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This paper studies the problem of approximating a function f in a Banach space $$\mathcal{X}$$ from measurements $$l_j(f)$$ , $$j=1,\ldots ,m$$ , where the $$l_j$$ are linear functionals from $$\mathcal{X}^*$$ . Quantitative results for such recovery problems require additional information about the sought after function f. These additional assumptions take the form of assuming that f is in a certain model class $$K\subset \mathcal{X}$$ . Since there are generally infinitely many functions in K which share these same measurements, the best approximation is the center of the smallest ball B, called the Chebyshev ball, which contains the set $$\bar{K}$$ of all f in K with these measurements. Therefore, the problem is reduced to analytically or numerically approximating this Chebyshev ball. Most results study this problem for classical Banach spaces $$\mathcal{X}$$ such as the $$L_p$$ spaces, $$1\le p\le \infty $$ , and for K the unit ball of a smoothness space in $$\mathcal{X}$$ . Our interest in this paper is in the model classes $$K=\mathcal{K}(\varepsilon ,V)$$ , with $$\varepsilon >0$$ and V a finite dimensional subspace of $$\mathcal{X}$$ , which consists of all $$f\in \mathcal{X}$$ such that $$\mathrm{dist}(f,V)_\mathcal{X}\le \varepsilon $$ . These model classes, called approximation sets, arise naturally in application domains such as parametric partial differential equations, uncertainty quantification, and signal processing. A general theory for the recovery of approximation sets in a Banach space is given. This theory includes tight a priori bounds on optimal performance and algorithms for finding near optimal approximations. It builds on the initial analysis given in Maday et al. (Int J Numer Method Eng 102:933-965, 2015) for the case when $$\mathcal{X}$$ is a Hilbert space, and further studied in Binev et al. (SIAM UQ, 2015). It is shown how the recovery problem for approximation sets is connected with well-studied concepts in Banach space theory such as liftings and the angle between spaces. Examples are given that show how this theory can be used to recover several recent results on sampling and data assimilation. [ABSTRACT FROM AUTHOR]

The main purpose of this paper is to give the numerical integration scheme for a fast Petrov-Galerkin method for solving the generalized airfoil equation, considered in a recent paper (Cai, J. Complex. 25:420-436, ). This scheme leads to a fully discrete sparse linear system. We show that it requires a nearly linear computational cost to get this system, and the approximate solution of the resulting linear system preserves the optimal convergent order. Numerical experiments are presented to confirm the theoretical estimates. [ABSTRACT FROM AUTHOR]

The numerical method of lines is a technique for solving partial differential equations by discretising in all but one dimension. In this paper the solution of the approximate problem is extended outside the domain using the boundary condition. This leads to functional differential-algebraic equations. Sufficient conditions for the well-posedness, stability and convergence of the resulting method of lines are given. [ABSTRACT FROM AUTHOR]

The paper deals with the weakly penalized exponentially fitted incomplete interior penalty (EF-IIPG0) scheme for advection-diffusion problems. In the first part of the paper, the $$M$$ -matrix property on conforming weakly-acute meshes is discussed. In the second part, an a posteriori error estimate is derived. The estimator, especially designed for the advection dominated case, controls the energy norm as well as a semi-norm associated with the advective derivative, taking full advantage of the formulation on non-matching grids. The paper is supplemented by numerical experiments, where the estimator is used as local error indicator for marking the triangles to be refined in an adaptive strategy. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the superconvergence of a variational discretization approximation for parabolic optimal control problems with control constraints. The state and the adjoint state are approximated by piecewise linear functions and the control is not directly discretized. The time discretization is based on difference methods. We derive the superconvergence between the numerical solution and elliptic projection for the state and the adjoint state and present a numerical example for illustrating our theoretical results. [ABSTRACT FROM AUTHOR]

