Given a linear self-adjoint differential operator \(\mathscr {L}\) along with a discretization scheme (like Finite Differences, Finite Elements, Galerkin Isogeometric Analysis, etc.), in many numerical applications it is crucial to understand how good the (relative) approximation of the whole spectrum of the discretized operator \(\mathscr {L}\,^{(n)}\) is, compared to the spectrum of the continuous

This paper deals with asymptotic stability of differential-algebraic equations with multiple delays and numerical solutions generated by Runge–Kutta methods combined with Lagrange interpolation. We study the solvability and asymptotic stability of delay differential-algebraic equations and present some sufficient conditions for the zero solution to be asymptotically stable. A sufficient and necessary

In a recent work, we developed three new compact numerical quadrature formulas for finite-range periodic supersingular integrals \(I[f]={\mathop\int{\!\!\!\!\!\!=}}^{\,\,b}_{\!\!a} f(x)\,dx\), where \(f(x)=g(x)/(x-t)^3,\) assuming that \(g\in C^\infty [a,b]\) and f(x) is T-periodic, \(T=b-a\). With \(h=T/n\), these numerical quadrature formulas read $$\begin{aligned} {\widehat{T}}{}^{(0)}_n[f]&=h\sum

An energy conservative discontinuous Galerkin scheme for a generalised third order KdV type equation is designed. Based on the conservation principle, we propose techniques that allow for the derivation of optimal a priori bounds for the linear KdV equation and a posteriori bounds for the linear and modified KdV equation. Extensive numerical experiments showcasing the good long time behaviour of the

Matrix decompositions play a prominent role in the theoretical study and numerical calculation of split quaternion mechanics. This paper, by means of a complex representation of split quaternion matrices, introduces Gaussian elimination of split quaternion matrices, and obtains a complex structure-preserving algorithm for split quaternion matrix LDU decomposition. Numerical examples show that the complex

The displacement and gradient of a finite element solution are of primary interest and a high accuracy displacement and gradient approximation are always desirable in scientific computing. In this paper, we introduce a new state of the art way to reconstruct a high accuracy approximated gradient. The proposed method inherits the advantages of both the extrapolation technique and the interpolation post-processing

The present paper is concerned with a class of penalized Signorini problems also called normal compliance models. These nonlinear models approximate the Signorini problem and are characterized both by a penalty parameter \(\varepsilon \) and by a “power parameter” \(\alpha \ge 1\), where \(\alpha = 1\) corresponds to the standard penalization. We choose a continuous conforming linear finite element

We consider the approximation of the inverse of the finite element stiffness matrix in the data sparse \({\mathcal{H}}\)-matrix format. For a large class of shape regular but possibly non-uniform meshes including algebraically graded meshes, we prove that the inverse of the stiffness matrix can be approximated in the \({\mathcal{H}}\)-matrix format at an exponential rate in the block rank. Since the

We develop a geometrically intrinsic formulation of the arbitrary-order Virtual Element Method (VEM) on polygonal cells for the numerical solution of elliptic surface partial differential equations (PDEs). The PDE is first written in covariant form using an appropriate local reference system. The knowledge of the local parametrization allows us to consider the two-dimensional VEM scheme, without any

In this article we develop a second-order high resolution finite difference method for a general size-structured population model coupled with environmental dynamics where the individual growth rate may change sign. The model consists of a first order hyperbolic partial differential equation describing the population density of individuals coupled with a system of ordinary differential equations describing

A new finite element method is presented for a general class of singularly perturbed reaction-diffusion problems \(-\varepsilon ^2\varDelta u +bu=f\) posed on bounded domains \(\varOmega \subset \mathbb {R}^k\) for \(k\ge 1\), with the Dirichlet boundary condition \(u=0\) on \(\partial \varOmega\), where \(0 <\varepsilon \ll 1\). The method is shown to be quasioptimal (on arbitrary meshes and for arbitrary

A correction to this paper has been published: https://doi.org/10.1007/s10092-021-00422-9

