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Optimal Order (optimal + order)
Selected AbstractsA hybridizable discontinuous Galerkin method for linear elasticityINTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING, Issue 8 2009S.-C. Soon Abstract This paper describes the application of the so-called hybridizable discontinuous Galerkin (HDG) method to linear elasticity problems. The method has three significant features. The first is that the only globally coupled degrees of freedom are those of an approximation of the displacement defined solely on the faces of the elements. The corresponding stiffness matrix is symmetric, positive definite, and possesses a block-wise sparse structure that allows for a very efficient implementation of the method. The second feature is that, when polynomials of degree k are used to approximate the displacement and the stress, both variables converge with the optimal order of k+1 for any k,0. The third feature is that, by using an element-by-element post-processing, a new approximate displacement can be obtained that converges at the order of k+2, whenever k,2. Numerical experiments are provided to compare the performance of the HDG method with that of the continuous Galerkin (CG) method for problems with smooth solutions, and to assess its performance in situations where the CG method is not adequate, that is, when the material is nearly incompressible and when there is a crack. Copyright © 2009 John Wiley & Sons, Ltd. [source] Weak imposition of boundary conditions for the Navier,Stokes equations by a penalty methodINTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN FLUIDS, Issue 4 2009Atife Caglar Abstract We prove convergence of the finite element method for the Navier,Stokes equations in which the no-slip condition and no-penetration condition on the flow boundary are imposed via a penalty method. This approach has been previously studied for the Stokes problem by Liakos (Weak imposition of boundary conditions in the Stokes problem. Ph.D. Thesis, University of Pittsburgh, 1999). Since, in most realistic applications, inertial effects dominate, it is crucial to extend the validity of the method to the nonlinear Navier,Stokes case. This report includes the analysis of this extension, as well as numerical results validating their analytical counterparts. Specifically, we show that optimal order of convergence can be achieved if the computational boundary follows the real flow boundary exactly. Copyright © 2008 John Wiley & Sons, Ltd. [source] Performance optimization of object comparisonINTERNATIONAL JOURNAL OF INTELLIGENT SYSTEMS, Issue 10 2009Axel Hallez Comparing objects can be considered as a hierarchical process. Separate aspects of objects are compared to each other, and the results of these comparisons are combined into a single result in one or more steps by aggregation operators. The set of operators used to compare the objects and the way these operators are related with each other is called the comparison scheme. If a threshold is applied to the final result of the object comparison, the mathematical properties of the operators in the comparison scheme can be used to derive thresholds on the intermediate results. These derived threshold can be used to break of a comparison early, thus offering a reduction of the comparison cost. Using this information, we show that the order in which the operators are evaluated has an influence on the average cost of comparing two objects. Next, we proceed with a study of the properties that allow us to find an optimal order, such that this average cost is minimized. Finally, we provide an algorithm that calculates an optimal order efficiently. Although specifically developed for object comparison, the algorithm can be applied to all kinds of selection processes that involve the combination of several test results. © 2009 Wiley Periodicals, Inc. [source] Characteristic-mixed covolume methods for advection-dominated diffusion problemsNUMERICAL LINEAR ALGEBRA WITH APPLICATIONS, Issue 9 2006Zhangxin Chen Abstract Characteristic-mixed covolume methods for time-dependent advection-dominated diffusion problems are developed and studied. The diffusion term in these problems is discretized using covolume methods applied to the mixed formulation of the problems on quadrilaterals, and the temporal differentiation and advection terms are treated by characteristic tracking schemes. Three characteristic tracking schemes are studied in the context of mixed covolume methods: the modified method of characteristics, the modified method of characteristics with adjusted advection, and the Eulerian,Lagrangian localized adjoint method. The proposed methods preserve the conceptual and computational merits of both characteristics-based schemes and the mixed covolume methods. Existence and uniqueness of a solution to the discrete problem arising from the methods is shown. Stability and convergence properties of these methods are also obtained; unconditionally stable results and error estimates of optimal order are established. Copyright © 2006 John Wiley & Sons, Ltd. [source] Eigenvalue estimates for preconditioned saddle point matricesNUMERICAL LINEAR ALGEBRA WITH APPLICATIONS, Issue 4 2006Owe Axelsson Abstract New accurate eigenvalue bounds for symmetric matrices of saddle point form are derived and applied for both unpreconditioned and preconditioned versions of the matrices. The estimates enable a better understanding of how preconditioners should be chosen. The preconditioners provide efficient iterative solution of the corresponding linear systems with, for some important applications, an optimal order of computational complexity. The methods are applied for Stokes problem and for linear elasticity problems. Copyright © 2005 John Wiley & Sons, Ltd. [source] Robust parameter-free algebraic multilevel preconditioningNUMERICAL LINEAR ALGEBRA WITH APPLICATIONS, Issue 6-7 2002Y. Notay Abstract To precondition large sparse linear systems resulting from the discretization of second-order elliptic partial differential equations, many recent works focus on the so-called algebraic multilevel methods. These are based on a block incomplete factorization process applied to the system matrix partitioned in hierarchical form. They have been shown to be both robust and efficient in several circumstances, leading to iterative solution schemes of optimal order of computational complexity. Now, despite the procedure is essentially algebraic, previous works focus generally on a specific context and consider schemes that use classical grid hierarchies with characteristic mesh sizes h,2h,4h, etc. Therefore, these methods require some extra information besides the matrix of the linear system and lack of robustness in some situations where semi-coarsening would be desirable. In this paper, we develop a general method that can be applied in a black box fashion to a wide class of problems, ranging from 2D model Poisson problems to 3D singularly perturbed convection,diffusion equations. It is based on an automatic coarsening process similar to the one used in the AMG method, and on coarse grid matrices computed according to a simple and cheap aggregation principle. Numerical experiments illustrate the efficiency and the robustness of the proposed approach. Copyright © 2002 John Wiley & Sons, Ltd. [source] The simple random walk and max-degree walk on a directed graphRANDOM STRUCTURES AND ALGORITHMS, Issue 3 2009Ravi Montenegro Abstract We bound total variation and L, mixing times, spectral gap and magnitudes of the complex valued eigenvalues of general (nonreversible nonlazy) Markov chains with a minor expansion property. The resulting bounds for the (nonlazy) simple and max-degree walks on a (directed) graph are of the optimal order. It follows that, within a factor of two or four, the worst case of each of these mixing time and eigenvalue quantities is a walk on a cycle with clockwise drift. © 2008 Wiley Periodicals, Inc. Random Struct. Alg., 2009 [source] | |||||||||||||