Two-sided a posteriori error estimates for linear elliptic problems with mixed boundary conditions

The paper is devoted to verification of accuracy of approximate solutions obtained in computer simulations. This problem is strongly related to a posteriori error estimates, giving computable bounds for computational errors and detecting zones in the solution domain where such errors are too large and certain mesh refinements should be performed. A mathematical model consisting of a linear elliptic (reaction-diffusion) equation with a mixed Dirichlet/Neumann/Robin boundary condition is considered in this work. On the base of this model, we present simple technologies for straightforward constructing computable upper and lower bounds for the error, which is understood as the difference between the exact solution of the model and its approximation measured in the corresponding energy norm. The estimates obtained are completely independent of the numerical technique used to obtain approximate solutions and are “flexible” in the sense that they can be, in principle, made as close to the true error as the resources of the used computer allow. This work was supported by the Academy Research Fellowship No. 208628 from the Academy of Finland.     

Existence of solutions for nonlinear singular boundary value problems

Existence of solutions to the two-point boundary value problem (p(t)y')' = q(t)f(t, y,p(t)y'), y(l) = 0, lim_t→0+ p(t)y'(t) = 0 is established under a variety of conditions. Here p(0) = 0 is allowed, and q is not assumed to be continuous at 0, so the problem may be doubly singular. In addition, the Dirichlet problem for this differential equation is investigated     

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     

New interpolation error estimates and a posteriori error analysis for linear parabolic interface problems

We derive residual‐based a posteriori error estimates of finite element method for linear parabolic interface problems in a two‐dimensional convex polygonal domain. Both spatially discrete and fully discrete approximations are analyzed. While the space discretization uses finite element spaces that are allowed to change in time, the time discretization is based on the backward Euler approximation. The main ingredients used in deriving a posteriori estimates are new Clément type interpolation estimates and an appropriate adaptation of the elliptic reconstruction technique introduced by (Makridakis and Nochetto, SIAM J Numer Anal 4 (2003), 1585–1594). We use only an energy argument to establish a posteriori error estimates with optimal order convergence in the ‐norm and almost optimal order in the ‐norm. The interfaces are assumed to be of arbitrary shape but are smooth for our purpose. Numerical results are presented to validate our derived estimators. © 2016 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 33: 570–598, 2017     

A posteriori error estimates of spectral method for optimal control problems governed by parabolic equations

In this paper,we investigate the Legendre Galerkin spectral approximation of quadratic optimal control problems governed by parabolic equations.A spectral approximation scheme for the parabolic optimal control problem is presented.We obtain a posteriori error estimates of the approximated solutions for both the state and the control.     

A note on functional a posteriori estimates for elliptic optimal control problems

In this work, new results on functional type a posteriori estimates for elliptic optimal control problems with control constraints are presented. More precisely, we derive new, sharp, guaranteed, and fully computable lower bounds for the cost functional in addition to the already existing upper bounds. Using both, the lower and the upper bounds, we arrive at two‐sided estimates for the cost functional. We prove that these bounds finally lead to sharp, guaranteed and fully computable upper estimates for the discretization error in the state and the control of the optimal control problem. First numerical tests are presented confirming the efficiency of the a posteriori estimates derived. © 2016 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 33: 403–424, 2017     

Functional a posteriori error estimates for discontinuous Galerkin approximations of elliptic problems

In this article, we develop functional a posteriori error estimates for discontinuous Galerkin (DG) approximations of elliptic boundary‐value problems. These estimates are based on a certain projection of DG approximations to the respective energy space and functional a posteriori estimates for conforming approximations developed by S. Repin (see e.g., Math Comp 69 (2000) 481–500). On these grounds, we derive two‐sided guaranteed and computable bounds for the errors in “broken” energy norms. A series of numerical examples presented confirm the efficiency of the estimates. © 2008 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2009     

Local a posteriori estimates for pointwise gradient errors in finite element methods for elliptic problems

We prove local a posteriori error estimates for pointwise gradient errors in finite element methods for a second-order linear elliptic model problem. First we split the local gradient error into a computable local residual term and a weaker global norm of the finite element error (the ``pollution term'). Using a mesh-dependent weight, the residual term is bounded in a sharply localized fashion. In specific situations the pollution term may also be bounded by computable residual estimators. On nonconvex polygonal and polyhedral domains in two and three space dimensions, we may choose estimators for the pollution term which do not employ specific knowledge of corner singularities and which are valid on domains with cracks. The finite element mesh is only required to be simplicial and shape-regular, so that highly graded and unstructured meshes are allowed.

     

A posteriori error estimates for a compressible Stokes system

A linearized compressible viscous Stokes system is considered. The a posteriori error estimates are defined and compared with the true error. They are shown to be globally upper and locally lower bounds for the true error of the finite element solution. Some numerical examples are given, showing an efficiency of the estimator. © 2004 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 20: 412–431, 2004.     

