Error Estimates For Approximation Research Articles

Error estimates of divergence-free generalized moving least squares (Div-Free GMLS) derivatives approximations in Sobolev spaces

The divergence-free generalized moving least squares (Div-Free GMLS) approximation has recently been utilized to solve some incompressible fluid flows problems. In our recent work (Mohammadi and Dehghan (2021) [28]), we have presented its formulation more precisely, and also the error estimates of derivatives have been carried out in L∞(Ω), where Ω⊂Rd is a bounded set satisfying an interior cone condition. However, the error estimates of this vector-valued approximation in Sobolev spaces are not done. So, in this paper we make the error estimates of Div-Free GMLS derivatives approximations in Lq(Ω), where 1≤q≤∞, using a stable local divergence-free polynomial reproduction property. Note that, the method is a direct approximants of exact derivatives of a divergence-free vector field, which possesses the optimal rates of convergence. This vector-valued technique can also be developed to find the numerical solution of the incompressible fluid flows problems easier than the other available mesh-dependent methods. Finally, we have shown how the proposed approximation can recover the velocity field variable of the well-known Darcy's problem in a two-dimensional space.

Space−time spectral approximations of a parabolic optimal control problem with an L2‐norm control constraint

AbstractThis article deals with the spectral approximation of an optimal control problem governed by a parabolic partial differential equation (PDE) with an ‐norm control constraint. The investigations employ the space−time spectral method, which is, more precisely, a dual Petrov‐Galerkin spectral method in time and a spectral method in space to discrete the continuous system. As a global method, it uses the global polynomials as the trial functions for discretization of PDEs. After obtaining the optimality condition of the continuous system and that of its spectral discrete surrogate, we establish a priori and a posteriori error estimates for the spectral approximation in detail. Three numerical examples in different spatial dimensions then confirm the theoretical results and also show the efficiency as well as a good precision of the adopted space−time spectral method.

A Posteriori Error Estimates for a Local Discontinuous Galerkin Approximation of Semilinear Second-Order Elliptic Problems on Cartesian Grids

A Posteriori Error Estimates for a Local Discontinuous Galerkin Approximation of Semilinear Second-Order Elliptic Problems on Cartesian Grids

Sharp error estimates for spatial-temporal finite difference approximations to fractional sub-diffusion equation without regularity assumption on the exact solution

Finite difference method as a popular numerical method has been widely used to solve fractional diffusion equations. In the general spatial error analyses, an assumption $$u\in C^{4}({\bar{\varOmega }})$$ is needed to preserve $${\mathcal {O}}(h^{2})$$ convergence when using central finite difference scheme to solve fractional sub-diffusion equation with Laplace operator, but this assumption is somewhat strong, where u is the exact solution and h is the mesh size. In this paper, a novel analysis technique is proposed to show that the spatial convergence rate can reach $${\mathcal {O}}(h^{\min (\sigma +\frac{1}{2}-\epsilon ,2)})$$ in both $$l^{2}$$ -norm and $$l^{\infty }$$ -norm in one-dimensional domain when the initial value and source term are both in $${\dot{H}}^{\sigma }(\varOmega )$$ but without any regularity assumption on the exact solution, where $$\sigma \ge 0$$ and $$\epsilon >0$$ being arbitrarily small. After making slight modifications on the scheme, acting on the initial value and source term, the spatial convergence rate can be improved to $${\mathcal {O}}(h^{2})$$ in $$l^{2}$$ -norm and $${\mathcal {O}}(h^{\min (\sigma +\frac{3}{2}-\epsilon ,2)})$$ in $$l^{\infty }$$ -norm. It is worth mentioning that our spatial error analysis is applicable to high dimensional cube domain by using the properties of tensor product. Moreover, two kinds of averaged schemes are provided to approximate the Riemann–Liouville fractional derivative, and $${\mathcal {O}}(\tau ^{2})$$ convergence is obtained for all $$\alpha \in (0,1)$$ . Finally, some numerical experiments verify the effectiveness of the built theory.

Method of Particular Solutions for Second-Order Differential Equation with Variable Coefficients via Orthogonal Polynomials

In this paper, with classic Legendre polynomials, a method of particular solutions (MPS, for short) is proposed to solve a kind of second-order differential equations with a variable coefficient on a unit interval. The particular solutions, satisfying the natural Dirichlet boundary conditions, are constructed with orthogonal Legendre polynomials for the variable coefficient case. Meanwhile, we investigate the a-priori error estimates of the MPS approximations. Two a-priori error estimations in H 1 - and L ∞ -norms are shown to depict the convergence order of numerical approximations, respectively. Some numerical examples and convergence rates are provided to validate the merits of our proposed meshless method.

