To reduce computational cost,we study some two-scale finite element approximations on sparse grids for elliptic partial differential equations of second order in a general *** any tensor product domain ?R^d with d =...
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To reduce computational cost,we study some two-scale finite element approximations on sparse grids for elliptic partial differential equations of second order in a general *** any tensor product domain ?R^d with d = 2,3,we construct the two-scale finite element approximations for both boundary value and eigenvalue problems by using a Boolean sum of some existing finite element approximations on a coarse grid and some univariate fine grids and hence they are cheaper *** applications,we obtain some new efficient finite element discretizations for the two classes of problem:The new two-scale finite element approximation on a sparse grid not only has the less degrees of freedom but also achieves a good accuracy of approximation.
In recent years,a nonoverlapping domain decomposition iterative procedure,which is based on using Robin-type boundary conditions as information transmission conditions on the subdomain interfaces,has been developed an...
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In recent years,a nonoverlapping domain decomposition iterative procedure,which is based on using Robin-type boundary conditions as information transmission conditions on the subdomain interfaces,has been developed and *** is known that the convergence rate of this method is 1-O(h),where h is mesh *** this paper,the convergence rate is improved to be 1-O(h1/2 H-1/2)sometime by choosing suitable parameter,where H is the subdomain *** examples are constructed to show that our convergence estimates are sharp,which means that the convergence rate cannot be better than 1-O(h1/2H-1/2)in a certain case no matter how parameter is chosen.
This paper deals with boundary value problems for linear uniformly elliptic systems. First the general linear uniformly elliptic system of the first order equations is reduced to complex form, and then the compound bo...
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This paper deals with boundary value problems for linear uniformly elliptic systems. First the general linear uniformly elliptic system of the first order equations is reduced to complex form, and then the compound boundary value problem for the complex equations of the first order is discussed. The approximate solutions of the boundary value problem are found by the variation-difference method, and the error estimates for the approximate solutions are *** the approximate method of the oblique derivative problem for linear uniformly elliptic equations of the second or der is introduced.
We propose a multigrid method to solve the molecular mechanics model(molecular dynamics at zero temperature).The Cauchy-Born elasticity model is employed as the coarse grid operator and the elastically deformed state ...
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We propose a multigrid method to solve the molecular mechanics model(molecular dynamics at zero temperature).The Cauchy-Born elasticity model is employed as the coarse grid operator and the elastically deformed state as the initial guess of the molecular mechanics *** efficiency of the algorithm is demonstrated by three examples with homogeneous deformation,namely,one dimensional chain under tensile deformation and aluminum under tension and shear *** method exhibits linear-scaling computational complexity,and is insensitive to parameters arising from iterative *** addition,we study two examples with inhomogeneous deformation:vacancy and nanoindentation of *** results are still satisfactory while the linear-scaling property is lost for the latter example.
In this paper, we consider the problem of finding sparse solutions for underdetermined systems of linear equations, which can be formulated as a class of L_0 norm minimization problem. By using the least absolute resi...
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In this paper, we consider the problem of finding sparse solutions for underdetermined systems of linear equations, which can be formulated as a class of L_0 norm minimization problem. By using the least absolute residual approximation, we propose a new piecewis, quadratic function to approximate the L_0 ***, we develop a piecewise quadratic approximation(PQA) model where the objective function is given by the summation of a smooth non-convex component and a non-smooth convex component. To solve the(PQA) model,we present an algorithm based on the idea of the iterative thresholding algorithm and derive the convergence and the convergence rate. Finally, we carry out a series of numerical experiments to demonstrate the performance of the proposed algorithm for(PQA). We also conduct a phase diagram analysis to further show the superiority of(PQA) over L_1 and L_(1/2) regularizations.
Trust region (TR) algorithms are a class of recently developed algorithms for nonlinear optimization. A new family of TR algorithms for unconstrained optimization, which is the extension of the usual TR method, is pre...
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Trust region (TR) algorithms are a class of recently developed algorithms for nonlinear optimization. A new family of TR algorithms for unconstrained optimization, which is the extension of the usual TR method, is presented in this paper. When the objective function is bounded below and continuously, differentiable, and the norm of the Hesse approximations increases at most linearly with the iteration number, we prove the global convergence of the algorithms. Limited numerical results are reported, which indicate that our new TR algorithm is competitive.
In this paper we are concerned with a domain decomposition method with nonmatching grids for Raviart-Thomas finite elements. In this method, the normal complement of the resulting approximation is not continuous acros...
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In this paper we are concerned with a domain decomposition method with nonmatching grids for Raviart-Thomas finite elements. In this method, the normal complement of the resulting approximation is not continuous across the interface. To handle such non-conformity, a new matching condition will be introduced. Such matching condition still results in a symmetric and positive definite stiffness matrix. It will be shown that the approximate solution generated by the domain decomposition possesses the optimal energy error estimate.
A full multigrid method with coarsening by a factor-of-three to distributed control problems constrained by Stokes equations is *** optimal control problem with cost functional of velocity and/or pressure tracking-typ...
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A full multigrid method with coarsening by a factor-of-three to distributed control problems constrained by Stokes equations is *** optimal control problem with cost functional of velocity and/or pressure tracking-type is considered with Dirichlet boundary *** optimality system that results from a Lagrange multiplier framework,form a linear system connecting the state,adjoint,and control *** investigate multigrid methods with finite difference discretization on staggered grids.A coarsening by a factor-of-three is used on staggered grids that results nested hierarchy of staggered grids and simplified the inter-grid transfer operators.A distributive-Gauss-Seidel smoothing scheme is employed to update the stateand adjoint-variables and a gradient update step is used to update the control *** experiments are presented to demonstrate the effectiveness and efficiency of the proposed multigrid framework to tracking-type optimal control problems.
In our previous paper[1] two weighted NND dtherence schemes were presentedby using proper weighted functions instead of minmod functions. As a result,the WNNDschemes enhance accuracy and yield smoother numrical auxes....
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In our previous paper[1] two weighted NND dtherence schemes were presentedby using proper weighted functions instead of minmod functions. As a result,the WNNDschemes enhance accuracy and yield smoother numrical auxes. In this paper two weightedENN schemes based on the ENN scheme[2] are constructed. The ENN scheme and WENNschemes are uniformly second-order accuracy and can achieve third-order accuracy in certainsmooth regions
A general local C-m(m greater than or equal to 0) tetrahedral interpolation scheme by polynomials of degree 4m + 1 plus low order rational functions from the given data is proposed. The scheme can have either 4m + 1 o...
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A general local C-m(m greater than or equal to 0) tetrahedral interpolation scheme by polynomials of degree 4m + 1 plus low order rational functions from the given data is proposed. The scheme can have either 4m + 1 order algebraic precision if C-2m data at vertices and C-m data on faces are given or k + E[k/3] + 1 order algebraic precision if C-k (k less than or equal to 2m) data are given at vertices. The resulted interpolant and its partial derivatives of up to order m are polynomials on the boundaries of the tetrahedra.
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