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Abstract: Abstract It has been known that the traditional scaling argument cannot be directly applied to the error analysis of immersed finite elements (IFE) because, in general, the spaces on the reference element associated with the IFE spaces on different interface elements via the standard affine mapping are not the same. By analyzing a mapping from the involved Sobolev space to the IFE space, this article is able to extend the scaling argument framework to the error estimation for the approximation capability of a class of IFE spaces in one spatial dimension. As demonstrations of the versatility of this unified error analysis framework, the manuscript applies the proposed scaling argument to obtain optimal IFE error estimates for a typical first-order linear hyperbolic interface problem, a second-order elliptic interface problem, and the fourth-order Euler-Bernoulli beam interface problem, respectively. PubDate: 2024-02-29

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Abstract: Abstract In this work we revisit the convergence analysis of the Subspace Iteration Method (SIM) for the computation of approximations of a matrix A by matrices of rank h. Typically, the analysis of convergence of these low-rank approximations has been obtained by first estimating the (angular) distance between the subspaces produced by the SIM and the dominant subspaces of A. It has been noticed that this approach leads to upper bounds that overestimate the approximation error in case the hth singular value of A lies in a cluster of singular values. To overcome this difficulty we introduce a substitute for dominant subspaces, which we call admissible subspaces. We develop a proximity analysis of subspaces produced by the SIM to admissible subspaces; in turn, this analysis allows us to obtain novel estimates for the approximation error by low-rank matrices obtained by the implementation of the deterministic SIM. Our results apply in the case when the hth singular value of A belongs to a cluster of singular values. Indeed, our approach allows us to consider the case when the hth and the \((h+1)\) st singular values of A coincide, which does not seem to be covered by previous works in the deterministic setting. PubDate: 2024-02-28

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Abstract: Abstract This paper presents a quadrature method for evaluating layer potentials in two dimensions close to periodic boundaries, discretized using the trapezoidal rule. It is an extension of the method of singularity swap quadrature, which recently was introduced for boundaries discretized using composite Gauss–Legendre quadrature. The original method builds on swapping the target singularity for its preimage in the complexified space of the curve parametrization, where the source panel is flat. This allows the integral to be efficiently evaluated using an interpolatory quadrature with a monomial basis. In this extension, we use the target preimage to swap the singularity to a point close to the unit circle. This allows us to evaluate the integral using an interpolatory quadrature with complex exponential basis functions. This is well-conditioned, and can be efficiently evaluated using the fast Fourier transform. The resulting method has exponential convergence, and can be used to accurately evaluate layer potentials close to the source geometry. We report experimental results on a simple test geometry, and provide a baseline Julia implementation that can be used for further experimentation. PubDate: 2024-02-27

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Abstract: Abstract We derive computable formulas for the structured backward errors of a complex number \(\lambda \) when considered as an approximate eigenvalue of rational matrix functions that carry a symmetry structure. We consider symmetric, skew-symmetric, Hermitian, skew-Hermitian, \(*\) -palindromic, T-even, T-odd, \(*\) -even, and \(*\) -odd structures. Numerical experiments show that the backward errors with respect to structure-preserving and arbitrary perturbations are significantly different. PubDate: 2024-02-17

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Abstract: Abstract A collocation method for the numerical solution of Volterra integro-differential equations with weakly singular kernels, based on piecewise polynomials of fractional order, is constructed and analysed. Typical exact solutions of this class of problems have a weak singularity at the initial time \(t=0\) . A rigorous error analysis of our method shows that, with an appropriate choice of the fractional-order polynomials and a suitably graded mesh, one can attain optimal orders of convergence to the exact solution and its derivative, and certain superconvergence results are also derived. In particular, our analysis shows that on a uniform mesh our method attains a higher order of convergence than standard piecewise polynomial collocation. Numerical examples are presented to demonstrate the sharpness of our theoretical results. PubDate: 2024-02-12

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Abstract: Abstract This paper is devoted to explore the convolution quadrature based on a class of two-point Hermite collocation methods. Incorporating derivatives into the numerical scheme enhances the accuracy while preserving stability, which is confirmed by the convergence analysis for the discretization of the initial value problem. Moreover, we employ the resulting quadrature to evaluate a class of highly oscillatory integrals. The frequency-explicit convergence analysis demonstrates that the proposed convolution quadrature surpasses existing convolution quadratures, achieving the highest convergence rate with respect to the oscillation among them. Numerical experiments involving convolution integrals with smooth, weakly singular, and highly oscillatory Bessel kernels illustrate the reliability and efficiency of the proposed convolution quadrature. PubDate: 2024-02-09

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Abstract: Abstract A posteriori error estimates are established for a two-step dual finite element method for singularly perturbed reaction–diffusion problems. The method can be considered as a modified least-squares finite element method. The least-squares functional is the basis for our residual-type a posteriori error estimators, which are shown to be reliable and efficient with respect to the error in an energy-type norm. Moreover, guaranteed upper bounds for the errors in the computed primary and dual variables are derived; these bounds are then used to drive an adaptive algorithm for our finite element method, yielding any desired accuracy. Our theory does not require the meshes generated to be shape-regular. Numerical experiments show the effectiveness of our a posteriori estimators. PubDate: 2024-02-05

