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Abstract: We consider time-harmonic Maxwell’s equations set in a heterogeneous medium with perfectly conducting boundary conditions. Given a divergence-free right-hand side lying in \(L^2\) , we provide a frequency-explicit approximability estimate measuring the difference between the corresponding solution and its best approximation by high-order Nédélec finite elements. Such an approximability estimate is crucial in both the a priori and a posteriori error analysis of finite element discretizations of Maxwell’s equations, but the derivation is not trivial. Indeed, it is hard to take advantage of high-order polynomials given that the right-hand side only exhibits \(L^2\) regularity. We proceed in line with previously obtained results for the simpler setting of the scalar Helmholtz equation and propose a regularity splitting of the solution. In turn, this splitting yields sharp approximability estimates generalizing known results for the scalar Helmholtz equation and showing the interest of high-order methods. PubDate: 2022-05-16
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Abstract: For solving a class of complex symmetric linear systems, we improve the combination method of real part and imaginary part (CRI) by introducing two optimization techniques—minimum residual and block successive overrelaxation acceleration—and obtain two new iteration methods: minimum residual CRI (MRCRI) and modified CRI (MCRI). Theoretical analysis implies that the new methods are convergent under suitable conditions. Numerical experiments are used to confirm the effectiveness of the MRCRI and MCRI methods, and experiments of parameter sensitivity show that the MRCRI method is more effective than the CRI and PMHSS methods. PubDate: 2022-05-05
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Abstract: Abstract By defining two important terms called basic perturbation vectors and obtaining their linear bounds, we obtain the linear componentwise perturbation bounds for unitary factors and upper triangular factors of the generalized Schur decomposition. The perturbation bounds for the diagonal elements of the upper triangular factors and the generalized invariant subspace are also derived. From the former, we present an upper bound and a condition number of the generalized eigenvalue. Furthermore, with numerical iterative method, the nonlinear componentwise perturbation bounds of the generalized Schur decomposition are also provided. Numerical examples are given to test the obtained bounds. Among them, we compare our upper bound and condition number of the generalized eigenvalue with their counterparts given in the literature. Numerical results show that they are very close to each other but our results don’t contain the information on the left and right generalized eigenvectors. PubDate: 2022-04-20
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Abstract: Abstract In this article, we consider the discretization of nonlocal coupled parabolic problem within the framework of the virtual element method. The presence of nonlocal coefficients not only makes the computation of the Jacobian more expensive in Newton’s method, but also destroys the sparsity of the Jacobian. In order to resolve this problem, an equivalent formulation that has very simple Jacobian is proposed. We derive the error estimates in the \(L^2\) and \(H^1\) norms. To further reduce the computational complexity, a linearized scheme without compromising the rate of convergence in different norms is proposed. Finally, the theoretical results are justified through numerical experiments over arbitrary polygonal meshes. PubDate: 2022-03-17
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Abstract: Abstract Symmetric second derivative general linear methods (SGLMs) have been already introduced for the numerical solution of time-reversible differential equations. To construct suitable high order methods for such problems, the newly developed composition theory has been successfully used for structure-preserving methods. In this paper, composite symmetric SGLMs are introduced using the generalization of composite theory for general linear methods. Then, we construct symmetric methods of order six by the composition of symmetric SGLMs of order four. Numerical results of the constructed methods verify the theoretical order of accuracy and illustrate that the invariants of motion over long time intervals for reversible Hamiltonian systems are well preserved. PubDate: 2022-03-14
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Abstract: Abstract The memory-less SR1 with generalized secant equation (MM-SR1gen) is presented and developed together with its numerical performances for solving a collection of 800 unconstrained optimization problems with the number of variables in the range [1000, 10000]. The convergence of the MM-SR1gen method is proved under the classical assumptions. Comparison between the MM-SR1gen versus the memory-less SR1 method, versus the memory-less BFGS method and versus the BFGS in implementation of Shanno and Phua from CONMIN show that MM-SR1gen is more efficient and more robust than these algorithms. By solving five applications from MINPACK-2 collection, each of them with 40,000 variables, we have the computational evidence that MM-SR1gen is more efficient than memory-less SR1 and than memory-less BFGS. The conclusion of this study is that the memory-less SR1 method with generalized secant equation is a rapid and reliable method for solving large-scale minimizing problems. Besides, it is shown that the accuracy of the Hessian approximations along the iterations in quasi-Newton methods is not as crucial in these methods as it is believed. PubDate: 2022-03-11