Consistency results for the dual-wind discontinuous Galerkin method

This paper further analyzes the dual-wind discontinuous Galerkin (DWDG) method for approximating Poisson’s problem by directly examining the relationship between the Laplacian and the underlying discrete Laplacian. DWDG methods are derived from the DG differential calculus framework that defines discrete differential operators to replace continuous differential operators. We establish a priori error estimates for the DWDG approximation of Δ. Since the DWDG method does not satisfy a Galerkin orthogonality condition, we also explore the relationship between the DWDG approximation and the Ritz projection defined to satisfy an exact Galerkin orthogonality condition linked to the DWDG method. Numerical experiments are provided to validate the theoretical results.

Fractional Step Scheme to Approximate a Non-Linear Second-Order Reaction–Diffusion Problem with Inhomogeneous Dynamic Boundary Conditions

Two main topics are addressed in the present paper, first, a rigorous qualitative study of a second-order reaction–diffusion problem with non-linear diffusion and cubic-type reactions, as well as inhomogeneous dynamic boundary conditions. Under certain assumptions about the input data: gd(t,x), gfr(t,x), U0(x) and ζ0(x), we prove the well-posedness (the existence, a priori estimates, regularity and uniqueness) of a solution in the space Wp1,2(Q)×Wp1,2(Σ). Here, we extend previous results, enabling new mathematical models to be more suitable to describe the complexity of a wide class of different physical phenomena of life sciences, including moving interface problems, material sciences, digital image processing, automatic vehicle detection and tracking, the spread of an epidemic infection, semantic image segmentation including U-Net neural networks, etc. The second goal is to develop an iterative splitting scheme, corresponding to the non-linear second-order reaction–diffusion problem. Results relating to the convergence of the approximation scheme and error estimation are also established. On the basis of the proposed numerical scheme, we formulate the algorithm alg-frac_sec-ord_dbc, which represents a delicate challenge for our future works. The benefit of such a method could simplify the process of numerical computation.

Advantages of Using Unweighted Approximation Error Measures for Model Fit Assessment

Fit indices are highly frequently used for assessing the goodness of fit of latent variable models. Most prominent fit indices, such as the root-mean-square error of approximation (RMSEA) or the comparative fit index (CFI), are based on a noncentrality parameter estimate derived from the model fit statistic. While a noncentrality parameter estimate is well suited for quantifying the amount of systematic error, the complex weighting function involved in its calculation makes indices derived from it challenging to interpret. Moreover, noncentrality-parameter-based fit indices yield systematically different values, depending on the indicators’ level of measurement. For instance, RMSEA and CFI yield more favorable fit indices for models with categorical as compared to metric variables under otherwise identical conditions. In the present article, approaches for obtaining an approximation discrepancy estimate that is independent from any specific weighting function are considered. From these unweighted approximation error estimates, fit indices analogous to RMSEA and CFI are calculated and their finite sample properties are investigated using simulation studies. The results illustrate that the new fit indices consistently estimate their true value which, in contrast to other fit indices, is the same value for metric and categorical variables. Advantages with respect to interpretability are discussed and cutoff criteria for the new indices are considered.

Trotter–Kato approximations of semilinear stochastic evolution equations in Hilbert spaces

This article deeply studies a class of semilinear stochastic evolution equations in real separable Hilbert spaces. The main goal here is to consider Trotter–Kato approximations for mild solutions of such equations. After proving the existence and uniqueness of mild solutions, the Trotter–Kato approximation system is introduced and studied in detail. Based on this, the weak convergence of probability measures induced by mild solutions of Trotter–Kato approximation equations and the approximation error estimator are established. Finally, as an application, the classical limit theorem about the parameter dependence of this kind of equations is exhibited. To deal with the general additive diffusion term, which not only depends on the state but also depends on the probability distribution of the solution process at that time, we adopt some specific techniques of measure theory and random functional analysis.

