Error Estimates For Approximation Research Articles

Spectral Galerkin Approximation and Error Analysis Based on a Mixed Scheme for Fourth‐Order Problems in Complex Regions

ABSTRACTIn this article, an efficient spectral Galerkin method, which is based on a mixed scheme, is proposed and studied for solving fourth‐order problems in complex regions. The fundamental idea behind this approach is to transform the initial problem into an equivalent form in cylindrical coordinates and to reshape the computational domain into a product‐type rectangular one, which facilitates the utilization of spectral methods. However, when considering the equivalent fourth‐order form directly in cylindrical coordinates, it introduces intricate pole conditions and variable coefficients, posing challenges to both theoretical analysis and algorithm implementation. To address this, we employ the orthogonality of Fourier series to further decompose it into a sequence of decoupled two‐dimensional fourth‐order eigenvalue problems. For each such problem, we introduce an auxiliary function to transform it into an equivalent second‐order coupled system. Building on this, we formulate a mixed variational formulation and discrete scheme, and prove the error estimates for eigenvalue and eigenfunction approximations. Furthermore, we extend this algorithm to the two‐dimensional complex domains. Finally, a series of numerical examples are presented, and the numerical results validate the effectiveness of the algorithm and the correctness of the theoretical results.

A Cooperative Control Method for Wide-Range Maneuvering of Autonomous Aerial Refueling Controllable Drogue

In the realm of autonomous aerial refueling missions for unmanned aerial vehicles (UAVs), the controllable drogue represents a novel approach that significantly enhances both the safety and efficiency of aerial refueling operations. This paper delves into the issue of wide-range maneuverability control for the controllable drogue. Initially, a dynamic model for the variable-length hose–drogue system is presented. Based on this, a cooperative control framework that synergistically utilizes both the hose and the drogue is designed to achieve wide-range maneuverability of the drogue. To address the delay in hose retrieval and release, an open-loop control strategy based on neural networks is proposed. Furthermore, a closed-loop control method utilizing fuzzy approximation and adaptive error estimation is designed to tackle the challenges posed by modeling inaccuracies and uncertainties in aerodynamic parameters. Comparative simulation results show that the proposed control strategy can make the drogue maneuvering range reach more than 6 m. And it can accurately track the time-varying trajectory under the influence of model uncertainty and wind disturbance with an error of less than 0.1 m throughout. This method provides an effective means for achieving wide-range maneuverability control of the controllable drogue in autonomous aerial refueling missions.

On estimation of the uniform approximation error by interpolating multilinear spline with l p distances

UDC 517.5 We consider the estimation of the uniform approximation error on classes of functions by interpolating multilinear spline with l p distances for p > 3.

Non-coercive Neumann Boundary Control Problems

The article examines a linear-quadratic Neumann control problem that is governed by a non-coercive elliptic equation. Due to the non-self-adjoint nature of the linear control-to-state operator, it is necessary to independently study both the state and adjoint state equations. The article establishes the existence and uniqueness of solutions for both equations, with minimal assumptions made about the problem’s data. Next, the regularity of these solutions is studied in three frameworks: Hilbert–Sobolev spaces, Sobolev–Slobodeckiĭ spaces, and weighted Sobolev spaces. These regularity results enable a numerical analysis of the finite element approximation of both the state and adjoint state equations. The results cover both convex and non-convex domains and quasi-uniform and graded meshes. Finally, the optimal control problem is analyzed and discretized. Existence and uniqueness of the solution, first-order optimality conditions, and error estimates for the finite element approximation of the control are obtained. Numerical experiments confirming these results are included.

Enhancing interpolation and approximation error estimates using a novel Taylor-like formula

In this paper, we present an approach to enhance interpolation and approximation error estimates. Based on a previously derived first-order Taylor-like formula, we demonstrate its applicability in improving the P1-interpolation error estimate. Following the same principles, we have also developed a novel numerical scheme for the heat equation that provides a better error estimate compared to the classical implicit finite differences scheme. Numerical illustrations confirm this behavior.

Stability of Fixed Points of Partial Contractivities and Fractal Surfaces

In this paper, a large class of contractions is studied that contains Banach and Matkowski maps as particular cases. Sufficient conditions for the existence of fixed points are proposed in the framework of b-metric spaces. The convergence and stability of the Picard iterations are analyzed, giving error estimates for the fixed-point approximation. Afterwards, the iteration proposed by Kirk in 1971 is considered, studying its convergence, stability, and error estimates in the context of a quasi-normed space. The properties proved can be applied to other types of contractions, since the self-maps defined contain many others as particular cases. For instance, if the underlying set is a metric space, the contractions of type Kannan, Chatterjea, Zamfirescu, Ćirić, and Reich are included in the class of contractivities studied in this paper. These findings are applied to the construction of fractal surfaces on Banach algebras, and the definition of two-variable frames composed of fractal mappings with values in abstract Hilbert spaces.

