查看更多>>摘要：Some extragradient-type algorithms with inertial effect for solving strongly pseudo-monotone variational inequalities have been proposed and investigated recently. While the convergence of these algorithms was established, it is unclear if the linear rate is guaranteed. In this paper, we provide R-linear convergence analysis for two extragradient-type algorithms for solving strongly pseudo-monotone, Lipschitz continuous variational inequality in Hilbert spaces. The linear convergence rate is obtained without the prior knowledge of the Lipschitz constants of the variational inequality mapping and the stepsize is bounded from below by a positive number. Some numerical results are provided to show the computational effectiveness of the algorithms. (C) 2021 Elsevier B.V. All rights reserved.
查看更多>>摘要：Invariance transformations of polyadic decompositions of matrix multiplication tensors define an equivalence relation on the set of such decompositions. In this paper, we present an algorithm to efficiently decide whether two polyadic decompositions of a given matrix multiplication tensor are equivalent. With this algorithm, we analyze the equivalence classes of decompositions of several matrix multiplication tensors. This analysis is relevant for the study of fast matrix multiplication as it relates to the question of how many essentially different fast matrix multiplication algorithms there exist. This question has been first studied by de Groote, who showed that for the multiplication of 2 x2 matrices with 7 active multiplications, all algorithms are essentially equivalent to Strassen's algorithm. In contrast, the results of our analysis show that for the multiplication of larger matrices (e.g., 2 x3 by 3 x2 or 3 x3 by 3 x3 matrices), two decompositions are very likely to be essentially different. We further provide a necessary criterion for a polyadic decomposition to be equivalent to a polyadic decomposition with integer entries. Decompositions with specific integer entries, e.g., powers of two, provide fast matrix multiplication algorithms with better efficiency and stability properties. This condition can be tested algorithmically and we present the conclusions obtained for the decompositions of small/medium matrix multiplication tensors. (C)& nbsp;2021 Elsevier B.V. All rights reserved.
查看更多>>摘要：In this work, we introduce a new semi-discrete scheme based on the so-called no-flow curves and its numerical analysis for solving initial value problems that involve one-dimensional (1D) scalar hyperbolic conservation laws and of the form, u(t) + H(u)(x) = 0, x is an element of R, t > 0, u(x, 0) = u(0)(x). In addition, we present a two-dimensional (2D) version of the semi-discrete Lagrangian-Eulerian scheme to show that the proposed method can be applied to multidimensional problems. From an improved 1D weak numerical asymptotic analysis, we found that the solutions provided by the novel semi-discrete scheme satisfy a maximum principle property and a Kruzhkov entropy condition. We also highlight the possibility of using the no-flow curves as a new desingularization analysis technique for the construction of computationally stable numerical fluxes in the locally conservative form for nonlinear hyperbolic problems. We provide nontrivial 1D and 2D numerical examples with nonlinear wave interaction to illustrate the effectiveness and capabilities of the proposed approach and verify the theory. According to the results, the scheme handles discontinuous solutions (shocks) with low numerical dissipation quite well and shows a very good resolution of rarefaction waves with no spurious glitch effect in the vicinity of the sonic points. We also consider a test case for nonstrictly hyperbolic conservation laws with a resonance point (coincidence of eigenvalues) modeling three-phase flow in a porous media transport problem. (C)& nbsp;2021 Elsevier B.V. All rights reserved.
Wang, ChengWise, Steven M.Yue, XingyeZhou, Shenggao...
13页
查看更多>>摘要：In this paper, we provide a theoretical analysis for an iteration solver to implement a finite difference numerical scheme for the Poisson-Nernst-Planck (PNP) system, based on the Energetic Variational Approach (EnVarA), in which a non-constant mobility H-1 gradient flow is formulated. In particular, the nonlinear and singular nature of the logarithmic energy potentials has always been the essential difficulty. In the numerical design, the mobility function is explicitly updated, for the sake of unique solvability analysis. The logarithmic and the electric potential diffusion terms, which come from the gradient of convex energy functional parts, are implicitly computed. The positivity-preserving property for all the concentrations, an unconditional energy stability, and the optimal rate error estimate have been established in a recent work. A modified Newton iteration for the nonlinear and logarithmic part, combined with a linear iteration for the electric potential part, is proposed to implement the given numerical scheme, in which a non-constant linear elliptic equation needs to be solved at each iteration stage. A theoretical analysis is presented in this article, and a linear convergence is proved for such an iteration, with an asymptotic error constant in the same order of the time step size. A numerical test is also presented in this article, which demonstrates the linear convergence rate of the proposed iteration solver. (C)& nbsp;2021 Elsevier B.V. All rights reserved.
查看更多>>摘要：In this work, for the phase-field model of lipid vesicles with the property of accurate volume conservation, we construct an effective fully-discrete numerical scheme, in which, the time marching method is based on a novel splitting type technique, and space is discretized by using the Spectral-Galerkin method. The advantage of this scheme is its high efficiency and ease of implementation. Specifically, although the model is highly nonlinear, just by solving two independent linear biharmonic equations with constant coefficients at each time step, the scheme can achieve the second-order accuracy in time, spectral accuracy in space, and unconditional energy stability. The essence of the scheme is to introduce several additional auxiliary variables and use the specially designed ordinary differential equations to reformulate the system. In this way, energy stability can be obtained unconditionally, while avoiding the calculation of variable-coefficient systems. We strictly prove that the energy stability in the fully-discrete form that the scheme holds and give a detailed implementation process. Numerical experiments in 2D and 3D are further carried out to verify the convergence rate, energy stability, and effectiveness of the developed algorithm.(C)& nbsp;2021 Elsevier B.V. All rights reserved.& nbsp;
查看更多>>摘要：This paper provides a Nystrom method for the numerical solution of Volterra integral equations whose kernels contain singularities of algebraic type. It is proved that the method is stable and convergent in suitable weighted spaces. An error estimate is also given as well as several numerical tests are presented. (C) 2021 Elsevier B.V. All rights reserved.
查看更多>>摘要：In this article, we present and analyze a weak Galerkin finite element method (WG-FEM) for the coupled Navier-Stokes/temperature (or Boussinesq) problems. In this WG-FEM, discontinuous functions are applied to approximate the velocity, temperature, and the normal derivative of temperature on the boundary while piecewise constants are used to approximate the pressure. The stability, existence and uniqueness of solution of the associated WG-FEM are proved in detail. An optimal a priori error estimate is then derived for velocity in the discrete H-1 and L-2 norms, pressure in the L-2 norm, temperature in the discrete H-1 and L-2 norms, and the normal derivative of temperature in H-1/2 norm. Finally, to complete this study some numerical tests are presented which illustrate that the numerical errors are consistent with theoretical results. (c) 2021 Elsevier B.V. All rights reserved.
Medina, Emmanuel Y. Y.Toledo, Elson M. M.Igreja, IuryRocha, Bernardo M. M....
16页
查看更多>>摘要：In this paper, we present a stabilized mixed hybrid discontinuous Galerkin finite element method for solving fourth-order parabolic problems such as the Cahn-Hilliard equation used to describe the dynamics of avascular tumor growth. The proposed method is compared to other methods previously studied in this context. Several numerical experiments are presented to show convergence, performance and accuracy comparisons with other methods. Besides, we also show the ability of the method to solve a complex case of avascular tumor growth where the behavior of tumor cells towards nutrient gradients drives fingering instabilities. The results show the great potential of the presented method for the efficient and accurate solution of Cahn-Hilliard problems which includes applications on complex tumor growth problems. (c) 2021 Elsevier B.V. All rights reserved.
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