Error Estimates For Approximation Research Articles

New estimates of Gauss-Jacobi and trapezium type inequalities for strongly $(h_{1},h_{2})$-preinvex mappings via general fractional integrals

In this paper, authors discover two interesting identities regarding Gauss--Jacobi and trapezium type integral inequalities. By using the first lemma as an auxiliary result, some new bounds with respect to Gauss--Jacobi type integral inequalities for a new class of functions called strongly $(h_{1},h_{2})$--preinvex of order $sigma>0$ with modulus $mu>0$ via general fractional integrals are established. Also, using the second lemma, some new estimates with respect to trapezium type integral inequalities for strongly $(h_{1},h_{2})$--preinvex functions of order $sigma>0$ with modulus $mu>0$ via general fractional integrals are obtained. It is pointed out that some new special cases can be deduced from main results. Some applications to special means for different real numbers and new approximation error estimates for the trapezoidal are provided as well. These results give us the generalizations of some previous known results. The ideas and techniques of this paper may stimulate further research in the fascinating field of inequalities.

Efficient multivariate approximation on the cube

We combine a periodization strategy for weighted L_{2}-integrands with efficient approximation methods in order to approximate multivariate non-periodic functions on the high-dimensional cube left[ -frac{1}{2},frac{1}{2}right] ^{d}. Our concept allows to determine conditions on the d-variate torus-to-cube transformations {psi :left[ -frac{1}{2},frac{1}{2}right] ^{d}rightarrow left[ -frac{1}{2},frac{1}{2}right] ^{d}} such that a non-periodic function is transformed into a smooth function in the Sobolev space {mathcal {H}}^{m}(mathbb {T}^{d}) when applying psi . We adapt L_{infty }(mathbb {T}^{d})- and L_{2}(mathbb {T}^{d})-approximation error estimates for single rank-1 lattice approximation methods and adjust algorithms for the fast evaluation and fast reconstruction of multivariate trigonometric polynomials on the torus in order to apply these methods to the non-periodic setting. We illustrate the theoretical findings by means of numerical tests in up to d=5 dimensions.

A posteriori error estimates and an adaptive finite element solution for the system of unsteady convection-diffusion-reaction equations in fluidized beds

The a posteriori error estimates for finite element approximations to the governing equations of heat and mass transfer in fluidized beds are derived in this work. These are a system of five time dependent coupled nonlinear convection-diffusion-reaction equations. Based on the variational formulation, computable residual based a posteriori error estimates are obtained. The time discretization has been done using the implicit Euler method. The a posteriori error estimates for all the five variables are derived using the total residual and error indicators due to spatial discretization, time discretization and linearization. An adaptive finite element solution of these model equations is computed and the performance is illustrated.

Lq-error estimates for approximation of irregular functionals of random vectors

AbstractIn Avikainen (2009, On irregular functionals of SDEs and the Euler scheme. Finance Stoch., 13, 381–401) the author showed that, for any $p,q \in [1,\infty )$, and any function $f$ of bounded variation in $\mathbb{R}$, it holds that $ \mathbb{E}[|f(X)-f(\widehat{X})|^{q}] \leq C(p,q) \mathbb{E}[|X-\widehat{X}|^{p}]^{\frac{1}{p+1}} $, where $X$ is a one-dimensional random variable with a bounded density, and $\widehat{X}$ is an arbitrary random variable. In this article we will provide multi-dimensional versions of this estimate for functions of bounded variation in $\mathbb{R}^{d}$, Orlicz–Sobolev spaces, Sobolev spaces with variable exponents and fractional Sobolev spaces. The main idea of our arguments is to use the Hardy–Littlewood maximal estimates and pointwise characterizations of these function spaces. We apply our main results to analyze the numerical approximation for some irregular functionals of the solution of stochastic differential equations.

Error estimates of finite volume method for Stokes optimal control problem

In this paper, we discuss a priori error estimates for the finite volume element approximation of optimal control problem governed by Stokes equations. Under some reasonable assumptions, we obtain optimal L^{2}-norm error estimates. The approximate orders for the state, costate, and control variables are O(h^{2}) in the sense of L^{2}-norm. Furthermore, we derive H^{1}-norm error estimates for the state and costate variables. Finally, we give some conclusions and future works.

A Priori Error Estimates for the Finite Element Approximation of a Nonsmooth Optimal Control Problem Governed by a Coupled Semilinear PDE-ODE System

A Priori Error Estimates for the Finite Element Approximation of a Nonsmooth Optimal Control Problem Governed by a Coupled Semilinear PDE-ODE System

О методе Прагера-Синга оценки погрешности аппроксимации

The classic method by Prager and Synge ("hypercircle" method) for a posteriori error estimation in addressed from the viewpoint of the extension of the applicability domain (the range of analyzed problems and methods of the realization). The nonintrusive version of the approximation error estimation that implements the method by Prager and Synge for the arbitrary PDE system is presented. The semiheuristical variant of the Prager and Synge method may be related with the modern approaches for approximation error estimation on the ensemble of solutions obtained by the algorithms of different inner structure.