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Abstract: Abstract Complex valued systems with an indefinite matrix term arise in important applications such as for certain time-harmonic partial differential equations such as the Maxwell’s equation and for the Helmholtz equation. Complex systems with symmetric positive definite matrices can be solved readily by rewriting the complex matrix system in two-by-two block matrix form with real matrices which can be efficiently solved by iteration using the preconditioned square block (PRESB) preconditioning method and preferably accelerated by the Chebyshev method. The appearances of an indefinite matrix term causes however some difficulties. To handle this we propose different forms of matrix splitting methods, with or without any parameters involved. A matrix spectral analyses is presented followed by extensive numerical comparisons of various forms of the methods. PubDate: 2022-03-10
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Abstract: Abstract In a wide range of applications, second-order ordinary differential equation (ODE) appears frequently. If the second-order ODE is stiff, the implicit Runge–Kutta–Nyström (RKN) method is often used to obtain numerical solutions. In addition, there are often some inherent properties in these problems, such as symmetry, symplecticness and exponentially fitting. Considering these properties, two three-stage implicit modified RKN integrators are obtained in this paper. The new six-order integrators called ISSEFMRKN integrate exactly differential systems whose solutions can be expressed as linear combinations of functions from the set \(\{\exp (\lambda t), \exp (-\lambda t)\mid \lambda \in \mathbb {R}~\text {or}~\lambda \in \mathbb {C}\}\) . Furthermore, their final stages are also exact for \(y=t^2\) or \(y\in \{\exp (2\lambda t), \exp (-2\lambda t)\}\) . The numerical results show that the new methods are more accurate than some highly accurate codes in the literature. PubDate: 2022-03-02 DOI: 10.1007/s10092-022-00456-7
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Abstract: Abstract Uncertainty quantification plays an important role in problems that involve inferring a parameter of an initial value problem from observations of the solution. Conrad et al. (Stat Comput 27(4):1065–1082, 2017) proposed randomisation of deterministic time integration methods as a strategy for quantifying uncertainty due to the unknown time discretisation error. We consider this strategy for systems that are described by deterministic, possibly time-dependent operator differential equations defined on a Banach space or a Gelfand triple. Our main results are strong error bounds on the random trajectories measured in Orlicz norms, proven under a weaker assumption on the local truncation error of the underlying deterministic time integration method. Our analysis establishes the theoretical validity of randomised time integration for differential equations in infinite-dimensional settings. PubDate: 2022-02-25 DOI: 10.1007/s10092-022-00457-6
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Abstract: Abstract In this paper, we develop efficient and accurate algorithms for evaluating both \(\varphi _l(A)\) and \(\varphi _l(A)b,\) where \(\varphi _l(\cdot )\) is the function related to the exponential defined by \(\varphi _l(z)\equiv \sum \nolimits ^{\infty }_{k=0}\frac{z^k}{(l+k)!}\) , A is an \(N\times N\) matrix and b is an N dimensional vector. Such matrix functions play a key role in a class of numerical methods well-known as exponential integrators. The algorithms use the modified scaling and squaring procedure combined with truncated Taylor series. A quasi-backward error analysis is presented to find the optimal value of the scaling and the degree of the Taylor approximation. Some useful techniques are employed for reducing the computational cost. Numerical comparisons with state-of-the-art algorithms show that the algorithms perform well in both accuracy and efficiency. PubDate: 2022-01-21 DOI: 10.1007/s10092-021-00453-2
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Abstract: Abstract Classical rational interpolation usually enjoys better approximation properties than polynomial interpolation because it avoids wild oscillations and exhibits exponential convergence for approximating analytic functions. We develop a rational interpolation operator, which not only preserves the advantage of classical rational interpolation, but also has a finite Lebesgue constant. In particular, it is convergent for approximating any continuous function, and the convergence rate of the interpolants approximating a function is obtained using the modulus of continuity. PubDate: 2022-01-17 DOI: 10.1007/s10092-021-00454-1