Eigenvalue problems with eigenparameter appearing in the boundary conditions usually have complicated characteristic determinant where zeros cannot be explicitly computed. Sampling theory is one of the most important mathematical tools used in communication engineering since it enables engineers to reconstruct signals from some of their sampled data. The sinc Gaussian sampling technique derived by Qian (Proc Am Math Soc 131:1169-1176, ) establishes a sampling technique which converges faster than the classical sampling technique. Schmeisser and Stenger (Sampl Theory Signal Image Process 6:199-221, ) studied the associated error analysis. In the present paper we apply a sinc Gaussian technique to compute approximate values of the eigenvalues of Sturm-Liouville problems with eigenvalue parameter in one or two boundary conditions. The error of this method decays exponentially in terms of the number of involved samples. Therefore the accuracy of the new technique is higher than the classical sinc method. Numerical worked examples with tables and illustrative figures are given at the end of the paper. [ABSTRACT FROM AUTHOR]

In this paper, a kinetic method to compute approximative solutions of the one-dimensional open channel flow equations with a varying cross-sectional area and a varying bottom profile is proposed. The scheme preserves the steady states at rest, keeps the water height non-negative and is able to handle dry channel beds. These three properties are essential indicators for the quality of a numerical scheme. The ability of the scheme to treat varying cross-sectional areas along the channel is an important advancement in comparison to previous papers. Further the same equations are considered on a network by coupling the equations at the nodes. The main difference to the majority of common numerical hydraulic models is the usage of the energy equality as coupling condition which seems to be more realistic than conditions based on the water height. Moreover, we set up a numerical method for computing subcritical flow on such networks. Several examples are treated to illustrate the results. [ABSTRACT FROM AUTHOR]

In this paper, we present an alternative approach to a priori $$L^\infty $$ -error estimates for the piecewise linear finite element approximation of the classical obstacle problem. Our approach is based on stability results for discretized obstacle problems and on error estimates for the finite element approximation of functions under pointwise inequality constraints. As an outcome, we obtain the same order of convergence proven in several works before. In contrast to prior results, our estimates can, for example, also be used to study the situation where the function space is discretized but the obstacle is not modified at all. [ABSTRACT FROM AUTHOR]

Discontinuous weighted least-squares (DWLS) approximation is modification of a weighted least-squares method that requires a local support (a reconstruction stencil) to approximate a function at a given point. A DWLS method is often employed in computational problems where a function is approximated on an irregular computational grid. It has recently been revealed that the method provides inaccurate approximation on irregular grids and conventional weighting of distant points captured by a reconstruction stencil on an irregular coarse mesh does not improve the accuracy of the approximation. Thus in our paper we further investigate the impact of distant points on the accuracy of DWLS approximation and design new weight coefficients for DWLS reconstruction that allow one to obtain more accurate reconstruction results. Our approach is based on a concept of numerically distant points originally developed in author's previous works, as a new weight function calculates the distance between two points in the data space. [ABSTRACT FROM AUTHOR]

In this paper we propose and analyze an exponentially fitted simplicial finite element method for the numerical approximation of solutions to diffusion-convection equations with tensor-valued diffusion coefficients. The finite element method is first formulated using exponentially fitted finite element basis functions constructed on simplicial elements in arbitrary dimensions. Stability of the method is then proved by showing that the corresponding bilinear form is coercive. Upper error bounds for the approximate solution and the associated flux are established. [ABSTRACT FROM AUTHOR]

This paper deals with a finite element numerical quadrature method. It is applied for a class of second-order self-adjoint elliptic operators defined on a bounded domain in the plane. Isoparametric finite element transformations and triangular Lagrange finite elements are used. We establish the rate of convergence for approximate eigenvalues and eigenfunctions of second-order elliptic eigenvalue problems, obtained by a numerical quadrature finite element approximation. Thus the relationship between possible quadrature formulas and the optimal and almost optimal precision of the method is established. The emphasis of the paper is on the error analysis of the approximate eigenpairs. Numerical results confirming the theory are presented. [ABSTRACT FROM AUTHOR]