IMPROVING A POSTERIORI SPATIAL DISCRETIZATION ERROR ESTIMATION FOR SN TRANSPORT BY SINGULAR CHARACTERISTICS TRACKING

Previously, we have developed a novel spatial discretization error estimator, the “residual source” estimator, in which an error transport problem, analogous to the discretized transport equation, is solved to acquire an estimate of the error, with a residual term acting as a fixed source. Like all error estimators, the residual source estimator suffers inaccuracy and imprecision in the proximity of singular characteristics, lines across which the solution is irregular. Estimator performance worsens as the irregularities become more pronounced, especially so if the true solution itself is discontinuous. This work introduces a modification to the residual approximation procedure that seeks to reduce the adverse effects of the singular characteristics on the error estimate. A partial singular characteristic tracking scheme is implemented to reduce the portion of the error in the numerical solution born by irregularities in the true solution. This treated numerical solution informs the residual approximations. The partial singular characteristic tracking scheme greatly enhances the numerical solution for a problem with prominent singular characteristics. The residual approximation and resultant residual source error estimate are likewise improved by the scheme, which only incurs the computational cost of an extra inner iteration.

Оценка погрешности дискретной аппроксимации стационарной задачи радиационно-кондуктивного теплообмена в системе абсолютно черных стержней квадратного сечения

Studying heat transfer processes in periodic media containing vacuum interlayers or cavities, heat through which is transferred by radiation, is of significant interest for applications. Direct numerical solution of such problems involves considerable computational efforts and becomes almost impossible for systems containing a large number of heat conducting elements, especially for 2D and 3D structures. Therefore, it is of issue to develop effective approximate solution methods for such problems. This publication continues a series of studies on developing and substantiating special discrete and asymptotic approximations of radiant-and-conduction heat transfer problems in periodic systems of heat conducting elements separated by vacuum. In this study, the stationary radiant-and-conduction heat transfer problem in a system of absolutely black square rods is considered. The sought quantity is the absolute temperature, which is found from the solution of the boundary-value problem for the stationary heat conduction equation with nonlinear nonlocal boundary conditions describing radiant heat transfer between the rods through vacuum interlayers. A special discrete approximation of this problem leading to the system of linear algebraic equations with respect to the fourth power of the temperature is presented. The solution of this system as approximation of the mean temperature over the rod cross-section is described. The discrete approximation error estimate as a function of the square rod side length (the small parameter of the problem) and the thermal conductivity coefficient has been obtained. The obtained error estimate proves applicability of the discrete approximation for materials with a high thermal conductivity coefficient.

Approximation of positive operators by analytic vectors

We give the estimates of approximation errors while approximating of a positive operator $A$ in a Banach space by analytic vectors. Our main results are formulated in the form of Bernstein and Jackson type inequalities with explicitly calculated constants. We consider the classes of invariant subspaces ${\mathcal E}_{q,p}^{\nu,\alpha}(A)$ of analytic vectors of $A$ and the special scale of approximation spaces $\mathcal {B}_{q,p,\tau}^{s,\alpha}(A)$ associated with the complex degrees of positive operator. The approximation spaces are determined by $E$-functional, that plays a similar role as the module of smoothness. We show that the approximation spaces can be considered as interpolation spaces generated by $K$-method of real interpolation. The constants in the Bernstein and Jackson type inequalities are expressed using the normalization factor.

Error Estimate for Discrete Approximation of the Radiative-Conductive Heat Transfer Problem in a System of Absolutely Black Rods

We prove an error estimate of order $$ O\left(\sqrt{\varepsilon }/\uplambda \right) $$ for a discrete approximation of the radiative-conductive heat transfer problem in a system of absolutely black rods bundled in a square box; here, e is the rod diameter and λ is the heat transfer coefficient.

Pointwise error estimates for 𝐶⁰ interior penalty approximation of biharmonic problems

The aim of this paper is to derive pointwise global and local best approximation type error estimates for biharmonic problems using the C 0 C^0 interior penalty method. The analysis uses the technique of dyadic decompositions of the domain, which is assumed to be a convex polygon. The proofs require local energy estimates and new pointwise Green’s function estimates for the continuous problem which has independent interest.

Error estimates for Galerkin finite element approximations of time-fractional nonlocal diffusion equation

This paper is concerned to study the well-posedness, the Mittag–Leffler stability of solutions of time-fractional nonlocal reaction–diffusion equation in bounded domain We use the Faedo–Galerkin approximation method with initial data in to show a solution in Further, we construct the suitable Lyapunov function to ensure that a solution of the proposed model is the Mittag–Leffler stable. Furthermore, we fully discretize the Galerkin finite element method for the proposed time-fractional model in two-space dimension. Here, time-fractional derivative is given in Caputo's sense and discretized using approximation scheme. Error analysis of the proposed numerical method is performed and error bounds are obtained for the error measured in norm. All the theoretical results are validated with several constructive numerical examples.