On the numerical approximation of Boussinesq/Boussinesq systems for internal waves

AbstractThe present paper is concerned with the numerical approximation of a three‐parameter family of Boussinesq systems. The systems have been proposed as models of the propagation of long internal waves along the interface of a two‐layer system of fluids with rigid‐lid condition for the upper layer and under a Boussinesq regime for the flow in both layers. We first present some theoretical properties of the systems on well‐posedness, conservation laws, Hamiltonian structure, and solitary‐wave solutions, using the results for analogous models for surface wave propagation. Then the corresponding periodic initial‐value problem is discretized in space by the spectral Fourier Galerkin method and for each system, error estimates for the semidiscrete approximation are proved. The spectral semidiscretizations are numerically integrated in time by a fourth‐order Runge–Kutta‐composition method based on the implicit midpoint rule. Numerical experiments illustrate the accuracy of the fully discrete scheme, in particular its ability to simulate accurately solitary‐wave solutions of the systems.

Parameter‐robust mixed element method for poroelasticity with Darcy‐Forchheimer flow

AbstractIn this paper, the Biot's consolidation model is considered, and Darcy‐Forchheimer equation is used to describe the relationship between the fluid velocity and pressure. The model is nonlinear and the unknown variables are the solid displacement, the fluid velocity and pressure. The Bernardi‐Raugel element is used for displacement and the RT mixed element is used for the fluid velocity and pressure. The parameter‐robust BR‐RT0‐P0 method presented here is uniformly stable not only with respect to the Lamé constant , but also with respect to the constrained specific storage coefficient . We demonstrate the well‐posedness of the BR‐RT0‐P0 finite element solutions, and give the error estimates of the finite element approximations using the monotonicity of the nonlinear Forchheimer term. The error estimates are divided into two cases, that is, is positive and is nonnegative. Numerical experiments are presented to verify theoretical analysis. Moreover we focus on the variation of pressure by several poroelasticity problems.

An accurate operational matrix method based on Lagrange polynomials for solving fractional-order pantograph delay and Riccati differential equations

This paper introduces the fractional-order Lagrange polynomials approach to solve initial value problems for pantograph delay and Riccati differential equations involving fractional-order derivatives. The fractional derivative is determined as per the idea of Caputo. First, operational matrices of fractional integration with fractional-order Lagrange polynomials have been constructed using the Laplace transform. Then, we use these operational matrices and the collocation method to convert the given initial value problem to a system of algebraic equations. Subsequently, we use Newton’s iterative approach to solve the resultant system of algebraic equations. Error estimates for the function approximation also have been discussed. Finally, some numerical examples supported the theoretical findings by demonstrating the applicability and accuracy of the proposed strategy.

Coupling of Finite and Boundary Elements for Singularly Nonlinear Transmission and Contact Problems

Abstract This article discusses the well-posedness and error analysis of the coupling of finite and boundary elements for interface problems in nonlinear elasticity. It concerns 𝑝-Laplacian-type Hencky materials with an unbounded stress-strain relation, as they arise in the modelling of ice sheets, non-Newtonian fluids or porous media. We propose a functional analytic framework for the numerical analysis and obtain a priori and a posteriori error estimates for Galerkin approximations to the resulting boundary/domain variational inequality.

Pointwise error estimates for linear finite element approximation to elliptic Dirichlet problems in smooth domains

Pointwise error estimates for linear finite element approximation to elliptic Dirichlet problems in smooth domains

The long time error estimates for the second order backward difference approximation to sub-diffusion equations with boundary time delay and feedback gain

We consider time fractional diffusion initial boundary value problem with boundary time delay and feedback gain involving two Caputo fractional derivatives in time. When both the time delay and feedback gain are taken as appropriately small such that the characteristic equation has no solution in the location of the right half plane of the imaginary axis, we derive the solution’s existence, uniqueness and representation. The second-order backward difference time discrete schemes are investigated in time discretization. The lt∞(0,∞;L2) error analysis of the numerical schemes is derived when both the time delay and feedback gain are appropriately small such that the characteristic equation of the discrete problem has no solution in a strip of the right half plane of the imaginary axis, and the order of boundary Caputo fractional derivative is equal to the order or half order of interior domain Caputo fractional derivative. The numerical examples illustrate its accuracy, efficiency and robustness, and are consistent with the theoretical results.