Under very mild additional assumptions, translates of conditionally positive definite radial basis functions allow unique interpolation to scattered multivariate data, because the interpolation matrices have a symmetric and positive definite dominant part. In many applications, the data density varies locally according to the signal behaviour, and then the translates should get different scalings that match the local data density. Furthermore, if there is a local anisotropy in the data, the radial basis functions should possibly be distorted into functions with ellipsoids as level sets. In such cases, the symmetry and the definiteness of the matrices are no longer guaranteed. However, this brief note is the first paper to provide sufficient conditions for the unique solvability of such interpolation processes. The basic technique is a simple matrix perturbation argument combined with the Ball–Narcowich–Ward stability results. [ABSTRACT FROM AUTHOR]

This paper is concerned with high-order numerical methods for a class of fractional mobile/immobile convection-diffusion equations. The convection coefficient of the equation may be spatially variable. In order to overcome the difficulty caused by variable coefficient problems, we first transform the original equation into a special and equivalent form, which is then discretized by a fourth-order compact finite difference approximation for the spatial derivative and a second-order difference approximation for the time first derivative and the Caputo time fractional derivative. The local truncation error and the solvability of the resulting scheme are discussed in detail. The (almost) unconditional stability and convergence of the method are proved using a discrete energy analysis method. A Richardson extrapolation algorithm is presented to enhance the temporal accuracy of the computed solution from the second-order to the third-order. Applications using two model problems give numerical results that demonstrate the accuracy of the new method and the high efficiency of the Richardson extrapolation algorithm. [ABSTRACT FROM AUTHOR]

Although numerical methods of nonlinear stochastic differential delay equations (SDDEs) have been discussed by several authors, there is so far little work on the numerical approximation of SDDE with coefficients of polynomial growth. The main aim of the paper is to investigate convergence in probability of the Euler-Maruyama (EM) approximate solution for SDDE with one-sided polynomial growing drift coefficient and polynomial growing diffusion coefficient. Moreover, we prove the existence-and-uniqueness of almost surely exponentially stable global solution for this nonlinear stochastic delay system. Finally, a computer simulation confirms the efficiency of our numerical method. [ABSTRACT FROM AUTHOR]

This paper addresses the problem of capturing nondominated points on non-convex pareto frontier, which are encountered in E-[0,1] convex multi-objective programming problems. We use E-[0,1] map to transfer non-convex pareto frontier to convex pareto frontier, then An algorithm to find a piecewise linear approximation of the nondominated set of convex pareto frontier are applied. Finally, the inverse map of E-[0,1] is used to get the nondominated set of non-convex pareto frontier. [ABSTRACT FROM AUTHOR]

This paper is concerned with numerical methods for a class of two-dimensional fractional convection-subdiffusion equations with a time Caputo fractional derivative of order $$\alpha (0<\alpha <1)$$ . We first transform the original equation into a special and equivalent form, which is then discretized by a fourth-order compact finite difference approximation in the spatial directions and by an alternating direction implicit (ADI) approximation in the temporal direction. The resulting compact ADI scheme is uniquely solvable and unconditionally stable. The optimal error estimates in the weighted $$L^{\infty }$$ , $$H^{1}$$ and $$L^{2}$$ norms are obtained, and show that the compact ADI method has the temporal accuracy of order $$\min \{1+\alpha ,2-\alpha \}$$ and the fourth-order spatial accuracy. Applications using three model problems give numerical results that demonstrate the accuracy and the effectiveness of this new method. [ABSTRACT FROM AUTHOR]