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Abstract: Abstract This paper is concerned with numerical solution of the two-dimensional Klein-Gordon-Dirac equations by a fourth-order compact finite difference method in space and an energy-preserving Crank-Nicolson-type discretization in time. For convenience of illustrating the conservative properties and investigating the convergence results, we convert the component-wise form of the proposed scheme into an equivalent matrix-vector form. By using the mathematical induction argument and standard energy method, we establish the optimal error estimates under condition \(\tau \le \frac{1}{ \mathrm {ln}(h) }\) with time step \(\tau\) and mesh size h. Compared with the condition \(\tau =O(h^{\frac{1}{2}})\) required by existing results in literature for two-dimensional case, this greatly relaxes the dependence of the time step on the grid size. The convergence order of the scalar \(\phi\) and the 2-spinor \(\psi\) is of \(O(\tau ^{2}+h^{4})\) in the maximum norm and the discrete \(H^{1}\) -norm, respectively. In addition, by using the orthogonal diagonalization technique, a fast solver is designed to solve the proposed scheme. Several numerical results are reported to verify the error estimates and the discrete conservation laws. PubDate: 2021-12-21 DOI: 10.1007/s10092-021-00452-3
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Abstract: Abstract Given a square matrix A and a polynomial p, the Crouzeix ratio is the norm of the polynomial on the field of values of A divided by the 2-norm of the matrix p(A). Crouzeix’s conjecture states that the globally minimal value of the Crouzeix ratio is 0.5, regardless of the matrix order and polynomial degree, and it is known that 1 is a frequently occurring locally minimal value. Making use of a heavy-tailed distribution to initialize our optimization computations, we demonstrate for the first time that the Crouzeix ratio has many other locally minimal values between 0.5 and 1. Besides showing that the same function values are repeatedly obtained for many different starting points, we also verify that an approximate nonsmooth stationarity condition holds at computed candidate local minimizers. We also find that the same locally minimal values are often obtained both when optimizing over real matrices and polynomials, and over complex matrices and polynomials. We argue that minimization of the Crouzeix ratio makes a very interesting nonsmooth optimization case study, illustrating among other things how effective the BFGS method is for nonsmooth, nonconvex optimization. Our method for verifying approximate nonsmooth stationarity is based on what may be a novel approach to finding approximate subgradients of max functions on an interval. Our extensive computations strongly support Crouzeix’s conjecture: in all cases, we find that the smallest locally minimal value is 0.5. PubDate: 2021-12-18 DOI: 10.1007/s10092-021-00448-z
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Abstract: Abstract We consider the numerical computation of \(I[f]=\int \!\!\!\!\!=^b_a f(x)\,dx\) , the Hadamard finite part of the finite-range singular integral \(\int ^b_a f(x)\,dx\) , \(f(x)=g(x)/(x-t)^{m}\) with \(a<t<b\) and \(m\in \{1,2,\ldots \},\) assuming that (i) \(g\in C^\infty (a,b)\) and (ii) g(x) is allowed to have arbitrary integrable singularities at the endpoints \(x=a\) and \(x=b\) . We first prove that \(\int \!\!\!\!\!=^b_a f(x)\,dx\) is invariant under any legitimate variable transformation \(x=\psi (\xi )\) , \(\psi :[\alpha ,\beta ]\rightarrow [a,b]\) , hence there holds \(\int \!\!\!\!\!=^\beta _\alpha F(\xi )\,d\xi =\int \!\!\!\!\!=^b_a f(x)\,dx\) , where \(F(\xi )=f(\psi (\xi ))\,\psi '(\xi )\) . Based on this result, we next choose \(\psi (\xi )\) such that \(\mathcal{{F}}(\xi )\) , the \({{\mathcal {T}}}\) -periodic extension of \(F(\xi )\) , \({{\mathcal {T}}}=\beta -\alpha \) , is sufficiently smooth, and prove, with the help of some recent extension/generalization of the Euler–Maclaurin expansion, that we can apply to \(\int \!\!\!\!\!=^\beta _\alpha F(\xi )\,d\xi \) the quadrature formulas derived for periodic singular integrals developed in an earlier work of the author: [A. Sidi, “Unified compact numerical quadrature formulas for Hadamard finite parts of singular integrals of periodic functions.” Calcolo, 58, 2021. Article number 22]. We give a whole family of numerical quadrature formulas for \(\int \!\!\!\!\!=^\beta _\alpha F(\xi )\,d\xi \) for each m, which we denote \({\widehat{T}}^{(s)}_{m,n}[\mathcal{{F}}]\) . Letting \(G(\xi )=(\xi -\tau )^m F(\xi )\) , with \(\tau \in (\alpha ,\beta )\) determined from \(t=\psi (\tau )\) , and letting \(h={\mathcal {T}}/n\) , for \(m=3\) , for example, we have the three formulas $$\begin{aligned} {\widehat{T}}^{(0)}_{3,n}[\mathcal{{F}}]&=h\sum ^{n-1}_{j=1}\mathcal{{F}}(\tau +jh)-\frac{\pi ^2}{3}\,G'(\tau )\,h^{-1} +\frac{1}{6}\,... PubDate: 2021-12-18 DOI: 10.1007/s10092-021-00446-1
