A posteriori error estimates of discontinuous Galerkin methods for non-standard Volterra integro-differential equations

^** Email: jingtang{at}lsec.cc.ac.cn^*** Email: hermann{at}math.mun.ca In this paper we establish a posteriori error estimates forthe discontinuous Galerkin (DG) method applied to linear, semilinearand non-standard (non-linear) Volterra integro-differentialequations. We also present an analysis of the DG method withquadrature for the memory term. Numerical experiments basedon three integro-differential equations are used to illustratevarious aspects of the error analysis.     

Richardson extrapolation and defect correction of mixed finite element methods for integro-differential equations in porous media

Asymptotic error expansions in the sense of L ^∞-norm for the Raviart-Thomas mixed finite element approximation by the lowest-order rectangular element associated with a class of parabolic integro-differential equations on a rectangular domain are derived, such that the Richardson extrapolation of two different schemes and an interpolation defect correction can be applied to increase the accuracy of the approximations for both the vector field and the scalar field by the aid of an interpolation postprocessing technique, and the key point in deriving them is the establishment of the error estimates for the mixed regularized Green’s functions with memory terms presented in R. Ewing at al., Int. J. Numer. Anal. Model 2 (2005), 301–328. As a result of all these higher order numerical approximations, they can be used to generate a posteriori error estimators for this mixed finite element approximation. This project was supported in part by the Special Funds for Major State Basic Research Project (2007CB8149), the National Natural Science Foundation of China (10471103 and 10771158), the Social Science Foundation of the Ministry of Education of China (Numerical methods for convertible bonds, 06JA630047), the NSERC, Tianjin Natural Science Foundation (07JCYBJC14300), and Tianjin University of Finance and Economics.     

An h-p Petrov-Galerkin finite element method for linear Volterra integro-differential equations

We analyze an h-p version Petrov-Galerkin finite element method for linear Volterra integrodifferential equations. We prove optimal a priori error bounds in the L ²- and H ¹-norm that are explicit in the time steps, the approximation orders and in the regularity of the exact solution. Numerical experiments confirm the theoretical results. Moreover, we observe that the numerical scheme superconverges at the nodal points of the time partition.     

Numerical Solutions for second-kind Volterra integral equations by Galerkin methods

In this paper, we study the global convergence for the numerical solutions of nonlinear Volterra integral equations of the second kind by means of Galerkin finite element methods. Global superconvergence properties are discussed by iterated finite element methods and interpolated finite element methods. Local superconvergence and iterative correction schemes are also considered by iterated finite element methods. We improve the corresponding results obtained by collocation methods in the recent papers [6] and [9] by H. Brunner, Q. Lin and N. Yan. Moreover, using an interpolation post-processing technique, we obtain a global superconvergence of the O(h ^2r )-convergence rate in the piecewise-polynomial space of degree not exceeding (r–1). As a by-product of our results, all these higher order numerical methods can also provide an a posteriori error estimator, which gives critical and useful information in the code development.     

Boundary value methods for Volterra integral and integro-differential equations

New and effective quadrature rules generated by boundary value methods are introduced. We employ the introduced quadrature rules to construct quadrature methods for the second kind Volterra integral equations and Volterra integro-differential equations. These methods are shown to be effective and possess excellent convergence properties. The nonlinear multigrid method is applied to solve the discrete systems derived from the introduced numerical scheme. Numerical simulations are presented and confirm the efficiency and accuracy of the methods.     

Tension spline collocation methods for singularly perturbed Volterra integro-differential and Volterra integral equations

We consider the numerical discretization of singularly perturbed Volterra integro-differential equations (VIDE)

(*)

and Volterra integral equations (VIE)

(**)

by tension spline collocation methods in certain tension spline spaces, where is a small parameter satisfying 0<1, and q₁, q₂, g and K are functions sufficiently smooth on their domains to ensure that Eqs. (*) and (**) posses a unique solution.We give an analysis of the global convergence properties of a new tension spline collocation solution for 0<1 for singularly perturbed VIDE and VIE; thus, extending the existing theory for =1 to the singularly perturbed case.     

Stability of Runge-Kutta-Pouzet methods for Volterra integro-differential equations with delays

This paper is concerned with the study of the stability of Runge-Kutta-Pouzet methods for Volterra integro-differential equations with delays. We are interested in the comparison between the analytical and numerical stability regions. First, we focus on scalar equations with real coefficients. It is proved that all Gauss-Pouzet methods can retain the asymptotic stability of the analytical solution. Then, we consider the multidimensional case. A new stability condition for the stability of the analytical solution is given. Under this condition, the asymptotic stability of Gauss-Pouzet methods is investigated.      

Convergence and stability of extended block boundary value methods for Volterra delay integro-differential equations

In this paper, we construct a class of extended block boundary value methods (B₂VMs) for Volterra delay integro-differential equations and analyze the convergence and stability of the methods. It is proven under the classical Lipschitz condition that an extended B₂VM is convergent of order p if the underlying boundary value methods (BVM) has consistent order p. The analysis shows that a B₂VM extended by an A-stable BVM can preserve the delay-independent stability of the underlying linear systems. Moreover, under some suitable conditions, the extended B₂VMs can also keep the delay-dependent stability of the underlying linear systems. In the end, we test the computational effectiveness by applying the introduced methods to the Volterra delay dynamical model of two interacting species, where the theoretical precision of the methods is further verified.     