This paper deals with the numerical computation of null controls for the wave equation with a potential. The goal is to compute approximations of controls that drive the solution from a prescribed initial state to zero at a large enough controllability time. In [ Cîndea, Fernández-Cara & Münch, Numerical controllability of the wave equation through primal methods and Carleman estimates, 2013], a so called primal method is described leading to a strongly convergent approximation of boundary controls: the controls minimize quadratic weighted functionals involving both the control and the state and are obtained by solving the corresponding optimality condition. In this work, we adapt the method to approximate the control of minimal square-integrable norm. The optimality conditions of the problem are reformulated as a mixed formulation involving both the state and its adjoint. We prove the well-posedeness of the mixed formulation (in particular the inf-sup condition) then discuss several numerical experiments. The approach covers both the boundary and the inner controllability. For simplicity, we present the approach in the one dimensional case. [ABSTRACT FROM AUTHOR]

In this paper, an adaptive sizing BFGS method for unconstrained optimization is proposed, whose scaling factor is automatically chosen by the signature of some terms of the approximate model's curvature. The scaling factor is always chosen less than or equal to one as required by the convergence property of the self-scaling BFGS method of Al-Baali (Comput Optim Appl 9:191-203, ), while the choosing strategy can ensure the sufficient positive definiteness of the updating matrices. Under mild conditions, the global convergence properties of Al-Baali on convex functions are proved. Numerical results on some test problems show the proposed method is competitive with its counterparts provided that the scaling factor is not too small. [ABSTRACT FROM AUTHOR]

In this paper we give a theory for the approximation of eigenvalue problems in mixed form by non-conforming methods. We then apply this theory to analyze the problem of determining the vibrational modes of a linear elastic structure using the classical Hellinger-Reissner mixed formulation. We show that a numerical method based on the lowest-order Arnold-Winther non-conforming space provides a spectrally correct approximation of the eigenvalue/eigenvector pairs. Moreover, the method is proven to converge with optimal order. [ABSTRACT FROM AUTHOR]

In this paper, we study two new spline methods for the Birkhoff interpolation problem proposed in Costabile and Longo (Calcolo 47:49-63, ). Our methods are based on cubic spline and quintic spline respectively. The new methods are easy to be implemented. They are effective not only in approximating the original function but also in approximating its derivatives. The respective remainders and error bounds are given. Numerical tests are performed. Numerical results confirm the theoretical analysis and show that our methods are very practical. [ABSTRACT FROM AUTHOR]

In this paper, we present a new family of two-step iterative methods for solving nonlinear equations. The order of convergence of the new family without memory is four requiring three functional evaluations, which implies that this family is optimal according to Kung and Traubs conjecture Kung and Traub (J Appl Comput Math 21:643-651, ). Further accelerations of convergence speed are obtained by varying a free parameter in per full iteration. This self-accelerating parameter is calculated by using information available from the current and previous iteration. The corresponding R-order of convergence is increased form 4 to $$\frac{5+\sqrt{17}}{2}\approx 4.5616, \frac{5+\sqrt{21}}{2}\approx 4.7913$$ and 5. The increase of convergence order is attained without any additional calculations so that the family of the methods with memory possesses a very high computational efficiency. Another advantage of the new methods is that they remove the severe condition $$f^{\prime }(x)$$ in a neighborhood of the required root imposed on Newtons method. Numerical comparisons are made to show the performance of our methods, as shown in the illustration examples. [ABSTRACT FROM AUTHOR]

The main purpose of this paper is to propose the Legendre spectral-collocation method to solve the Volterra integral equations of the second kind with non-vanishing delay. We divide the definition domain into several subintervals according to the primary discontinuous points associated with the delay. In each subinterval, where the solution is smooth enough, we can apply Legendre spectral-collocation method to approximate the solution. The provided convergence analysis shows that the numerical errors decay exponentially. Numerical examples are presented to confirm this theoretical predict. [ABSTRACT FROM AUTHOR]