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Abstract: Abstract We propose and analyze a new mixed finite element method for the coupling of the Stokes equations with a transport problem modelled by a scalar nonlinear convection–diffusion problem. Our approach is based on the introduction of the Cauchy fluid stress and two vector unknowns involving the gradient and the total flux of the concentration. The introduction of these further unknowns lead to a mixed formulation in a Banach space framework in both Stokes and transport equations, where the aforementioned stress tensor and vector unknowns, together with the velocity and the concentration, are the main unknowns of the system. In this way, and differently from the techniques previously developed for this and related coupled problems, no augmentation procedure needs to be incorporated now into the formulation nor into the solvability analysis. The resulting non-augmented scheme is then written equivalently as a fixed-point equation, so that the well-known Banach theorem, combined with Babuška–Brezzi’s theory in Banach spaces, classical results on nonlinear monotone operators and certain regularity assumptions, are applied to prove the unique solvability of the continuous system. As for the associated Galerkin scheme, whose solvability is established similarly to the continuous case by using the Brouwer fixed-point theorem, we employ Raviart–Thomas approximations of order \(k \ge 0\) for the stress and total flux, and discontinuous piecewise polynomials of degree k for the velocity, concentration, and concentration gradient. With this choice of spaces, momentum is conserved in both Stokes and transport equations if the external forces belong to the piecewise constants and concentration discrete space, respectively, which constitutes one of the main features of our approach. Finally, we derive optimal a priori error estimates and provide several numerical results illustrating the good performance of the scheme and confirming the theoretical rates of convergence. PubDate: 2021-12-11 DOI: 10.1007/s10092-021-00451-4
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Abstract: Abstract This paper discusses finite elements defined by Ciarlet’s triple on grids that consist of general quadrilaterals not limited in parallelograms. Specifically, two finite elements are established for the \(H^1\) and \(H(\mathrm rot)\) elliptic problems, respectively. An \({\mathcal {O}}(h)\) order convergence rate in energy norm for both of them and an \({\mathcal {O}}(h^2)\) order convergence in \(L^2\) norm for the \(H^1\) scheme are proved under the \({\mathcal {O}}(h^2)\) asymptotic-parallelogram assumption on the grids. Further, these two finite element spaces, together with the space of piecewise constant functions, formulate a discretized de Rham complex on general quadrilateral grids. The finite element spaces consist of piecewise polynomial functions, and, thus, are nonconforming on general quadrilateral grids. Indeed, a rigorous analysis is given in this paper that it is impossible to construct a practically useful finite element by Ciarlet’s triple that can formulate a finite element space which consists of continuous piecewise polynomial functions on a grid that may include arbitrary quadrilaterals. PubDate: 2021-12-07 DOI: 10.1007/s10092-021-00447-0
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Abstract: Abstract We explore the convergence rate of the Kačanov iteration scheme for different models of shear-thinning fluids, including Carreau and power-law type explicit quasi-Newtonian constitutive laws. It is shown that the energy difference contracts along the sequence generated by the iteration. In addition, an a posteriori computable contraction factor is proposed, which improves, on finite-dimensional Galerkin spaces, previously derived bounds on the contraction factor in the context of the power-law model. Significantly, this factor is shown to be independent of the choice of the cut-off parameters whose use was proposed in the literature for the Kačanov iteration applied to the power-law model. Our analytical findings are confirmed by a series of numerical experiments. PubDate: 2021-11-28 DOI: 10.1007/s10092-021-00444-3
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Abstract: Abstract The goal of this paper is to solve the tensor least-squares (TLS) problem associated with multilinear system \(\mathcal {A}{} \mathbf{x} ^{m-1}=\mathbf{b}\) , where \(\mathcal {A}\) is an mth-order n-dimensional tensor and \(\mathbf{b}\) is a vector in \(\mathbb {R}^{n}\) , which has practical applications in numerical PDEs, data mining, tensor complementary problems, higher order statistics and so on. We transform the TLS problem into a multi-block optimization problem with consensus constraints, and propose an alternating linearized method with proximal regularization for it. Under some mild assumptions, it is shown that every limit point of the sequence generated by this method is a stationary point. Moreover, when the tensor \(\mathcal {A}\) can be constructed in the tensor-train format explicitly, the total number of operations with respect to the method mentioned above decreases from the order of \(\mathcal {O}(n^{m-1})\) to \(\mathcal {O}((m-1)^2nr^2)+\mathcal {O}(mnr^3)\) , alleviating the curse-of-dimensionality. As an application, the inverse iteration methods, derived from the proposed methods, for solving the tensor eigenvalue problems are presented. Some numerical examples are provided to illustrate the feasibility of our algorithms. PubDate: 2021-11-26 DOI: 10.1007/s10092-021-00450-5
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