Convergence of a two-layer scheme for equations of gas dynamics in Eulerian variables with geo-physical applications

This paper deals with the convergence of a completely conservative, two-layer difference scheme for equations of gas dynamics in Eulerian variables. The convergence of the difference solution to the smooth solution of the original periodic Cauchy problem of order τ²+h ² at layer-by-layer norm L ₂ is proved, provided that the mesh step sizes are sufficiently small and that τ=h ^1+? (?=constant>0). Several modifications of the proposed method were used for the numerical solution of a one-dimensional mathematical model (on the basis of the shallow water theory), which describes crash events produced by dam collapse.     

A predictor-corrector smoothing Newton method for solving the mixed complementarity problem with a P 0-function

The mixed complementarity problem (denoted by MCP(F)) can be reformulated as the solution of a nonsmooth system of equations. In the paper, based on a perturbed mid function, we contract a new smoothing function. The existence and continuity of a smooth path for solving the mixed complementarity problem with a P ₀ function are discussed. Then we presented a predictor-corrector smoothing Newton algorithm to solve the MCP with a P ₀-function. The global convergence of the proposed algorithm is verified under mild conditions. And by using the smooth and semismooth technique, the local superlinear convergence of the method is proved under some suitable assumptions.     

The effect of numerical integration in finite element approximations to degenerate evolution equations

This paper studies some effects of numerical integration on the rate of convergence of finite element approximations to a degenerate initial, boundary value problem. It is shown that if the quadrature scheme and the finite element space satisfy certain accuracy and consistency conditions then the approximation converges to the true solution at an optimal rate in theW ₂ ¹ norm. When a specific choice of starting data is made, convergence is also optimal inL ². Partially supported by grant MCS-8202025 from the National Science Foundation.     

Denoising of Smooth Images Using L 1-Fitting

In this paper, denoising of smooth (H¹₀-regular) images is considered. The purpose of the paper is basically twofold. First, to compare the denoising methods based on L¹- and L²-fitting. Second, to analyze and realize an active-set method for solving the non-smooth optimization problem arising from the former approach. More precisely, we formulate the algorithm, proof its convergence, and give an efficient numerical realization. Several numerical experiments are presented, where the convergence of the proposed active-set algorithm is studied and the denoising properties of the methods based on L¹- and L²-fitting are compared. Also a heuristic method for determining the regularization parameter is presented and tested.     

Multiscale enrichment of a finite volume element method for the stationary Navier–Stokes problem

In this paper, we introduce a finite volume element method for the Navier–Stokes problem. This method is based on the multiscale enrichment and uses the lowest finite element pair P ₁/P ₀. The stability and convergence of the optimal order in H ¹-norm for velocity and L ²-norm for pressure are obtained. Using a dual problem for the Navier–Stokes problem, we establish the convergence of the optimal order in L ²-norm for the velocity.     

Non-smooth SOR for L 1-Fitting: Convergence Study and Discussion of Related Issues

In this article, the denoising of smooth (H ¹-regular) images is considered. To reach this objective, we introduce a simple and highly efficient over-relaxation technique for solving the convex, non-smooth optimization problems resulting from the denoising formulation. We describe the algorithm, discuss its convergence and present the results of numerical experiments, which validate the methods under consideration with respect to both efficiency and denoising capability. Several issues concerning the convergence of an Uzawa algorithm for the solution of the same problem are also discussed.     

二阶收敛的光滑正则化压缩感知信号重构方法

目的压缩感知信号重构过程是求解不定线性系统稀疏解的过程。针对不定线性系统稀疏解3种求解方法不够鲁棒的问题:最小化l₀-范数属于NP问题,最小化l₁-范数的无解情况以及最小化l_p-范数的非凸问题,提出一种基于光滑正则凸优化的方法进行求解。方法为了获得全局最优解并保证算法的鲁棒性,首先,设计了全空间信号l₀-范数凸拟合函数作为优化的目标函数;其次,将n元函数优化问题转变为n个一元函数优化问题;最后,求解过程中利用快速收缩算法进行求解,使收敛速度达到二阶收敛。结果该算法无论在仿真数据集还是在真实数据集上,都取得了优于其他3种类型算法的效果。在仿真实验中,当信号维数大于150维时,该方法重构时间为其他算法的50%左右,具有快速性;在真实数据实验中,该方法重构出的信号与原始信号差的F-范数为其他算法的70%,具有良好的鲁棒性。结论本文算法为二阶收敛的凸优化算法,可确保快速收敛到全局最优解,适合处理大型数据,在信息检索、字典学习和图像压缩等领域具有较大的潜在应用价值。     