A jump-diffusion model for the pricing of options leads to a partial integro-differential equation (PIDE). Discretizing the PIDE by certain method, we get a sequence of systems of linear equations, where the coefficient matrices are Toeplitz matrices. In this paper, we decompose the coefficient matrix as the sum of a tridiagonal matrix and a near low-rank matrix, and approximate the near low-rank matrix by low-rank matrices. Then we introduce a stationary iterative method for the approximate systems of linear equations. Comparison of the performance of our algorithm to that proposed in Pang et al. (Linear Algebra Appl. 434:2325-2342, ) is presented. [ABSTRACT FROM AUTHOR]

The aim of this paper is to discuss the properties of the bubble stabilized discontinuous Galerkin method with respect to mesh geometry. First we show that on certain non-conforming meshes the bubble stabilized discontinuous Galerkin method allows for hanging nodes/edges. Then we consider the case of conforming meshes and present a post-processing algorithm based on the Crouzeix-Raviart method to obtain the Bubble Stabilized Discontinuous Galerkin (BSDG) method. Although finally the post-processed solution does not coincide with the BSDG-solution in general, they satisfy the same (approximation) properties and are close to each other. Moreover, the post-processed solution has continuous flux over the edges. [ABSTRACT FROM AUTHOR]

In this paper, the variations of both subunitary polar factors and Hermitian positive semidefinite polar factors in the polar decomposition are studied. New perturbation bounds of both polar factors are given without the restriction that A and its perturbed matrix à have the same rank. These improve recent results. [ABSTRACT FROM AUTHOR]

The method of Shepard is an efficient method for interpolation of very large scattered data sets; unfortunately, it has poor reproduction qualities and high computational cost. In this paper we introduce a new operator which diminishes these drawbacks. This operator results from the combination of the Shepard operator with a new interpolation operator, recently proposed by Costabile and Dell’Accio, and generalizes to two variate functions the Shepard-Bernoulli operator introduced in [2]. We study this combined operator and give error bounds in terms of the modulus of continuity of high order and of the mesh length. We improve the accuracy and computational efficiency using a method introduced by Franke and Nielson. Keywords: Shepard operator, Bernoulli operator, interpolation of scattered data, error estimations. Mathematics Subject Classification (2000): 41A05, 41A25, 41A80. [ABSTRACT FROM AUTHOR]

Domain decomposition techniques provide powerful tools for the numerical approximation of partial differential equations. In this paper, we analyze the Nitsche method for the Lamé operator, establish a priori error estimates and compare this method with the mortar method using dual Lagrange multiplier spaces. Both methods can be applied to non-matching triangulations. We use a multigrid algorithm to solve the algebraic systems. Although we have a mesh dependent bilinear form, optimal-cycle convergence rates can be obtained. Numerical results for the two methods with linear and quadratic finite elements illustrate the performance and flexibility of these nonconforming discretization techniques. [ABSTRACT FROM AUTHOR]

In this paper, we revisit the old problem of compact finite difference approximations of the homogeneous Dirichlet problem in dimension 1. We design a large and natural set of schemes of arbitrary high order, and we equip this set with an algebraic structure. We give some general criteria of convergence and we apply them to obtain two new results. On the one hand, we use Padé approximant theory to construct, for each given order of consistency, the most efficient schemes and we prove their convergence. On the other hand, we use diophantine approximation theory to prove that almost all of these schemes are convergent at the same rate as the consistency order, up to some logarithmic correction. [ABSTRACT FROM AUTHOR]

In this paper, we introduce two modifications of the forward-backward splitting method with a new step size rule for inclusion problems in real Hilbert spaces. The modifications are based on Mann and viscosity-ideas. Under standard assumptions, such as Lipschitz continuity and monotonicity (also maximal monotonicity), we establish strong convergence of the proposed algorithms. We present two numerical examples, the first in infinite dimensional spaces, which illustrates mainly the strong convergence property of the algorithm. For the second example, we illustrate the performances of our scheme, compared with the classical forward-backward splitting method for the problem of recovering a sparse noisy signal. Our result extend some related works in the literature and the primary experiments might also suggest their potential applicability. [ABSTRACT FROM AUTHOR]

