Applications of Mathematics最新文献

We propose a symmetric interior penalty discontinuous Galerkin (DG) method for nonlinear fully coupled quasi-static thermo-poroelasticity problems. Firstly, a fully implicit nonlinear discrete scheme is constructed by adopting the DG method for the spatial approximation and the backward Euler method for the temporal discretization. Subsequently, the existence and uniqueness of the solution of the numerical scheme is proved, and then we derive the a priori error estimate for the three variables, i.e., the displacement, the pressure and the temperature. Lastly, we carry out numerical experiments to confirm the theoretical findings of our suggested approach.

We study the convergence of two-step Ulm-Chebyshev-like method for solving the inverse singular value problems. We focus on the case when the given singular values are positive and multiple. This work extends the result of W. Ma (2022). We show that the new method is cubically convergent. Moreover, numerical experiments are given in the last section, which show that the proposed method is practical and efficient.

We discuss Lyapunov stability/instability of both lower and upper equilibria of free damped pendulum with periodically oscillating suspension point. We recall the results of Bogolyubov and Kapitza, provide new effective criteria of stability/instability of the equilibria of pendulum equation, and give the exact and complete proofs. The criteria obtained are formulated in terms of positivity/negativity of Green’s functions of the periodic boundary value problems for linearized equations. Furthermore, we show that if both lower and upper equilibria are stable, then the pendulum considered may possess a periodic motion that corresponds to the “quasistatic solution” of Bogolyubov as well as to the “quasistatic balance” of Kapitza.

FETI (finite element tearing and interconnecting) based domain decomposition methods are well-established massively parallel methods for solving huge linear systems arising from discretizing partial differential equations. The first steps of FETI decompose the domain into nonoverlapping subdomains, discretize the subdomains using matching grids, and interconnect the adjacent variables by multipoint constraints. However, the multipoint constraints enforcing identification of the corners’ variables do not have a unique representation and their proper choice and modification can improve the performance of FETI. Here, we briefly review the main options, including orthogonal, fully redundant, or localized constraints, and use the basic linear algebra and spectral graph theory to examine the quantitative effect of their choice on the effective control of the feasibility error and rate of convergence of FETI.

We investigate the recovery of k-sparse signals using the ℓ₁-ℓ₂ minimization model with prior support set information. The prior support set information, which is believed to contain the indices of nonzero signal elements, significantly enhances the performance of compressive recovery by improving accuracy, efficiency, reducing complexity, expanding applicability, and enhancing robustness. We assume k-sparse signals x with the prior support T which is composed of g true indices and b wrong indices, i.e., ∣T∣ = g+b ⩽ k. First, we derive a new condition based on RIP of order 2α (α = k − g) to guarantee signal recovery via ℓ₁-ℓ₂ minimization with partial support information. Second, we also derive the high order RIP with tα for some t ⩾ 3 to guarantee signal recovery via ℓ₁-ℓ₂ minimization with partial support information.

We propose some approaches for the generation of conforming simplicial partitions with various regularity properties for polytopic domains that are products or a union of products, thus generalizing our earlier results. The techniques presented can be used for finite element simulations of higher-dimensional problems.

Conjugate gradient methods are widely used for solving large-scale unconstrained optimization problems, because they do not need the storage of matrices. Based on the self-scaling memoryless Broyden-Fletcher-Goldfarb-Shanno (SSML-BFGS) method, new conjugate gradient algorithms CG-DESCENT and CGOPT have been proposed by W. Hager, H. Zhang (2005) and Y. Dai, C. Kou (2013), respectively. It is noted that the two conjugate gradient methods perform more efficiently than the SSML-BFGS method. Therefore, C. Kou, Y. Dai (2015) proposed some suitable modifications of the SSML-BFGS method such that the sufficient descent condition holds. For the sake of improvement of modified SSML-BFGS method, in this paper, we present an efficient SSML-BFGS-type three-term conjugate gradient method for solving unconstrained minimization using Ford-Moghrabi secant equation instead of the usual secant equations. The method is shown to be globally convergent under certain assumptions. Numerical results compared with methods using the usual secant equations are reported.

We introduce a new scaling parameter for the Dai-Kou family of conjugate gradient algorithms (2013), which is one of the most numerically efficient methods for unconstrained optimization. The suggested parameter is based on eigenvalue analysis of the search direction matrix and minimizing the measure function defined by Dennis and Wolkowicz (1993). The corresponding search direction of conjugate gradient method has the sufficient descent property and the extended conjugacy condition. The global convergence of the proposed algorithm is given for both uniformly convex and general nonlinear objective functions. Also, numerical experiments on a set of test functions of the CUTER collections and the practical problem of the manipulator of robot movement control show that the proposed method is effective.

We consider pressure-driven flow between adjacent surfaces, where the fluid is assumed to have constant density. The main novelty lies in using implicit algebraic constitutive relations to describe the fluid’s response to external stimuli, enabling the modeling of fluids whose responses cannot be accurately captured by conventional methods. When the implicit algebraic constitutive relations cannot be solved for the Cauchy stress in terms of the symmetric part of the velocity gradient, the traditional approach of inserting the expression for the Cauchy stress into the equation for the balance of linear momentum to derive the governing equation for the velocity becomes inapplicable. Instead, a non-standard system of first-order equations governs the flow. This system is highly complex, making it important to develop simplified models. Our primary contribution is the development of a framework for achieving this. Additionally, we apply our findings to a fluid that exhibits an S-shaped curve in the shear stress versus shear rate plot, as observed in some colloidal solutions.

We present a precise anisotropic interpolation error estimate for the Morley finite element method (FEM) and apply it to fourth-order elliptic equations. We do not impose the shape-regularity mesh condition in the analysis. Anisotropic meshes can be used for this purpose. The main contributions of this study include providing a new proof of the term consistency. This enables us to obtain an anisotropic consistency error estimate. The core idea of the proof involves using the relationship between the Raviart-Thomas and Morley finite-element spaces. Our results indicate optimal convergence rates and imply that the modified Morley FEM may be effective for errors.