Residual‐based a posteriori error estimates for nonconforming finite element approximation to parabolic interface problems

AbstractIn this paper, we derive a residual‐based a posteriori error estimates for nonconforming finite element approximation to parabolic interface problems. The present approach does not involve the Helmholtz decomposition while analyzing the reliability of the estimator. The constants involved in the estimators are independent of the jump of the diffusion coefficient across the interface, and the quasi‐monotonocity assumption on the diffusion coefficient is relaxed. The reliability bound of the estimator consists of the element residual, the edge flux jump and the edge solution jump. The efficiency of the estimator is analyzed by employing a coarsening strategy introduced by Chen and Feng's study. We derive both global upper bound and a local lower bound for the error and an adaptive space–time algorithm is prescribed using the derived estimators. Numerical results illustrating the behavior of the estimators are provided.

Space-time error estimates for deep neural network approximations for differential equations

Over the last few years deep artificial neural networks (ANNs) have very successfully been used in numerical simulations for a wide variety of computational problems including computer vision, image classification, speech recognition, natural language processing, as well as computational advertisement. In addition, it has recently been proposed to approximate solutions of high-dimensional partial differential equations (PDEs) by means of stochastic learning problems involving deep ANNs. There are now also a few rigorous mathematical results in the scientific literature which provide error estimates for such deep learning based approximation methods for PDEs. All of these articles provide spatial error estimates for ANN approximations for PDEs but do not provide error estimates for the entire space-time error for the considered ANN approximations. It is the subject of the main result of this article to provide space-time error estimates for deep ANN approximations of Euler approximations of certain perturbed differential equations. Our proof of this result is based (i) on a certain ANN calculus and (ii) on ANN approximation results for products of the form [0,T]times mathbb {R}^{d}ni (t,x){kern -.5pt}mapsto {kern -.5pt} tx{kern -.5pt}in {kern -.5pt} mathbb {R}^{d} where T{kern -.5pt}in {kern -.5pt} (0,infty ), d{kern -.5pt}in {kern -.5pt} mathbb {N}, which we both develop within this article.

Unconditional stability and convergence analysis of fully discrete stabilized finite volume method for the time-dependent incompressible MHD flow

In this paper, we consider the stability and convergence of fully discrete stabilized finite volume method for the unsteady incompressible magnetohydrodynamics (MHD) equations on the quasi-uniform and regular triangulations. The spatial discretization is based on the lowest equal-order mixed finite element pair for the velocity, pressure and the magnetic fields, while the time discretization is based on the backward Euler semi-implicit scheme. In order to overcome the restriction of the discrete inf-sup condition, the local pressure projection stabilized method on the element is employed. This stabilized method has the advantages of parameter-free, no need to calculate the high-order derivatives or the edge-based data structures and it can be implemented at the element level. The unconditional stability of numerical scheme is established by using the Gronwall lemma, mathematical induction and choosing different test functions. Optimal error estimates of numerical approximations in various norms are also presented by constructing the corresponding dual problem. Finally, some numerical results are proposed to verify the established theoretical findings and show the performances of the considered numerical method.

Error estimates for spectral approximation of flow optimal control problem with <inline-formula><tex-math id="M1">$ L^2 $</tex-math></inline-formula>-norm control constraint

<p style='text-indent:20px;'>In this paper, we are concerned with the Galerkin spectral approximation of an optimal control problem governed by the Stokes equation with <inline-formula><tex-math id="M2">\begin{document}$ L^2 $\end{document}</tex-math></inline-formula>-norm constraint on the control variable. By means of the derived optimality conditions for both the original control system and its spectral approximation one, we establish a priori error estimates and then obtain a posteriori error estimator. A numerical example is, subsequently, executed to illustrate the effectiveness of method and the high performance of estimators. Furthermore, we conjecture that the similar conclusions should hold for optimal control of the Navier-Stokes equation. It is then confirmed by another numerical example.</p>

Space-time spectral methods for a fourth-order parabolic optimal control problem in three control constraint cases

<p style='text-indent:20px;'>In this paper, we are concerned with the space-time spectral discretization of an optimal control problem governed by a fourth-order parabolic partial differential equations (PDEs) in three control constraint cases. The dual Petrov-Galerkin spectral method in time and the spectral method in space are adopted to discrete the continuous system. By means of the obtained optimality condition for the continuous system and that of its spectral discrete system, we establish a priori error estimate for the spectral approximation in details. Four numerical examples are, subsequently, executed to confirm the theoretical results. The experiment results show the high efficiency and a good precision of the space-time spectral method for this kind of problems.</p>