On Error Estimates of a discontinuous Galerkin Method of the Boussinesq System of Equations

Abstract In this paper, we propose and analyze a discontinuous Galerkin finite element method for solving the transient Boussinesq incompressible heat conducting fluid flow equations. This method utilizes an upwind approach to handle the nonlinear convective terms effectively. We discuss new a priori bounds for the semidiscrete discontinuous Galerkin approximations. Furthermore, we establish optimal a priori error estimates for the semidiscrete discontinuous Galerkin velocity approximation in L 2 \mathbf{L}^{2} and energy norms, the temperature approximation in L 2 L^{2} and energy norms and pressure approximation in L 2 L^{2} -norm for t > 0 t>0 . Additionally, under the smallness assumption on the data, we prove uniform in time error estimates. We also consider a backward Euler scheme for full discretization and derive fully discrete error estimates. Finally, we provide numerical examples to support the theoretical conclusions.

Approximation error estimates by noise‐injected neural networks

One‐hidden‐layer feedforward neural networks are described as functions having many real‐valued parameters. Approximation properties of neural networks are established (universal approximation property), and the approximation error is related to the number of parameters in the network. The essentially optimal order of approximation error bounds was already derived in 1996. We focused on the numerical experiment that indicates the neural networks whose parameters contain stochastic perturbations gain better performance than ordinary neural networks and explored the approximation property of neural networks with stochastic perturbations. In this paper, we derived the quantitative order of variance of stochastic perturbations to achieve the essentially optimal approximation order and verified the justifiability of our theory by numerical experiments.

A priori estimates of the unsteady incompressible thermomicropolar fluid equations and its numerical analysis based on penalty finite element method

In this paper, the unsteady incompressible thermomicropolar fluid (UITF) equations are considered. Theoretically, some a priori regularity conclusions are presented firstly, which seem to be not available in the literatures. Numerically, a penalty finite element method (PFEM) for the UITF equations is studied, the Euler semi-implicit temporal semi-discrete method of the penalty UITF equations is proposed, the stability and the L2-H1 error estimates of the temporal discrete solutions are proved. Finally, the stability and the L2-H1 error estimates for the finite element fully-discrete approximation of the penalty UITF equations are rigorously proved. The accuracy and efficiency of the fully-discrete PFEM are demonstrated by some numerical examples.

Error Identities for Parabolic Equations with Monotone Spatial Operators

Abstract The article studies quantitative relations (error identities) that characterize distances between exact solutions of nonlinear evolutionary problems and functions considered as approximations. The restrictions imposed on such a function are minimal and actually come down to the condition that it belongs to the same functional class as the generalized solution of the problem under consideration. Functional identities of this type reflect the most general relations between deviations from exact solutions of parabolic initial boundary value problems and those data that can be observed in a numerical experiment. The identities contain no mesh dependent constants and are valid for any function in the admissible (energy) class regardless of the method by which it was constructed. Therefore, they can serve as basic tools for deriving fully reliable a posteriori estimates of approximation errors as well as for analysis of modeling errors. The corresponding examples are discussed in the paper.

Space-time error estimates for approximations of linear parabolic problems with generalized time boundary conditions

Abstract We first give a general error estimate for the nonconforming approximation of a problem for which a Banach–Nečas–Babuška (BNB) inequality holds. This framework covers parabolic problems with general conditions in time (initial value problems as well as periodic problems) under minimal regularity assumptions. We consider approximations by two types of space-time discretizations, both based on a conforming Galerkin method in space. The first one is the Euler $\theta -$scheme. In this case, we show that the BNB inequality is always satisfied, and may require an extra condition on the time step for $\theta \le \frac 1 2$. The second one is the time discontinuous Galerkin method, where the BNB condition holds without any additional condition.

Gradient preserving Operator Inference: Data-driven reduced-order models for equations with gradient structure

Hamiltonian Operator Inference has been introduced in Sharma et al. (2022) to learn structure-preserving reduced-order models (ROMs) for Hamiltonian systems. This approach constructs a low-dimensional model using only data and knowledge of the Hamiltonian function. Such ROMs can keep the intrinsic structure of the system, allowing them to capture the physics described by the governing equations. In this work, we extend this approach to more general systems that are either conservative or dissipative in energy, and which possess a gradient structure. We derive the optimization problems for inferring structure-preserving ROMs that preserve the gradient structure. We further derive an a priori error estimate for the reduced-order approximation. To test the algorithms, we consider semi-discretized partial differential equations with gradient structure, such as the parameterized wave and Korteweg–de-Vries equations, and equations of three-dimensional linear elasticity in the conservative case and the one- and two-dimensional Allen-Cahn equations in the dissipative case. The numerical results illustrate the accuracy, structure-preservation properties, and predictive capabilities of the gradient-preserving Operator Inference ROMs.