For the discrete linear system resulting from certain steady-state space-fractional diffusion equations, we construct a lopsided scaled HSS (LSHSS) iteration method and establish its convergence theory. From the LSHSS, we obtain the corresponding matrix splitting preconditioner. By further replacing the involved Toeplitz matrix with certain circulant matrix, we construct a fast LSHSS (FLSHSS) preconditioner

We consider the Lidstone–Euler interpolation problem and the associated Lidstone–Euler boundary value problem, in both theoretical and computational aspects. After a theorem of existence and uniqueness of the solution to the Lidstone–Euler boundary value problem, we present a numerical method for solving it. This method uses the extrapolated Bernstein polynomials and produces an approximating convergent

The paper proves the global convergence of a general block Jacobi method for the generalized eigenvalue problem \(\mathbf {A}x=\lambda \mathbf {B}x\) with symmetric matrices \(\mathbf {A}\), \(\mathbf {B}\) such that \(\mathbf {B}\) is positive definite. The proof is made for a large class of generalized serial strategies that includes important serial and parallel strategies. The sequence of matrix

Statistical solutions have recently been introduced as an alternative solution framework for hyperbolic systems of conservation laws. In this work, we derive a novel a posteriori error estimate in the Wasserstein distance between dissipative statistical solutions and numerical approximations obtained from the Runge-Kutta Discontinuous Galerkin method in one spatial dimension, which rely on so-called

In this paper, we investigate the oscillation properties of solutions of a class of highly oscillatory Volterra integral equations and develop a Hermite collocation method to approximate the solution of these equations. We begin our analysis by obtaining an asymptotic expansion for the solution of these equations using their resolvent representation. We then introduce a new Gaussian radial basis function

The aim of the present work is to derive error estimates for the Laplace eigenvalue problem in mixed form, implementing a virtual element method. With the aid of the theory for non-compact operators, we prove that the proposed method is spurious free and convergent. Optimal order of convergence for the eigenvalues and eigenfunctions are derived. Finally, we report numerical tests to confirm the theoretical

We derive novel error estimates for Hybrid High-Order (HHO) discretizations of Leray–Lions problems set in \(W^{1,p}\) with \(p\in (1,2]\). Specifically, we prove that, depending on the degeneracy of the problem, the convergence rate may vary between \((k+1)(p-1)\) and \((k+1)\), with k denoting the degree of the HHO approximation. These regime-dependent error estimates are illustrated by a complete

A finite element cochain complex on Cartesian meshes of any dimension based on the \(H^1\)-inner product is introduced. It yields \(H^1\)-conforming finite element spaces with exterior derivatives in \(H^1\). We use a tensor product construction to obtain \(L^2\)-stable projectors into these spaces which commute with the exterior derivative. The finite element complex is generalized to a family of

We use the scalar auxiliary variable (SAV) reformulation of the nonlinear Schrödinger (NLS) equation to construct structure-preserving SAV–Gauss methods for the NLS equation, namely \(L^2\)-conservative methods satisfying a discrete analogue of the energy (the Hamiltonian) conservation of the equation. This is in contrast to Gauss methods for the standard form of the NLS equation that are \(L^2\)-conservative

In this study, we discuss the superconvergence of the space-time discontinuous Galerkin method for the first-order linear nonhomogeneous hyperbolic equation. By using the local differential projection method to construct comparison function, we prove that the numerical solution is \((2n+1)\)-th order superconvergent at the downwind-biased Radau points in the discrete \(L^2\)-norm. As a by-product,

The velocity solution of the incompressible Stokes equations is not affected by changes of the right hand side data in form of gradient fields. Most mixed methods do not replicate this property in the discrete formulation due to a relaxation of the divergence constraint which means that they are not pressure-robust. A recent reconstruction approach for classical methods recovers this invariance property

The randomized Kaczmarz algorithm has received considerable attention recently because of its simplicity, speed, and the ability to approximately solve large-scale linear systems of equations. In this paper we propose randomized double and triple Kaczmarz algorithms to solve extended normal equations of the form \(\mathbf{A}^\top \mathbf{Ax}=\mathbf{A}^\top \mathbf{b}-\mathbf{c}\). The proposed algorithms

