查看更多>>摘要：We analytically and numerically solve the PCP equation, u(xx)u(yy) = 1, with homogeneous Dirichlet boundary conditions on the unit square. Chebyshev and Fourier spectral methods with low degree truncations yield moderate accuracy but the usual exponential rate of convergence of spectral methods is destroyed by the boundary singularities of the solution. In the sequel to this work, we will apply a variety of strategies including a change-of-coordinates and singular basis functions to recover spectral accuracy in spite of the boundary singularities. As preparation for this numerical study, we find explicit solutions to related problems to the two-dimensional PCP equation in a domain with a boundary that is an ellipse and the three-dimensional PCP equation in a cubic domain. We also analyze the boundary behavior of these solutions: all have complicated singularities with unbounded first derivatives. (C) 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 study concerns the Darcy flow problem in a two-dimensional fractured porous domain, in which the fracture is regarded as a one-dimensional interface, interacting with the surrounding media. In this paper, a finite volume element method (FVEM) is first proposed for the multi-dimensional fracture model, and error estimates for the pressure with optimal convergence are discussed. On this basis, a two-grid FVEM is developed for decoupling the multi-domain fracture model by a coarse grid approximation to the interface coupling conditions, and theoretical analysis demonstrates that approximation accuracy does not deteriorate under the two-grid decoupling technique. Finally, numerical experiments for FVEM and two-grid FVEM are presented to confirm the accuracy of theoretical analysis. (c) 2021 Elsevier B.V. All rights reserved.
查看更多>>摘要：The well-known approximation formula by Barone-Adesi and Whaley (BAW) for pricing American options works well for contingent claims in the current business environment with low rates, but it lacks precision for pricing options when interest rates are high or in case of turmoil. In this paper, we introduce a new closed formula that is the solution of a non-autonomous PDE instead of the classical ODE. Our improved solution performs well in case of high turbulence allowing traders and risk managers to run stress tests with an appropriate model. This is complemented by an analytical approximation of the critical stock price S* as well as of the implied volatility. When a shock comes, it might be helpful to have the right model to deal with it. (c) 2021 Elsevier B.V. All rights reserved.
查看更多>>摘要：This paper deals with the design and analysis of a high order numerical scheme for the nonlinear time-fractional generalized Fisher's equation (TFGFE). The Caputo fractional derivative (FD) of order alpha, (alpha is an element of (0, 1)) appearing in the model problem is approximated by means of L1 - 2 scheme. The discretization for the space derivative is made by a collocation method based on quintic B-spline (QBS) basis function. Convergence analysis of the method is established. Five examples are provided to demonstrate the efficiency and feasibility of the method. The influence of alpha on the solution profile of the TFGFE is examined. It is shown that our method is of O(Delta t(2) + Delta x(4)) accuracy, where Delta t and Delta x respectively represent the time and space step sizes. The results obtained are compared with those of other three methods. The CPU time (in seconds) is given in order to justify the computational efficiency of proposed numerical scheme. (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.
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.
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