Two-level Stabilized Finite Element Methods for the Steady Navier–Stokes Problem

In this article, the two-level stabilized finite element formulations of the two-dimensional steady Navier–Stokes problem are analyzed. A macroelement condition is introduced for constructing the local stabilized formulation of the steady Navier–Stokes problem. By satisfying this condition the stability of the Q₁–P₀ quadrilateral element and the P₁–P₀ triangular element are established. Moreover, the two-level stabilized finite element methods involve solving one small Navier–Stokes problem on a coarse mesh with mesh size H, a large Stokes problem for the simple two-level stabilized finite element method on a fine mesh with mesh size h=O(H²) or a large general Stokes problem for the Newton two-level stabilized finite element method on a fine mesh with mesh size h=O(|log h|^1/2H³). The methods we study provide an approximate solution (u^h,p^h) with the convergence rate of same order as the usual stabilized finite element solution, which involves solving one large Navier–Stokes problem on a fine mesh with mesh size h. Hence, our methods can save a large amount of computational time.     

On finite element approximation of general boundary value problems in nonlinear elasticity

This paper deals with the approximate solution of the general boundary value problem in nonlinear elasticity by the finite element displacement method. Under usual conditions which also guarantee the existence of locally unique solutions the quasi-optimal convergence inL ² andL ^∞ is shown for displacement fields and stresses. Furthermore a projective Newton method is considered which reduces the solution of the nonlinear continuous problem to the successive solution of a sequence of linearized problems of increasing dimension. It is proved that this procedure is well defined and also converges with quasi-optimal rates.     

The Fourier-Nitsche-mortaring for elliptic problems with reentrant edges

The Fourier method is combined with the Nitsche-finite-element method (as a mortar method) and applied to the Dirichlet problem of the Poisson equation in three-dimensional axisymmetric domains with reentrant edges generating singularities. The approximating Fourier method yields a splitting of the 3D problem into a set of 2D problems on the meridian plane of the given domain. For solving the 2D problems bearing corner singularities, the Nitsche-finite-element method with non-matching meshes and mesh grading near reentrant corners is applied. Using the explicit representation of some singularity function of non-tensor product type, the rate of convergence of the Fourier-Nitsche-mortaring is estimated in some H ¹-like norm as well as in the L ₂-norm for weak regularity of the solution. Finally, some numerical results are presented.      

Deriving a discrete approximate solution for a nonlinear dynamic system of two-phase soil media

A discrete approximate generalized solution is derived for a nonlinear differential model of the dynamics of two-phase soil media and its convergence is estimated for the corresponding generalized solution in the space W ₂ ¹ (Ω). Translated from Kibernetika i Sistemnyi Analiz, No. 4, pp. 69–80, July–August 2009     

Finite-difference method for computation of 3-D gas dynamics equations with artificial viscosity

A new numerical method for the solution of gas dynamics problems for three-dimensional (3D) systems in Eulerian variables is presented in the paper. The proposed method uses the approximation O(τ² + h _x ² + h _y ² + h _z ²) in the areas of the solution’s smoothness and beyond the compression waves; τ is the time step; and h _x , h _y , and h _z are space variable steps. In the proposed difference scheme, in addition to Lax-Wendroff corrections, artificial viscosity μ that monotonizes the scheme is introduced. The viscosity is obtained from the conditions of the maximum principle. The results of the computation of the 3D test problem for the Euler equation are presented.     

A method for non-differentiable optimization problems

We introduce a new method for solving a class of non-smooth unconstrained optimization problems. The method constructs a sequence<texlscub>x _k </texlscub>in ? ⁿ , and at the kth iteration, a search direction h _k is considered, where h _k is a solution to a variational inequality problem. A convergence theorem for our algorithm model and its discrete version can be easily proved. Furthermore, preliminary computational results show that the new method performs quite well and can compete with other methods.     

Iterative Stokes Solvers in the Harmonic Velte Subspace

We explore the prospects of utilizing the decomposition of the function space (H ¹ ₀) ⁿ (where n=2,3) into three orthogonal subspaces (as introduced by Velte) for the iterative solution of the Stokes problem. It is shown that Uzawa and Arrow-Hurwitz iterations – after at most two initial steps – can proceed fully in the third, smallest subspace. For both methods, we also compute optimal iteration parameters. Here, for two-dimensional problems, the lower estimate of the inf-sup constant by Horgan and Payne proves useful and provides an inclusion of the spectrum of the Schur complement operator of the Stokes problem. We further consider the conjugate gradient method in the third Velte subspace and derive a corresponding convergence estimate. Computational results show the effectiveness of this approach for discretizations which admit a discrete Velte decomposition. Received June 11, 1999; revised October 27, 2000     

Numerical analysis and computations of a high accuracy time relaxation fluid flow model

We present a numerical study based on continuous finite element analysis for a time relaxation regularization of Navier–Stokes equations. This regularization is based on filtering and deconvolution. We study the convergence of the regularized equations using a fully discretized filter and deconvolution algorithm. Velocity and pressure error estimates and the L ₂ Aubin–Nitsche lift technique are proved for the equilibrium problem, and this analysis is accompanied by the velocity error estimate for the time-dependent problem, too. Thus, optimal error estimates in L ₂ and H ¹ norms are derived and followed by their computational verification. Also, computational results of the vortex street are presented for the two-dimensional cylinder benchmark flow problem. Maximum drag and lift coefficients and difference in pressure between the front and back of the cylinder at the final time were investigated as well, showing that the time relaxation regularization can attain the benchmark values.     