On the generalized Burgers-Huxley equation: Existence, uniqueness, regularity, global attractors and numerical studies

<p style='text-indent:20px;'>In this work, we consider the forced generalized Burgers-Huxley equation and establish the existence and uniqueness of a global weak solution using a Faedo-Galerkin approximation method. Under smoothness assumptions on the initial data and external forcing, we also obtain further regularity results of weak solutions. Taking external forcing to be zero, a positivity result as well as a bound on the classical solution are also established. Furthermore, we examine the long-term behavior of solutions of the generalized Burgers-Huxley equations. We first establish the existence of absorbing balls in appropriate spaces for the semigroup associated with the solutions and then show the existence of a global attractor for the system. The inviscid limits of the Burgers-Huxley equations to the Burgers as well as Huxley equations are also discussed. Next, we consider the stationary Burgers-Huxley equation and establish the existence and uniqueness of weak solution by using a Faedo-Galerkin approximation technique and compactness arguments. Then, we discuss about the exponential stability of stationary solutions. Concerning numerical studies, we first derive error estimates for the semidiscrete Galerkin approximation. Finally, we present two computational examples to show the convergence numerically.

A Study of Error Estimation for Second Order Fredholm Integro-Differential Equations

In this work, we study efficient asymptotically correct a posteriori error estimates for the numerical approximation of second order Fredholm integro-differential equations. We use the defect correction principle to find the deviation of the error estimation and show that collocation method by using m degree piecewise polynomial provides order $$\mathcal{O}(h^{m+2})$$ for the deviation of the error. Also, the theoretical behavior is tested on examples and it is shown that the numerical results confirm theoretical analysis.

Improved error estimates for optimal control of the Stokes problem with pointwise tracking in three dimensions

This work is motivated by recent interest in the topic of pointwise tracking type optimal control problems for the Stokes problem. Pointwise tracking consists of point evaluations in the objective functional which lead to Dirac measures appearing as source terms of the adjoint problem. Considering bounds for the control allows for improved regularity results for the exact solution and improved approximation error estimates of its numerical counterpart. We show a sub-optimal convergence result in three dimensions that nonetheless improves the results known from the literature. Finally, we offer supporting numerical experiments and insights towards optimal approximation error estimates.

Strong approximation of monotone stochastic partial differential equations driven by multiplicative noise

We establish a general theory of optimal strong error estimation for numerical approximations of a second-order parabolic stochastic partial differential equation with monotone drift driven by a multiplicative infinite-dimensional Wiener process. The equation is spatially discretized by Galerkin methods and temporally discretized by drift-implicit Euler and Milstein schemes. By the monotone and Lyapunov assumptions, we use both the variational and semigroup approaches to derive a spatial Sobolev regularity under the $L_\omega^p L_t^\infty \dot H^{1+\gamma}$-norm and a temporal H\"older regularity under the $L_\omega^p L_x^2$-norm for the solution of the proposed equation with an $\dot H^{1+\gamma}$-valued initial datum for $\gamma\in [0,1]$. Then we make full use of the monotonicity of the equation and tools from stochastic calculus to derive the sharp strong convergence rates $O(h^{1+\gamma}+\tau^{1/2})$ and $O(h^{1+\gamma}+\tau^{(1+\gamma)/2})$ for the Galerkin-based Euler and Milstein schemes, respectively.

Finite element approach to linear parabolic pointwise control problems of incomplete data

In this paper we give a priori error estimates for finite element approximations of linear parabolic problems with pointwise control and incomplete data. We discretise the optimal control problem by using piecewise linear and continuous finite elements for the space discretisation of the state, and we use the backward Euler scheme for time discretisation. We prove a priori error estimates for the state, the adjoint-state as well as for the low-regret pointwise optimal control.

Finite Element Error Estimates on Geometrically Perturbed Domains

We develop error estimates for the finite element approximation of elliptic partial differential equations on perturbed domains, i.e. when the computational domain does not match the real geometry. The result shows that the error related to the domain can be a dominating factor in the finite element discretization error. The main result consists of H^1- and L^2-error estimates for the Laplace problem. Theoretical considerations are validated by a computational example.

EXtended HDG Methods for Second Order Elliptic Interface Problems

In this paper, we propose two arbitrary order eXtended hybridizable Discontinuous Galerkin (X-HDG) methods for second order elliptic interface problems in two and three dimensions. The first X-HDG method applies to any piecewise $C^2$ smooth interface. It uses piecewise polynomials of degrees $k$ $(k>= 1)$ and $k-1$ respectively for the potential and flux approximations in the interior of elements inside the subdomains, and piecewise polynomials of degree $ k$ for the numerical traces of potential on the inter-element boundaries inside the subdomains. Double value numerical traces on the parts of interface inside elements are adopted to deal with the jump condition. The second X-HDG method is a modified version of the first one and applies to any fold line/plane interface, which uses piecewise polynomials of degree $ k-1$ for the numerical traces of potential. The X-HDG methods are of the local elimination property, then lead to reduced systems which only involve the unknowns of numerical traces of potential on the inter-element boundaries and the interface. Optimal error estimates are derived for the flux approximation in $L^2$ norm and for the potential approximation in piecewise $H^1$ seminorm without requiring "sufficiently large" stabilization parameters in the schemes. In addition, error estimation for the potential approximation in $L^2$ norm is performed using dual arguments. Finally, we provide several numerical examples to verify the theoretical results.