Optimal error estimates for non-conforming approximations of linear parabolic problems with minimal regularity

Optimal error estimates for non-conforming approximations of linear parabolic problems with minimal regularity

Error estimates for finite element approximations of viscoelastic dynamics: The generalized Maxwell model

We prove error estimates for a finite element approximation of viscoelastic dynamics based on continuous Galerkin in space and time, both in energy norm and in L2 norm. The proof is based on an error representation formula using a discrete dual problem and a stability estimate involving the kinetic, elastic, and viscoelastic energies. To set up the dual error analysis and to prove the basic stability estimates, it is natural to formulate the problem as a first-order-in-time system involving evolution equations for the viscoelastic stress, the displacements, and the velocities. The equations for the viscoelastic stress can, however, be solved analytically in terms of the deviatoric strain velocity, and therefore, the viscoelastic stress can be eliminated from the system, resulting in a system for displacements and velocities.

Well-posedness and error estimates for coupled systems of nonlocal conservation laws

Abstract This article deals with the error estimates for numerical approximations of the entropy solutions of coupled systems of nonlocal hyperbolic conservation laws. The systems can be strongly coupled through the nonlocal coefficient present in the convection term. A fairly general class of fluxes is being considered, where the local part of the flux can be discontinuous at infinitely many points, with possible accumulation points. The aims of the paper are threefold: (1) Establishing existence of entropy solutions with rough local flux for such systems, by deriving a uniform $\operatorname {BV}$ bound on the numerical approximations; (2) Deriving a general Kuznetsov-type lemma (and hence uniqueness) for such systems with both smooth and rough local fluxes; (3) Proving the convergence rate of the finite volume approximations to the entropy solutions of the system as $1/2$ and $1/3$, with homogeneous (in any dimension) and rough local parts (in one dimension), respectively. Numerical experiments are included to illustrate the convergence rates.

Finite element error analysis of affine optimal control problems

This paper is concerned with error estimates for the numerical approximation for affine optimal control problems subject to semilinear elliptic PDEs. To investigate the error estimates, we focus on local minimizers that satisfy certain local growth conditions. The local growth conditions we consider in this paper appeared recently in the context of solution stability and contain the joint growth of the first and second variation of the objective functional. These growth conditions are especially meaningful for affine control constrained optimal control problems because the first variation can satisfy a local growth, which is not the case for unconstrained problems. The main results of this paper are the achievement of error estimates for the numerical approximations generated by a finite element scheme with piecewise constant controls or a variational discretization scheme. Even though the growth conditions considered are weaker than those appearing in the recent literature on finite element error estimates for affine problems, this paper substantially improves the existing error estimates for both the optimal controls and the states when a Hölder-type growth is assumed.

Estimation of approximation error of a function having its derivatives belonging to lipschitz class of order - α by extended legendre wavelet (ELW) method and its applications

Estimation of approximation error of a function having its derivatives belonging to lipschitz class of order - α by extended legendre wavelet (ELW) method and its applications

Approximation error of single hidden layer neural networks with fixed weights

Neural networks with finitely many fixed weights have the universal approximation property under certain conditions on compact subsets of the d-dimensional Euclidean space, where approximation process is considered. Such conditions were delineated in our paper [26]. But for many compact sets it is impossible to approximate multivariate functions with arbitrary precision and the question on estimation or efficient computation of approximation error arises. This paper provides an explicit formula for the approximation error of single hidden layer neural networks with two fixed weights.

Discontinuous Galerkin Two-Grid Method for the Transient Navier–Stokes Equations

Abstract In this paper, we apply a two-grid scheme to the DG formulation of the 2D transient Navier–Stokes model. The two-grid algorithm consists of the following steps: Step 1 involves solving the nonlinear system on a coarse mesh with mesh size 𝐻, and Step 2 involves linearizing the nonlinear system by using the coarse grid solution on a fine mesh of mesh size ℎ and solving the resulting system to produce an approximate solution with desired accuracy. We establish optimal error estimates of the two-grid DG approximations for the velocity and pressure in energy and L 2 L^{2} -norms, respectively, for an appropriate choice of coarse and fine mesh parameters. We further discretize the two-grid DG model in time, using the backward Euler method, and derive the fully discrete error estimates. Finally, numerical results are presented to confirm the efficiency of the proposed scheme.

On a New Approach for Stability and Controllability Analysis of Functional Equations

We consider a new approach to approximate stability analysis for a tri-additive functional inequality and to obtain the optimal approximation for permuting tri-derivations and tri-homomorphisms in unital matrix algebras via the vector-valued alternative fixed-point theorem, which is a popular technique of proving the stability of functional equations. We also present a small list of aggregation functions on the classical, well-known special functions to investigate the best approximation error estimates using a different concept of perturbation stability.