isolated values of transformation strains, this problem has a unique Young measure solution solely characterized by the boundary data that represents the simply laminated microstructure. We then present a theory of stability for such a microstructure, and apply it to the conforming finite element approximation to obtain the corresponding error estimates for the finite element energy minimizers. 1.
Preconditioning techniques for Newton’s method for the incompressible Navier–Stokes equations
, 2003
"... Newton’s method for the incompressible Navier–Stokes equations gives rise to large sparse nonsymmetric indefinite matrices with a socalled saddlepoint structure for which Schur complement preconditioners have proven to be effective when coupled with iterative methods of Krylov type. In this work ..."
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Cited by 14 (4 self)
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Newton’s method for the incompressible Navier–Stokes equations gives rise to large sparse nonsymmetric indefinite matrices with a socalled saddlepoint structure for which Schur complement preconditioners have proven to be effective when coupled with iterative methods of Krylov type. In this work we investigate the performance of two preconditioning techniques introduced originally for the Picard method for which both proved significantly superior to other approaches such as the Uzawa method. The first is a block preconditioner which is based on the algebraic structure of the system matrix. The other approach uses also a block preconditioner which is derived by considering the underlying partial differential operator matrix. Analysis and numerical comparison of the methods are presented.
An a posteriori error control framework for adaptive precision optimization using discontinuous Galerkin finite element method
, 2005
"... Professor Darmofal and the generous funding provided by NASA Langley (grant number NAG103035). Secondly, the effort put into Project X by faculty and students (past and present) have made it possible to carry out the computational demonstrations in higherorder DG. In particular, Krzysztof Fidkowsk ..."
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Cited by 14 (0 self)
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Professor Darmofal and the generous funding provided by NASA Langley (grant number NAG103035). Secondly, the effort put into Project X by faculty and students (past and present) have made it possible to carry out the computational demonstrations in higherorder DG. In particular, Krzysztof Fidkowski and Todd Oliver are to be acknowledged for their contributions towards the development of the flow solvers and also for providing some of the grids for the test cases demonstrated. Finally, thanks must go to thesis committee members Professors Peraire and Willcox as well as thesis readers Dr. Natalia Alexandrov and Dr. Steven Allmaras for the time they put into reading the thesis and providing the valuable feedbacks. 3 46 Adjoint approach to shape sensitivity 117 6.1 Introduction...............................
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.
A Finite Element Approach to the Immersed Boundary Method
, 2004
"... The immersed boundary method was introduced by Peskin in [31] to study the blood flow in the heart and further applied to many situations where a fluid interacts with an elastic structure. The basic idea is to consider the structure as a part of the fluid where additional forces are applied and addi ..."
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Cited by 13 (7 self)
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The immersed boundary method was introduced by Peskin in [31] to study the blood flow in the heart and further applied to many situations where a fluid interacts with an elastic structure. The basic idea is to consider the structure as a part of the fluid where additional forces are applied and additional mass is localized. The forces exerted by the structure on the fluid are taken into account as a source term in the NavierStokes equations and are mathematically described as a Dirac delta function lying along the immersed structure. In this paper we first review on various ways of modeling the elastic forces in different physical situations. Then we focus on the discretization of the immersed boundary method by means of finite elements which can handle the Dirac delta function variationally avoiding the introduction of its regularization. Practical computational aspects are described and some preliminary numerical experiment in two dimensions are reported.
Approximate factorization constraint preconditioners for saddlepoint matrices
 SIAM J. Sci. Comput
"... Abstract. We consider the application of the conjugate gradient method to the solution of large, symmetric indefinite linear systems. Special emphasis is put on the use of constraint preconditioners and a new factorization that can reduce the number of flops required by the preconditioning step. Res ..."
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Cited by 13 (2 self)
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Abstract. We consider the application of the conjugate gradient method to the solution of large, symmetric indefinite linear systems. Special emphasis is put on the use of constraint preconditioners and a new factorization that can reduce the number of flops required by the preconditioning step. Results concerning the eigenvalues of the preconditioned matrix and its minimum polynomial are given. Numerical experiments validate these conclusions.