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Abstract: Abstract For the well known parareal algorithm, we formulate and analyse a novel class of parareal exponential schemes with improved uniform accuracy for highly oscillatory system \(\ddot{q}+\frac{1}{\varepsilon ^2}M q =\frac{1}{\varepsilon ^{\mu }}f(q)\) with \(\mu =0\) or 1. The solution of this considered system propagates waves with wavelength at \(\mathcal {O} (\varepsilon )\) in time and the value of \(\mu \) corresponds to the strength of nonlinearity. This brings significantly numerical burden in scientific computation for highly oscillatory systems with \(0<\varepsilon \ll 1\) . The new proposed algorithm is formulated by using some reformulation approaches to the problem, Fourier pseudo-spectral methods, and parareal exponential integrators. The fast Fourier transform is incorporated in the implementation. We rigorously study the convergence, showing that for nonlinear systems, the algorithm has improved uniform accuracy \(\mathcal {O}\big ( \varepsilon ^{(2k+3)(1-\mu )}\Delta t^{2k+2}+\varepsilon ^{5(1-\mu )}\delta t^4\big )\) in the position and \(\mathcal {O}\big ( \varepsilon ^{(2k+3)(1-\mu )-1}\Delta t^{2k+2}+\varepsilon ^{4-5\mu }\delta t^4\big )\) in the momenta, where k is the number of parareal iterations, and \(\Delta t\) and \(\delta t\) are two time stepsizes used in the algorithm. The energy conservation is also discussed and the algorithm is shown to have an improved energy conservation. Numerical experiments are provided and the numerical results demonstrate the improved uniform accuracy and improved energy conservation of the obtained integrator through four Hamiltonian differential equations including nonlinear wave equations. PubDate: 2024-01-31

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Abstract: Abstract The exponential function that appears in the formula of the stability radius of linear time-delay differential systems is approximated by its Padé approximant. This reduces the computation of the level sets of singular values in the stability radius formula to the computation of imaginary eigenvalues of special matrix polynomials. Then a bisection method is used for computing lower and upper bounds on the stability radius. A rounding error analysis is presented. Several numerical examples are given to demonstrate the feasibility and efficiency of the bisection method. PubDate: 2024-01-30

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Abstract: Abstract We develop and study incremental algorithms for truncated higher-order singular value decompositions. By combining the SVD updating and different truncated higher-order singular value decompositions, two incremental algorithms are proposed. Not only the factor matrices but also the core tensor are updated in an incremental style. The costs of these algorithms are compared and the approximation errors are analyzed. Numerical results demonstrate that the proposed incremental algorithms have advantages in online computation. PubDate: 2024-01-08

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Abstract: Abstract Linear approximation methods suffer from Gibbs oscillations when approximating functions with jumps. Essentially non oscillatory subcell-resolution (ENO-SR) is a local technique avoiding oscillations and with a full order of accuracy, but a loss of regularity of the approximant appears. The goal of this paper is to introduce a new approach having both properties of full accuracy and regularity. In order to obtain it, we propose a three-stage algorithm: first, the data is smoothed by subtracting an appropriate non-smooth data sequence; then a chosen high order linear approximation operator is applied to the smoothed data and finally, an approximation with the proper jump or corner (jump in the first order derivative) discontinuity structure is reinstated by correcting the smooth approximation with the non-smooth element used in the first stage. This new procedure can be applied as subdivision scheme to design curves and surfaces both in point-value and in cell-average contexts. Using the proposed algorithm, we are able to construct approximations with high precision, with high piecewise regularity, and without smearing nor oscillations in the presence of discontinuities. These are desired properties in real applications as computer aided design or car design, among others. PubDate: 2024-01-04

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Abstract: Abstract In this article, a mixed dimensional elliptic partial differential equation is considered, posed in a bulk domain with a large number of embedded interfaces. In particular, well-posedness of the problem and regularity of the solution are studied. A fitted finite element approximation is also proposed and an a priori error bound is proved. For the solution of the arising linear system, an iterative method based on subspace decomposition is proposed and analyzed. Finally, numerical experiments are presented and rapid convergence using the proposed preconditioner is achieved, confirming the theoretical findings. PubDate: 2023-12-29

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Abstract: Abstract The randomized Kaczmarz algorithm is a simple iterative method for solving linear systems of equations. This study proposes a variant of the randomized Kaczmarz algorithm by combining block projection and weighted averaging techniques. Here, block projection quickly decreases iterative errors, and averaging reduces randomness and enables parallel computation simultaneously. Their combination can balance the convergence rate, convergence horizon, and computational complexity. In addition, three adaptive weights are designed to balance multiple block calculations and accelerate the proposed method. Exponential convergence is established for general linear systems (overdetermined or underdetermined, full-rank or deficient-rank, and consistent or inconsistent). Numerical simulations explain and verify the results. PubDate: 2023-12-12