In order to determine the extinction probability of a Markovian binary tree, it is necessary to find the minimal nonnegative solution of a quadratic vector equation. We apply the modified Newton-Shamanskii method for solving the equation. We show that the sequence of vectors generated by the modified Newton-Shamanskii method is monotonically increasing and converges to the minimal nonnegative solution of the equation. Numerical experiments show the effectiveness of our method. [ABSTRACT FROM AUTHOR]

A variable coefficient fourth-order partial differential equation in one space variable that arises in the study of transverse vibrations of a uniform flexible beam is examined. The numerical solution is obtained by using the sinc-Galerkin method. The method is then tested on three problems which have appeared in physical applications. The numerical results demonstrate the reliability and efficiency of the algorithm developed. [ABSTRACT FROM AUTHOR]

We consider the Cauchy problem for first order differential-functional equations. We present finite difference schemes to approximate viscosity solutions of this problem. The functional dependence in the equation is of the Hale type. It contains, as a particular case, equation with a retarded and deviated argument, and differential-integral equation. Numerical examples to illustrate the theory are presented. [ABSTRACT FROM AUTHOR]

Linear combinations of translates of a given basis function have long been successfully used to solve scattered data interpolation and approximation problems. We demonstrate how the classical basis function approach can be transferred to the projective space ℙ. To be precise, we use concepts from harmonic analysis to identify positive definite and strictly positive definite zonal functions on ℙ. These can then be applied to solve problems arising in tomography since the data given there consists of integrals over lines. Here, enhancing known reconstruction techniques with the use of a scattered data interpolant in the 'space of lines', naturally leads to reconstruction algorithms well suited to limited angle and limited range tomography. In the medical setting algorithms for such incomplete data problems are desirable as using them can limit radiation dosage. [ABSTRACT FROM AUTHOR]

radial basis function approximation takes the form where the coefficients a,..., a are real numbers, the centres b,..., b are distinct points in ℝ, and the function φ:ℝ→ℝ is radially symmetric. Such functions are highly useful in practice and enjoy many beautiful theoretical properties. In particular, much work has been devoted to the polyharmonic radial basis functions, for which φ is the fundamental solution of some iterate of the Laplacian. In this note, we consider the construction of a rotation-invariant signed (Borel) measure μ for which the convolution ψ= μ φ is a function of compact support, and when φ is polyharmonic. The novelty of this construction is its use of the Paley-Wiener theorem to identify compact support via analysis of the Fourier transform of the new kernel ψ, so providing a new form of kernel engineering. [ABSTRACT FROM AUTHOR]

A posteriori error estimates for the nonconforming P1 element are easily determined by the hypercircle method via Marini’s observation on the relation to the mixed method of Raviart–Thomas. Another tool is Ainsworth’s application of the hypercircle method to mixed methods. The relation on the finite element solutions is also extended to an a priori relation of the errors, and the errors of four different finite element methods can be compared. [ABSTRACT FROM AUTHOR]

We study the gap of discrete spectra of the Laplace operator in 1d for non-uniform meshes by analyzing the corresponding spectral symbol, which allows to show how to design the discretization grid for improving the gap behavior. The main tool is the study of a univariate monotonic version of the spectral symbol, obtained by employing a proper rearrangement via the GLT theory. We treat in detail the case of basic finite-difference approximations. In a second step, we pass to precise approximation schemes, coming from the celebrated Galerkin isogeometric analysis based on B-splines of degree p and global regularity Cp-1, and finally we address the case of finite-elements with global regularity C0 and local polynomial degree p. The surprising result is that the GLT approach allows a unified spectral treatment of the various schemes also in terms of the preservation of the average gap property, which is necessary for the uniform gap property. The analytical results are illustrated by a number of numerical experiments. We conclude by discussing some open problems. [ABSTRACT FROM AUTHOR]

