In this paper, we propose a mass- and modified-energy-conserving linearly implicit Crank–Nicolson finite element method for the rotating Gross–Pitaevskii equation (GPE), which arises in the modeling of rotating Bose–Einstein condensates. The construction combines a scalar auxiliary variable (SAV) reformulation of the rotating GPE with a linearly implicit Crank–Nicolson temporal discretization and a conforming Lagrange finite element discretization in space. At each time step, the scheme reduces to one linear elliptic system supplemented by an explicit scalar update, thereby avoiding any nonlinear iteration. The fully discrete solution preserves the discrete mass and the SAV modified energy exactly. We prove unique solvability at every time step and derive error bounds of order 𝒪(h^k+τ ^2) in the H^1 norm and 𝒪(h^k+1+τ ^2) in the L^2 norm, without a coupling condition between h and τ . The analysis also gives an 𝒪(h^k+1+τ ^2) bound for the difference between the SAV modified energy and the original discrete Hamiltonian and for the drift of the latter. Numerical experiments illustrate the predicted convergence rates and the long-time conservation of the discrete mass and the SAV modified energy.
The aim of this paper is to implement a high-order Chebyshev spectral collocation method for solving partial integro-differential equations (PIDEs) in multiple dimensions. Time marching methods and transform-based methods (non-time marching) can be used to discretize temporal terms. The combination of a small time step size in time marching methods and a high-order scheme in space with many degrees of freedom requires a considerable amount of computational time. To address this limitation, we propose an algorithm based on a combination of the Laplace transform and Chebyshev spectral collocation for solving second-order time-fractional PIDEs with weakly singular kernels. Unlike time-marching approaches, the transform-based method has a much lower computational complexity and can take advantage of parallel computing. Numerical experiments in one, two, and three dimensions validate the accuracy, efficiency, and applicability of the proposed approach.
In this paper, we present convergence theorems that ensure sufficient conditions for an extrapolated Tseng-type algorithm to converge without requiring monotonicity of the underlying operator. The extrapolation takes the form of a double-inertial technique, and the stepsize is designed within adaptive rules. We establish weak convergence for each of the algorithms and further obtain strong convergence results that ensure R-linear convergence rate of the proposed methods. Numerical experiments illustrate the performance of the proposed adaptive variant of the algorithm in two application settings. The first considers constrained minimization of a difference-of-convex function arising from negative ridge regularization, which is relevant in various technological and engineering fields. The second involves an application to a blood supply chain network with 44 paths and 5 demand points, where the perishable nature of the product necessitates efficient optimization to reduce waste and cost while managing risk. The results presented here complement recent results in the literature and provide alternative approaches to the existing methods with similar structures.
In this paper, we propose a family of hybrid derivative-free projection methods for solving pseudo-monotone nonlinear equations. The proposed hybrid search direction incorporates an arbitrary nonzero vector together with a hybrid adaptive denominator structure, and always satisfies both the sufficient descent condition and the trust-region-type property. Furthermore, a backtracking self-adaptive line search is developed to improve the iteration complexity of the proposed methods. Global convergence is established without assuming the Lipschitz continuity of the underlying mapping. Under the local Lipschitz continuity assumption, we prove that the proposed methods attain a favorable iteration-complexity bound. Moreover, under the local Lipschitz continuity and local error bound assumptions, we further establish a linear convergence rate without imposing any additional restrictions on the algorithmic parameters. Finally, two different choices of the nonzero vector and three different hybrid adaptive denominator structures are considered, resulting in a family of six hybrid derivative-free projection methods. Numerical experiments on benchmark nonlinear equations demonstrate the competitive numerical performance of the proposed methods. Furthermore, two representative variants are applied to regularized logistic regression and compressed sensing, illustrating their practical potential.
We investigate optimized Schwarz domain decomposition methods in time for the control of the 1D transport equation. In the case of an internal control over the whole domain, the optimization problem can be transformed into a system of two coupled PDEs. We then apply the time-domain decomposition (without overlap) strategy on this PDE system and extend it to its discretized counterpart. We show that while a judicious choice of parameters in the transmission conditions leads to convergence in a finite number of iterations for the continuous PDE system, no such choice exists for the discretized counterpart. Using Fourier analysis, we analyse three different iterations for the discrete problem: the fixed point iteration, the relaxed iteration and the preconditioned GMRES method. For each case, we are able to determine parameters for the transmission conditions that lead to fast convergence of the method. Unlike for elliptic problems, here we find that the optimal parameters for the three methods tend towards different limits under grid refinement, leading to different asymptotic convergence factors. We illustrate our results by numerical examples.
