查看更多>>摘要：A new conforming discontinuous Galerkin method, which is based on weak Galerkin finite element method, is introduced for solving second order elliptic interface problems with discontinuous coefficient. The numerical method studied in this paper has no stabilizer and fewer unknowns compared with the known weak Galerkin algorithms. The error estimates in H-1 and L-2 norms are established, which are the optimal order convergence. Numerical experiments demonstrate the performance of the method, confirm the theoretical results of accuracy. (c) 2022 Elsevier B.V. All rights reserved.
查看更多>>摘要：In this article, we study a posteriori error analysis of quadratic finite element method in the maximum norm for the elliptic obstacle problem. We discuss the reliability and the efficiency of the proposed a posteriori error estimator. In the analysis, regularized Green's function plays a crucial role and together with that, in obtaining the sign of the discrete Lagrange multiplier we have exploited the property that midpoint quadrature rules are exact for quadratic polynomials. Numerical results are performed to illustrate the convergence behavior of a posteriori error estimator through various test examples. (c) 2022 Elsevier B.V. All rights reserved.
查看更多>>摘要：We hybridized the rigorous coupled-wave approach (RCWA) with transformation optics to develop a hybrid coordinate-transform method for solving the time-harmonic Maxwell equations in a 2D domain containing a surface-relief grating. In order to prove that this method converges for the p-polarization state, we studied several different but related scattering problems. The imposition of generalized non-trapping conditions allowed us to prove a-priori estimates for these problems. To do this, we proved a Rellich identity and used density arguments to extend the estimates to more general problems. These a-priori estimates were then used to analyze the hybrid method. We obtained convergence rates with respect to two different parameters, the first being a slice thickness indicative of spatial discretization in the depth dimension, the second being the number of terms retained in the Rayleigh-Bloch expansions of the electric and magnetic field phasors with respect to the other dimension. Testing with a numerical example revealed faster convergence than our analysis predicted. The hybrid method does not suffer from the Gibbs phenomenon seen with the standard RCWA. (C) 2022 Elsevier B.V. All rights reserved.
查看更多>>摘要：We construct a family of embedded pairs for optimal explicit strong stability preserving Runge-Kutta methods of order 2 <= p <= 4 to be used to obtain numerical solution of spatially discretized hyperbolic PDEs. In this construction, the goals include non-defective property, large stability region, and small error values as defined in Dekker and Verwer (1984) and Kennedy et al. (2000). The new family of embedded pairs offer the ability for strong stability preserving (SSP) methods to adapt by varying the step-size. Through several numerical experiments, we assess the overall effectiveness in terms of work versus precision while also taking into consideration accuracy and stability. (c) 2022 The Author(s). Published by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).
查看更多>>摘要：We consider linear scalar wave equations with a hereditary integral term of the kind used to model viscoelastic solids. The kernel in this Volterra integral is a sum of decaying exponentials (The so-called Maxwell, or Zener model) and this allows the introduction of one of two types of families of internal variables, each of which evolve according to an ordinary differential equation (ODE). There is one such ODE for each decaying exponential, and the introduction of these ODEs means that the Volterra integral can be removed from the governing equation. The two types of internal variable are distinguished by whether the unknown appears in the Volterra integral, or whether its time derivative appears; we call the resulting problems the displacement and velocity forms. We define fully discrete formulations for each of these forms by using continuous Galerkin finite element approximations in space and an implicit `Crank-Nicolson' type of finite difference method in time. We prove stability and a priori bounds, and using the FEniCS environment, https://fenicsproject.org/ (The FEniCS project version 1.5, Archive of Numerical Software, 3 (100), 9-23, 2015.) give some numerical results. These bounds do not require Gronwall's inequality and so can be regarded to be of high quality, allowing confidence in long time integration without an a priori exponential build up of error. As far as we are aware this is the first time that these two formulations have been described together with accompanying proofs of such high quality stability and error bounds. The extension of the results to vector-valued viscoelasticity problems is straightforward and summarised at the end. The numerical results are reproducible by acquiring the python sources from https://github.com/Yongseok7717, or by running a custom built docker container (instructions are given). (c) 2022 Elsevier B.V. All rights reserved.
查看更多>>摘要：A projected residual algorithm for solving smooth convex optimization problems is presented. The proposed method is an extension of a residual algorithm for solving systems of nonlinear monotone equations introduced by La Cruz (2017), which uses in a systematic way the residual as a search direction combined with the Barzilai-Borwein's choice of the step size and a line search globalization strategy that does not impose the condition that the function value to decrease monotonically at every iteration. The global and R-sublinear convergence of the new method is established. With the aim of showing the advantages of the proposed global scheme an extensive set of numerical experiments including standard test problems and some specific applications are reported. (c) 2022 Elsevier B.V. All rights reserved.
查看更多>>摘要：We present important improvements and additions to a modern technique developed by Ixaru to solve the time-dependent two-dimensional Schrodinger equation with homogeneous Dirichlet boundary conditions over a rectangular domain. The algorithm, first described in Ixaru (2010), is based on the so-called Constant Perturbation technique. In this paper, we refine and extend the algorithm with important features. We focus in particular on new algorithms for the determination of the index of the eigenvalues, for the orthonormalization of eigenfunctions, for automatic step size selection and for the accurate computation of integrals. We provide the new developments with sufficient theoretical background and numerical experiments. (c) 2022 Elsevier B.V. All rights reserved.
Durmaz, Muhammet EnesCakir, MusaAmirali, IlhameAmiraliyev, Gabil M....
15页
查看更多>>摘要：This work presents a computational approximate to solve singularly perturbed Fredholm integro-differential equation with the reduced second type Fredholm equation. This problem is discretized by a finite difference approximate, which generates second-order uniformly convergent numerical approximations to the solution. Parameter-uniform approximations are generated using Shishkin type meshes. The performance of the numerical scheme is tested which supports the effectiveness of the technique. (c) 2022 Elsevier B.V. All rights reserved.
查看更多>>摘要：This paper aims at proving the convergence and quasi-optimality of an adaptive nonconforming finite element method for Stokes distributed control problems with pointwise control constraints. Nonconforming P-1/P-0 pair (Crouzeix-Raviart elements) and variational discretization are used to approximate the state equation and the control variable, respectively. A posteriori error estimates with upper and lower bounds are first derived for the state and adjoint variables. Then we prove the contraction property for the sum of the energy error of the state and adjoint state and the scaled error estimator on two consecutive adaptive meshes. The resulting linear convergence is finally used to show the quasi-optimal convergence rate of the adaptive algorithm. Additionally, some numerical results are provided to support our theoretical analysis. (c) 2022 Elsevier B.V. All rights reserved.
Shargatov, V. A.Chugainova, A. P.Kolomiytsev, G., V
18页
查看更多>>摘要：We consider traveling wave solutions of generalized Korteweg-de Vries-Burgers equation when the flux function has two inflection points. The dissipation coefficient mu depends only on the spatial coordinate in some moving coordinate system and increases monotonically from mu(1) to mu(2) in the narrow spatial region. Some external influence causes the change in the dissipation coefficient. The set of admissible shocks is defined. In order to determine which discontinuities are admissible, we study the nonlinear (global) stability of traveling wave solutions. Scenarios of the evolution of linearly unstable traveling waves are described, and asymptotics of unstable solutions are found. We find that a stable traveling wave solution and a solution with a time-dependent structure can correspond to the same admissible shock. (c) 2022 Elsevier B.V. All rights reserved.
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