Block boundary value methods for solving Volterra integral and integro-differential equations

Reducible quadrature rules generated by boundary value methods are considered in block version and applied to solve the second kind Volterra integral equations and Volterra integro-differential equations. These extended block boundary value methods are shown to possess both excellent stability properties and high accuracy for Volterra-type equations. Numerical experiments are presented and the efficiency, accuracy and stability of the schemes are confirmed.     

A novel numerical method and its convergence for nonlinear delay Volterra integro-differential equations

In this paper, we suggest a convergent numerical method for solving nonlinear delay Volterra integro-differential equations. First, we convert the problem into a continuous-time optimization problem and then use a shifted pseudospectral method to discrete the problem. Having solved the last problem, we can achieve the pointwise and continuous approximate solutions for the main delay Volterra integro-differential equations. Here, we analyze the convergence of the method and solve some numerical examples to show the efficiency of the method.     

Stability analysis of Runge-Kutta methods for nonlinear neutral delay integro-differential equations

The sufficient conditions for the stability and asymptotic stability of Runge-Kutta methods for nonlinear neutral delay integro-differential equations are derived. A numerical test that confirms the theoretical results is given in the end.     

Parallel computations and numerical simulations for nonlinear systems of Volterra integro-differential equations

We investigate thalamo-cortical systems that are modeled by nonlinear Volterra integro-differential equations of convolution type. We divide the systems into smaller subsystems in such a way that each of them is solved separately by a processor working independently of other processors results of which are shared only once in the process of computations. We solve the subsystems concurrently in a parallel computing environment and present results of numerical experiments, which show savings in the run time and therefore efficiency of our approach. For our numerical simulations, we apply different numbers np of processors and each case shows that the run time decreases with increasing np. The optimal speed-up is obtained with np = N, where N is the (moderate) number of equations in the thalamo-cortical model.     

The combined Laplace transform-Adomian decomposition method for handling nonlinear Volterra integro-differential equations

In this work, a combined form of the Laplace transform method with the Adomian decomposition method is developed for analytic treatment of the nonlinear Volterra integro-differential equations. The combined method is capable of handling both equations of the first and second kind. Illustrative examples will be examined to support the proposed analysis.     

A posteriori error estimates of mixed DG finite element methods for linear parabolic equations

In this article, we analyse a posteriori error estimates of mixed finite element discretizations for linear parabolic equations. The space discretization is done using the order λ?≥?1 Raviart–Thomas mixed finite elements, whereas the time discretization is based on discontinuous Galerkin (DG) methods (r?≥?1). Using the duality argument, we derive a posteriori l ^∞(L ²) error estimates for the scalar function, assuming that only the underlying mesh is static.     

A posteriori error estimation for a defect correction method applied to conduction convection problems

We present a posteriori error estimate for a defect correction method for approximating solutions of the stationary conduction convection problems in two dimension. The defect correction method is aiming at small viscosity ν. A reliable a posteriori error estimation is derived for the defect correction method. Finally, two numerical examples validate our theoretical results. The first example is a problem with known solution and the second example is a physical model of square cavity stationary flow. © 2012 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 2013     

B-Spline collocation method for linear and nonlinear Fredholm and Volterra integro-differential equations

A numerical method based on quintic B-spline has been developed to solve the linear and nonlinear Fredholm and Volterra integro-differential equations up to order 4. The solution and its derivatives are collocated by quintic B-spline and then the integral equation is approximated by the 4-points Gauss–Turán quadrature formula with respect to the weight function Legendre. The error analysis of proposed numerical method is studied theoretically. Numerical results are given to illustrate the efficiency of the proposed method which shows that our method can be applied for large values of N. The results are compared with the results obtained by other methods which show that our method is accurate.     

Wavelets method for solving systems of nonlinear singular fractional Volterra integro-differential equations

This paper presents a computational method for solving a class of system of nonlinear singular fractional Volterra integro-differential equations. First, existences of a unique solution for under studying problem is proved. Then, shifted Chebyshev polynomials and their properties are employed to derive a general procedure for forming the operational matrix of fractional derivative for Chebyshev wavelets. The application of this operational matrix for solving mentioned problem is explained. In the next step, the error analysis of the proposed method is investigated. Finally, some examples are included for demonstrating the efficiency of the proposed method.     

A posteriori error estimation and adaptive mesh-refinement techniques

We analyse three different a posteriori error estimators for elliptic partial differential equations. They are based on the evaluation of local residuals with respect to the strong form of the differential equation, on the solution of local problems with Neumann boundary conditions, and on the solution of local problems with Dirichlet boundary conditions. We prove that all three are equivalent and yield global upper and local lower bounds for the true error. Thus adaptive mesh-refinement techniques based on these estimators are capable to detect local singularities of the solution and to appropriately refine the grid near these singularities. Some numerical examples prove the efficiency of the error estimators and the mesh-refinement techniques.     

A posteriori error estimation for elliptic partial differential equations with small uncertainties

In this article, a finite element error analysis is performed on a class of linear and nonlinear elliptic problems with small uncertain input. Using a perturbation approach, the exact (random) solution is expanded up to a certain order with respect to a parameter that controls the amount of randomness in the input and discretized by finite elements. We start by studying a diffusion (linear) model problem with a random coefficient characterized via a finite number of random variables. The main focus of the article is the derivation of a priori and a posteriori error estimates of the error between the exact and approximate solution in various norms, including goal‐oriented error estimation. The analysis is then extended to a class of nonlinear problems. We finally illustrate the theoretical results through numerical examples, along with a comparison with the Stochastic Collocation method in terms of computational costs. © 2015 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 32: 175–212, 2016