The block Gauss–Seidel algorithm can significantly outperform the simple randomized Gauss–Seidel algorithm for solving overdetermined least-squares problems since it moves a large block of columns rather than a single column into working memory. Here, with the help of the maximum residual rule, we construct a two-step Gauss–Seidel (2SGS) algorithm, which selects two different columns simultaneously

The spectral collocation method is investigated for the system of nonlinear Riemann–Liouville fractional differential equations (FDEs). The main idea of the presented method is to solve the corresponding system of nonlinear weakly singular Volterra integral equations obtained from the system of FDEs. In order to carry out convergence analysis for the presented method, we investigate the regularity

We introduce and analyze a new mixed finite element method with reduced symmetry for the standard linear model in viscoelasticity. Following a previous approach employed for linear elastodynamics, the present problem is formulated as a second-order hyperbolic partial differential equation in which, after using the motion equation to eliminate the displacement unknown, the stress tensor remains as the

The Peaceman–Rachford splitting method (PRSM) is a preferred method for solving the two-block separable convex minimization problems with linear constraints at present. In this paper, we propose an inertial generalized proximal PRSM (abbreviated as IGPRSM) to improve computing efficiency, which unify the ideas of inertial proximal point and linearization technique. Both subproblems are linearized by

This paper is concerned with C0 (non-Lagrange) finite element approximations of the linear elliptic equations in non-divergence form and the Hamilton–Jacobi–Bellman (HJB) equations with Cordes coefficients. Motivated by the Miranda–Talenti estimate, a discrete analog is proved once the finite element space is C0 on the \((n-1)\)-dimensional subsimplex (face) and \(C^1\) on \((n-2)\)-dimensional subsimplex

The fractal interpolation functions with appropriate iterated function systems provide a method to perturb and approximate a continuous function on a compact interval I. This method produces a class of functions \(f^{\varvec{\alpha }}\in {\mathcal {C}}(I)\), where \(\varvec{\alpha }\) is a vector with functional components. The presence of scaling function in these fractal functions helps to get a

The Mittag-Leffler function is computed via a quadrature approximation of a contour integral representation. We compare results for parabolic and hyperbolic contours, and give special attention to evaluation on the real line. The main point of difference with respect to similar approaches from the literature is the way that poles in the integrand are handled. Rational approximation of the Mittag-Leffler

In this paper, based on the continuous collocation polynomial approximations, we derive and analyse a class of trigonometric collocation integrators for solving the highly oscillatory hyperbolic system. The symmetry, convergence and energy conservation of the continuous collocation polynomial approximations are rigorously analysed in details. Moreover, we also proved that the continuous collocation

We are interested in finding a solution to the tensor complementarity problem with a strong M-tensor, which we call the M-tensor complementarity problem. We propose a lower dimensional linear equation approach to solve that problem. At each iteration, only a lower dimensional system of linear equation needs to be solved. The coefficient matrices of the lower dimensional linear systems are independent

The need to estimate upper and lower bounds for matrix functions of the form \({\mathrm{trace}}(W^Tf(A)V)\), where the matrix \(A\in {{\mathbb {R}}}^{n\times n}\) is large and sparse, \(V,W\in {{\mathbb {R}}}^{n\times s}\) are block vectors with \(1\le s\ll n\) columns, and f is a function arises in many applications, including network analysis and machine learning. This paper describes the shifted

In this paper, we introduce a \(C^{0}\) virtual element method for the Helmholtz transmission eigenvalue problem, which is a fourth-order non-selfadjoint eigenvalue problem. We consider the mixed formulation of the eigenvalue problem discretized by the lowest-order virtual elements. This discrete scheme is based on a conforming \(H^{1}(\varOmega )\times H^{1}(\varOmega )\) discrete formulation, which

For backward differentiation formulae (BDF) applied to gradient flows of semiconvex functions, quadratic stability implies the existence of a Lyapunov functional. We compute the maximum time step which can be derived from quadratic stability for the 3-step BDF method (BDF3). Applications to the asymptotic behaviour of sequences generated by the BDF3 scheme are given.