Numerical solution of the two-yield elastoplastic minimization problem

Summary This paper concentrates on fast calculation techniques for the two-yield elastoplastic problem, a locally defined, convex but non-smooth minimization problem for unknown plastic-strain increment matrices P ₁ and P ₂. So far, the only applied technique was an alternating minimization, whose convergence is known to be geometrical and global. We show that symmetries can be utilized to obtain a more efficient implementation of the alternating minimization. For the first plastic time-step problem, which describes the initial elastoplastic transition, the exact solution for P ₁ and P ₂ can even be obtained analytically. In the later time-steps used for the computation of the further development of elastoplastic zones in a continuum, an extrapolation technique as well as a Newton-algorithm are proposed. Finally, we present a realistic example for the first plastic and the second time-steps, where the new techniques decrease the computation time significantly.      

On two-fluid equations for dispersed incompressible two-phase flow

This paper deals with the two-fluid formulation for dispersed two-phase flow. In one space dimension the equations with constant viscosity for both phases are shown to result in a locally in time well-posed periodic problem but without viscosity there are regions in phase space where the problem is non-hyperbolic. It is shown that the viscid problem linearized at constant states in the non-hyperbolic region is terribly unstable for small viscosity coefficients. By numerical experiments for the solution to a model problem it is demonstrated that for smooth initial data in the non-hyperbolic region, discontinuities seem to form. Second order artificial dissipation is added, in an effort to regularize the problem. Numerical examples are given, for different types of initial data. For initial data that is smooth in the non-hyperbolic region, the formation of jumps, or viscid layers, is strongly dependent on the amount of artificial dissipation. No convergence is obtained as the amount of artificial dissipation is diminished. On the other hand, if the initial data is smooth only in the hyperbolic region and with jumps through the non-hyperbolic region, then the jumps or viscid layers that later form, can damp the onset of new layers. In this case convergence in the L₂ sense seem to hold, the computed solutions to the regularized problem approach a weak solution as the artificial dissipation is diminished. Received: 30 January 2001 / Accepted: 30 May 2001     

Nonlocal quasi-Newton method for solving convex variational inequalities

Conclusion In the optimization problem [f ₀(x)│h_i(x)<-0,i=1,…,l] relaxation of the functionf ₀(x)+Nh⁺(x) does not produce, as we know [6, 7], α_k=1 in Newton's method with the auxiliary problem (5), (6), whereF(x)=f _0′(x). For this reason, Newton type methods based on relaxation off ₀(x)+Nh⁺(x) are not superlinearly convergent (so-called Maratos effect). The results of this article indicate that if (F(x)=f _0′(x), then replacement of the initial optimization problem with a larger equivalent problem (7) eliminates the Maratos effect in the proposed quasi-Newton method. This result is mainly of theoretical interest, because Newton type optimization methods in the space of the variablesx ∈R ⁿ are less complex. However to the best of our knowledge, the difficulties with nonlocal convergence arising in these methods (choice of parameters, etc.) have not been fully resolved [10, 11]. The discussion of these difficulties and comparison with the proposed method fall outside the scope of the present article, which focuses on solution of variational inequalities (1), (2) for the general caseF′(x)≠F′ ^T(x). Translated from Kibernetika i Sistemnyi Analiz, No. 6, pp. 78–91, November–December, 1994.     

An approximation method of the local type application to a problem of heart potential mapping

In this paper a local type method is proposed to smooth noisy data. Criteria of convergence and error bounds are given. An applciation is also presented for a biomedical problem in ?³, which is usually solved only in ?².     

Preconditioned conjugate gradient methods for the solution of indefinite least squares problems

The conjugate gradient (CG) method is considered for solving the large and sparse indefinite least squares (ILS) problem min  _x (b−Ax) ^T J(b−Ax) where J=diag (I _p ,−I _q ) is a signature matrix. However the rate of convergence becomes slow for ill-conditioned problems. The QR-based preconditioner is found to be effective in accelerating the convergence. Numerical results show that the sparse Householder QR-based preconditioner is superior to the CG method especially for sparse and ill-conditioned problems.