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Abstract: Abstract The well-known Ait-Sahalia-type interest model, arising in mathematical finance, has some typical features: polynomial drift that blows up at the origin, highly nonlinear diffusion, and positive solution. The known explicit numerical methods including truncated/tamed Euler–Maruyama (EM) applied to it do not preserve its positivity. The main interest of this work is to investigate the numerical conservation of positivity of the solution of generalised Ait-Sahalia-type model. By modifying the truncated EM method to generate positive sequences of numerical approximations, we obtain the rate of convergence of the numerical algorithm not only at time T but also over the time interval [0, T]. Numerical experiments confirm the theoretical results. PubDate: 2023-11-27 DOI: 10.1007/s10543-023-01000-x

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Abstract: Abstract We analyze the preservation properties of a family of reversible splitting methods when they are applied to the numerical time integration of linear differential equations defined in the unitary group. The schemes involve complex coefficients and are conjugated to unitary transformations for sufficiently small values of the time step-size. New and efficient methods up to order six are constructed and tested on the linear Schrödinger equation. PubDate: 2023-11-10 DOI: 10.1007/s10543-023-00998-4

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Abstract: Abstract An hp-version interior penalty discontinuous Galerkin method under nonconforming meshes is proposed to solve the quad-curl eigenvalue problem. We prove well-posedness of the numerical scheme for the quad-curl equation and then derive an error estimate in a mesh-dependent norm, which is optimal with respect to h but has different p-version error bounds under conforming and nonconforming tetrahedron meshes. The hp-version discrete compactness of the DG space is established for the convergence proof. The performance of the method is demonstrated by numerical experiments using conforming/nonconforming meshes and h-version/p-version refinement. The optimal h-version convergence rate and the exponential p-version convergence rate are observed. PubDate: 2023-11-10 DOI: 10.1007/s10543-023-00996-6

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Abstract: Abstract For several classes of mathematical models that yield linear systems, the splitting of the matrix into its Hermitian and skew Hermitian parts is naturally related to properties of the underlying model. This is particularly so for discretizations of dissipative Hamiltonian ODEs, DAEs and port-Hamiltonian systems where, in addition, the Hermitian part is positive definite or semi-definite. It is then possible to develop short recurrence optimal Krylov subspace methods in which the Hermitian part is used as a preconditioner. In this paper, we develop new, right preconditioned variants of this approach which, as their crucial new feature, allow the systems with the Hermitian part to be solved only approximately in each iteration while keeping the short recurrences. This new class of methods is particularly efficient as it allows, for example, to use few steps of a multigrid solver or a (preconditioned) CG method for the Hermitian part in each iteration. We illustrate this with several numerical experiments for large scale systems. PubDate: 2023-11-10 DOI: 10.1007/s10543-023-00999-3

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Abstract: Abstract In this work we develop the Gaussian quadrature rule for weight functions involving powers, exponentials and Bessel functions of the first kind. Besides the computation based on the use of the standard and the modified Chebyshev algorithm, here we present a very stable algorithm based on the preconditioning of the moment matrix. Numerical experiments are provided and a geophysical application is considered. PubDate: 2023-11-01 DOI: 10.1007/s10543-023-00997-5

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Abstract: Abstract In this paper, we propose and analyze a new structure-preserving finite element method for the stationary magnetohydrodynamic equations with magnetic-current formulation on Lipschitz domains. Using a mixed finite element approach, we discretize the hydrodynamic unknowns by inf-sup stable velocity-pressure finite element pairs, and the current density, the induced electric field and the magnetic field by using the edge-edge-face elements from a discrete de-Rham complex pair. To deal with the divergence-free condition of the magnetic field, we introduce an augmented term to the discrete scheme rather than a Lagrange multiplier in the existing schemes. Thanks to discrete differential forms and finite element exterior calculus, the proposed scheme preserves the divergence-free property exactly for the magnetic induction on the discrete level. The well-posedness of the discrete problem is further proved under the small data condition. Under weak regularity assumptions, we rigorously establish the error estimates of the finite element schemes. Numerical results are provided to illustrate the theoretical results and demonstrate the efficiency of the proposed method. PubDate: 2023-11-01 DOI: 10.1007/s10543-023-00995-7

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Abstract: Abstract We consider the application of implicit Runge–Kutta (IRK) methods to systems of implicit ordinary differential equations (ODEs). We are especially interested in the situation when stiffness arises. We show that the eigenvalues of major families of A-stable implicit Runge–Kutta methods are simple and have non-negative real part. We give necessary and sufficient conditions for the invertibility of approximate Jacobians of IRK methods. The main result of this note provides some sufficient conditions to ensure local contractivity, hence convergence, of modified Newton iterations of IRK methods for stiff ODEs with step size conditions independent of stiffness. PubDate: 2023-11-01 DOI: 10.1007/s10543-023-00994-8