In this paper we provide algorithms for computing the bidiagonal decomposition of the Wronskian matrices of the monomial basis of polynomials and of the basis of exponential polynomials. It is also shown that these algorithms can be used to perform accurately some algebraic computations with these Wronskian matrices, such as the calculation of their inverses, their eigenvalues or their singular values

The \(H^m\)-nonconforming virtual elements of any order k on any shape of polytope in \({\mathbb {R}}^n\) with constraints \(m> n\) and \(k\ge m\) are constructed in a universal way. A generalized Green’s identity for \(H^m\) inner product with \(m>n\) is derived, which is essential to devise the \(H^m\)-nonconforming virtual elements. By means of the local \(H^m\) projection and a stabilization term

We discuss the solution of eigenvalue problems associated with partial differential equations that can be written in the generalized form \({\mathsf {A}}x=\lambda {\mathsf {B}}x\), where the matrices \({\mathsf {A}}\) and/or \({\mathsf {B}}\) may depend on a scalar parameter. Parameter dependent matrices occur frequently when stabilized formulations are used for the numerical approximation of partial

To locate the eigenvalues of a given tensor, we present two classes of new Ky Fan type eigenvalue inclusion sets for tensors, which are tighter than those in Yang et al. (SIAM J Matrix Anal Appl 31:2517–2530, 2010) and He et al. (J Inequal Appl 114:1-9, 2014), respectively. Under certain conditions, the theoretical comparisons of the new proposed Ky Fan type eigenvalue inclusion sets for tensors are

In this paper we consider the Virtual Element discretization of a minimal surface problem, a quasi-linear elliptic partial differential equation modeling the problem of minimizing the area of a surface subject to a prescribed boundary condition. We derive an optimal error estimate and present several numerical tests assessing the validity of the theoretical results.

The proximal term plays a significant role in the literature of proximal Alternating Direction Method of Multipliers (ADMM), since (positive-definite or indefinite) proximal terms can promote convergence of ADMM and further simplify the involved subproblems. However, an overlarge proximal parameter decelerates the convergence numerically, though the convergence can be established with it. In this paper

In this paper, we study discrete projection methods for solving the Hammerstein integral equations on the half-line with a smooth kernel using piecewise polynomial basis functions. We show that discrete Galerkin/discrete collocation methods converge to the exact solution with order \({\mathcal {O}}(n^{-min\{r, d\}}),\) whereas iterated discrete Galerkin/iterated discrete collocation methods converge

In this work we present a new mixed finite element method for a class of steady-state natural convection models describing the behavior of non-isothermal incompressible fluids subject to a heat source. Our approach is based on the introduction of a modified pseudostress tensor depending on the pressure, and the diffusive and convective terms of the Navier–Stokes equations for the fluid and a vector

The Quasi-Newton method is one of the most effective methods using the first derivative for solving all unconstrained optimization problems. The Broyden family method plays an important role among the quasi-Newton algorithms. However, the study of the convergence of the classical Broyden family method is still not enough. While in the special case, BFGS method, there have been abundant achievements

Combining the quadratic equal-order stabilized method with the approach of local and parallel finite element computations and classical iterative methods for the discretization of the steady-state Navier–Stokes equations, three parallel iterative stabilized finite element methods based on fully overlapping domain decomposition are proposed and compared in this paper. In these methods, each processor

In this work, ideas previously introduced for a discontinuous Galerkin recovery method in one dimension, that involves a penalty stabilization term, are extended to an elliptic differential equation in several dimensions and different types of boundary conditions and meshes. Using standard arguments for other existing discontinuous Galerkin methods, we show results of existence and uniqueness of the