To model a liquid-gas mixture, in this article we propose a phase-field approach that might also provide a description of the expansion stage of a metal foam inside a hollow mold. We conceive the mixture as a two-phase incompressible-compressible fluid governed by a Navier-Stokes-Cahn-Hilliard system of equations, and we adapt the Lowengrub-Truskinowsky model to take into account the expansion of the gaseous phase. The resulting system of equations is characterized by a velocity field that fails to be divergence-free, by a logarithmic term for the pressure that enters in the Gibbs free-energy expression and by the viscosity that degenerates in the gas phase. In the second part of the article we propose an energy-based numerical scheme that, at the discrete level, preserves the mass conservation property and the energy dissipation law of the original system. We use a discontinuous Galerkin approximation for the spatial approximation and a modified midpoint based scheme for the time approximation. [ABSTRACT FROM AUTHOR]

We investigate two greedy strategies for finding an approximation to the minimum of a convex function E defined on a Hilbert space H. We prove convergence rates for these algorithms under suitable conditions on the objective function E. These conditions involve the behavior of the modulus of smoothness and the modulus of uniform convexity of E. [ABSTRACT FROM AUTHOR]

We establish a Poincaré-Wirtinger type inequality on some particular domains with a precise estimate of the constant depending only on the geometry of the domain. This type of inequality arises, for instance, in the analysis of finite volume (FV) numerical methods. As an application of our result, we prove uniform a priori bounds for the FV approximate solutions of the heat equation with Ventcell boundary conditions in the natural energy space defined as the set of those functions in $$H^1(\varOmega )$$ whose traces belong to $$H^1(\partial \varOmega )$$ . The main difficulty here comes from the fact that the approximation is performed on non-polygonal control volumes since the domain itself is non-polygonal. [ABSTRACT FROM AUTHOR]

Conforming piecewise polynomial spaces with respect to cubic meshes are constructed for the Stokes problem in arbitrary dimensions yielding exactly divergence-free velocity approximations. The derivation of the finite element pair is motivated by a smooth de Rham complex that is well-suited for the Stokes problem. We derive the stability and convergence properties of the new elements as well as the construction of reduced elements with less global unknowns. [ABSTRACT FROM AUTHOR]

Variable-step (VS) $$3$$ -stage Hermite-Birkhoff (HB) methods HB $$(p=k+1)$$ of order $$p=8$$ and 9 are constructed as a combination of linear $$k$$ -step methods of order $$(p-2)$$ and a diagonally implicit one-step $$3$$ -stage Runge-Kutta method of order 3 (DIRK3) for solving stiff ordinary differential equations. Forcing a Taylor expansion of the numerical solution to agree with an expansion of the true solution leads to multistep and Runge-Kutta type order conditions which are reorganized into linear confluent Vandermonde-type systems. This approach allows us to develop A-stable methods of order up to 5 and A( $$\alpha $$ )-stable methods of order up to 10. Fast algorithms are developed for solving these systems in O $$(p^2)$$ operations to obtain HB interpolation polynomials in terms of generalized Lagrange basis functions. The stepsize of these methods are controlled by a local error estimator. When programmed in C++, HB( $$p$$ ) of order $$p=8$$ and 9 compare favorably with existing Cash modified extended backward differentiation formulae of order 7 and 8, MEBDF(7-8), in solving problems often used to test higher order stiff ODE solvers on the basis of CPU time and error at the endpoint of the integration interval. [ABSTRACT FROM AUTHOR]

We propose and study an iterative substructuring method for an $$h$$ - $$p$$ Nitsche-type discretization, following the original approach introduced in Bramble et al. Math. Comp. 47(175):103-134, () for conforming methods. We prove quasi-optimality with respect to the mesh size and the polynomial degree for the proposed preconditioner. Numerical experiments assess the performance of the preconditioner and verify the theory. [ABSTRACT FROM AUTHOR]