New H(div) conforming finite element methods for incompressible flows are designed that involve the rotation form of the equations of motion and the Bernoulli function. With a specific choice of numerical fluxes, we recover the same velocity field as in Guzmán et al. (IMA J Numer Anal 37(4):1733–1771, 2016) for the incompressible Euler equation in the convection form. Error estimates are presented

Image restoration problem is often solved by minimizing a cost function which consists of data-fidelity terms and regularization terms. Half-quadratic regularization has the advantage that it can preserve image details well in the recovered images. In this paper, we consider solving the image restoration model which involves multiplicative half-quadratic regularization term. Newton method is employed

In this paper, a small and necessary revision on an assumption condition of Aminifard and Babaie-Kafaki (Calcolo, 2019. https://doi.org/10.1007/s10092-019-0312-9) is made. By a little modification, a new conjugate gradient method is proposed, in which the search directions satisfy the sufficient descent condition with the strong Wolfe line search. The main difference between two algorithms is that

The study of the long time conservation for numerical methods poses interesting and challenging questions from the point of view of geometric integration. In this paper, we analyze the long time energy and magnetic moment conservations of two-step symmetric methods for charged-particle dynamics. A two-step symmetric method is proposed and its long time behaviour is shown not only in a normal magnetic

In this paper, we study the Poisson noise removal problem with total variation regularization term. Using the dual formulation of total variation and Lagrange dual, we formulate the problem as a new constrained minimax problem. Then, a modified Chambolle-Pock first-order primal-dual algorithm is developed to compute the saddle point of the minimax problem. The main idea of this paper is using different

In this work we review some proposals to define the fractional Laplace operator in two or more spatial variables and we provide their approximations using finite differences or the so-called Matrix Transfer Technique. We study the structure of the resulting large matrices from the spectral viewpoint. In particular, by considering the matrix-sequences involved, we analyze the extreme eigenvalues, we

In this paper, we propose an alternative version of the randomized Kaczmarz method, which chooses each row of the coefficient matrix A with probability proportional to the square of the Euclidean norm of the residual of each corresponding equation. We prove that it converges with expected linear rate and the convergence rate of this method is better than the Strohmer and Vershynin’s RK method. Numerical

In this paper a new modified Nyström method is proposed to solve linear integral equations of the second kind with fixed singularities of Mellin convolution type. It is based on the Gauss–Radau quadrature formula with a suitable Jacobi weight. The stability and convergence of the method is proved in weighted spaces with uniform norm. Moreover, an error estimate of the numerical solution is given under

A wide variety of different (fixed-point) iterative methods for the solution of nonlinear equations exists. In this work we will revisit a unified iteration scheme in Hilbert spaces from our previous work [16] that covers some prominent procedures (including the Zarantonello, Kačanov and Newton iteration methods). In combination with appropriate discretization methods so-called (adaptive) iterative

In this article, we study the convergence properties of the streamline-diffusion finite element method (SDFEM) for singularly perturbed 1D parabolic convection–diffusion initial-boundary-value problems. To discretize the spatial domain, we use a layer-adaptive nonuniform grids obtained through the equidistribution principle, whereas uniform grid is used in the time direction. Here, we use the backward-Euler

For any real numbers \(a,\ b\), and c, we form the sequence of polynomials \(\{P_n(z)\}_{n=0}^\infty\) satisfying the four-term recurrence$$\begin{aligned} P_n(z)+azP_{n-1}(z)+bP_{n-2}(z)+czP_{n-3}(z)=0,\ n\in {\mathbb {N}}, \end{aligned}$$with the initial conditions \(P_0(z)=1\) and \(P_{-n}(z)=0\). We find necessary and sufficient conditions on \(a,\ b\), and c under which the zeros of \(P_n(z)\)

In this article, a new \(\alpha\)-fractal rational cubic spline is introduced with the help of the iterated function system (IFS) that contains rational functions. The numerator of the rational function contains a cubic polynomial and the denominator of the rational function contains a quadratic polynomial with three shape parameters. The convergence analysis of the \(\alpha\)-fractal rational cubic