A variational formulation of the standard marker-and-cell scheme for the approximation of the Navier-Stokes problem yields an extension of the scheme to general 2D and 3D domains and more general meshes. An original discretization of the trilinear form of the nonlinear convection term is proposed; it is designed so as to vanish for discrete divergence free functions. This property allows us to give a mathematical proof of the convergence of the resulting approximate solutions, for the nonlinear Navier-Stokes equations in both steady-state and time-dependent regimes, without any small data condition. Numerical examples (analytical steady and time-dependent ones, inclined driven cavity) confirm the robustness and the accuracy of this method. [ABSTRACT FROM AUTHOR]

In this work we approximate the solution of a quasilinear elliptic problem of monotone type by using the Mimetic Finite Difference (MFD) method. Under a suitable approximation assumption, we prove that the MFD approximate solution converges, with optimal rate, to the exact solution in a mesh-dependent energy norm. The resulting nonlinear discrete problem is then solved iteratively via linearization by applying the Kačanov method. The convergence of the Kačanov algorithm in the discrete mimetic framework is also proved. Several numerical experiments confirm the theoretical analysis. [ABSTRACT FROM AUTHOR]

We analyze an adaptive boundary element method for Symm's integral equation in 2D and 3D which incorporates the approximation of the Dirichlet data $$g$$ into the adaptive scheme. We prove quasi-optimal convergence rates for any $$H^{1/2}$$ -stable projection used for data approximation. [ABSTRACT FROM AUTHOR]

For a given $$\theta \in (a,b)$$, we investigate the question whether there exists a positive quadrature formula with maximal degree of precision which has the prescribed abscissa $$\theta $$ plus possibly $$a$$ and/or $$b$$, the endpoints of the interval of integration. This study relies on recent results on the location of roots of quasi-orthogonal polynomials. The above positive quadrature formulae are useful in studying problems in one-sided polynomial $$L_1$$ approximation. [ABSTRACT FROM AUTHOR]

Fractal interpolation is a modern technique in approximation theory to fit and analyze scientific data. We develop a new class of $$\mathcal C ^1$$- rational cubic fractal interpolation functions, where the associated iterated function system uses rational functions of the form $$\frac{p_i(x)}{q_i(x)},$$ where $$p_i(x)$$ and $$q_i(x)$$ are cubic polynomials involving two shape parameters. The rational cubic iterated function system scheme provides an additional freedom over the classical rational cubic interpolants due to the presence of the scaling factors and shape parameters. The classical rational cubic functions are obtained as a special case of the developed fractal interpolants. An upper bound of the uniform error of the rational cubic fractal interpolation function with an original function in $$\mathcal C ^2$$ is deduced for the convergence results. The rational fractal scheme is computationally economical, very much local, moderately local or global depending on the scaling factors and shape parameters. Appropriate restrictions on the scaling factors and shape parameters give sufficient conditions for a shape preserving rational cubic fractal interpolation function so that it is monotonic, positive, and convex if the data set is monotonic, positive, and convex, respectively. A visual illustration of the shape preserving fractal curves is provided to support our theoretical results. [ABSTRACT FROM AUTHOR]

In this article a priori error estimates are derived for the finite element discretization of optimal distributed control problems governed by the biharmonic operator. The state equation is discretized in primal mixed form using continuous piecewise biquadratic finite elements, while piecewise constant approximations are used for the control. The error estimates derived for the state variable as well as that for the control are order-optimal on general unstructured meshes. However, on uniform meshes not all error estimates are optimal due to the low-order control approximation. All theoretical results are confirmed by numerical tests. [ABSTRACT FROM AUTHOR]

We introduce a type of modified truncations of approximate approximation with Gaussian kernels, which can be applied to approximate functions on compact intervals. Our results improve the related results of Chen and Cao (Appl Math Comput 217:725-734, ), Müller and Varnhorn (J Approx Theory 145:171-181, ). Also, we construct approximate approximation operators in multivariate cases and obtain the estimates of approximation rate. [ABSTRACT FROM AUTHOR]