Sharp upper global a posteriori error estimates for nonlinear elliptic variational problems

The paper is devoted to the problem of verification of accuracy of approximate solutions obtained in computer simulations. This problem is strongly related to a posteriori error estimates, giving computable bounds for computational errors and detecting zones in the solution domain where such errors are too large and certain mesh refinements should be performed. A mathematical model embracing nonlinear elliptic variational problems is considered in this work. Based on functional type estimates developed on an abstract level, we present a general technology for constructing computable sharp upper bounds for the global error for various particular classes of elliptic problems. Here the global error is understood as a suitable energy type difference between the true and computed solutions. The estimates obtained are completely independent of the numerical technique used to obtain approximate solutions, and are sharp in the sense that they can be, in principle, made as close to the true error as resources of the used computer allow. The latter can be achieved by suitably tuning the auxiliary parameter functions, involved in the proposed upper error bounds, in the course of the calculations.     

Elliptic reconstruction and a posteriori error estimates for parabolic optimal control problems

In this article, a semidiscrete finite element method for parabolic optimal control problems is investigate. By using elliptic reconstruction, a posteriori error estimates for finite element discretizations of optimal control problem governed by parabolic equations with integral constraints are derived.     

Local error estimates for moderately smooth problems: Part I – ODEs and DAEs

The paper consists of two parts. In the first part, we propose a procedure to estimate local errors of low order methods applied to solve initial value problems in ordinary differential equations (ODEs) and index 1 differential-algebraic equations (DAEs). Based on the idea of defect correction we develop local error estimates for the case when the problem data is only moderately smooth. Numerical experiments illustrate the performance of the mesh adaptation based on the error estimation developed in this paper. In the second part of the paper, we will consider the estimation of local errors in context of stochastic differential equations with small noise. AMS subject classification (2000)  65L06, 65L80, 65L50, 65L05     

二阶线性常微分方程的两点边值问题的泰勒展开式解法

本文用泰勒展开公式求解二阶线性常微分方程的两点边值问题.首先将两点边值问题化为一个F redho lm积分方程,进一步通过泰勒展开公式化F redho lm积分方程为线性方程组,利用G ramm er法则可求得问题的近似解.     

Existence and convergence of solutions of singular boundary value problems for second order ordinary differential equations and applications

In this paper,we consider the singular boundary value problems for second order quasilinear ordinary differential equations and prove existence and convergence on second order perturbation terms.The result is applied to solve the Riemann problem for 2×2 hyperbolic conservation laws,which is a partial differential equation arising in applied mathematical area.     

Defect correction and a posteriori error estimation of Petrov-Galerkin methods for nonlinear Volterra integro-differential equations

We present two defect correction schemes to accelerate the Petrov-Galerkin finite element methods [19] for nonlinear Volterra integro-differential equations. Using asymptotic expansions of the errors, we show that the defect correction schemes can yield higher order approximations to either the exact solution or its derivative. One of these schemes even does not impose any extra regularity requirement on the exact solution. As by-products, all of these higher order numerical methods can also be used to form a posteriori error estimators for accessing actual errors of the Petrov-Galerkin finite element solutions. Numerical examples are also provided to illustrate the theoretical results obtained in this paper.     

Implicit a posteriori error estimates for the Maxwell equations

An implicit a posteriori error estimation technique is presented and analyzed for the numerical solution of the time-harmonic Maxwell equations using Nédélec edge elements. For this purpose we define a weak formulation for the error on each element and provide an efficient and accurate numerical solution technique to solve the error equations locally. We investigate the well-posedness of the error equations and also consider the related eigenvalue problem for cubic elements. Numerical results for both smooth and non-smooth problems, including a problem with reentrant corners, show that an accurate prediction is obtained for the local error, and in particular the error distribution, which provides essential information to control an adaptation process. The error estimation technique is also compared with existing methods and provides significantly sharper estimates for a number of reported test cases.

     

Cubic spline for a class of singular two-point boundary value problems

In this paper we have presented a method based on cubic splines for solving a class of singular two-point boundary value problems. The original differential equation is modified at the singular point then the boundary value problem is treated by using cubic spline approximation. The tridiagonal system resulting from the spline approximation is efficiently solved by Thomas algorithm. Some model problems are solved, and the numerical results are compared with exact solution.     

An adaptive strategy for elliptic problems including a posteriori controlled boundary approximation

We derive a posteriori error estimates for the approximation of linear elliptic problems on domains with piecewise smooth boundary. The numerical solution is assumed to be defined on a Finite Element mesh, whose boundary vertices are located on the boundary of the continuous problem. No assumption is made on a geometrically fitting shape.

A posteriori error estimates are given in the energy norm and the -norm, and efficiency of the adaptive algorithm is proved in the case of a saturated boundary approximation. Furthermore, a strategy is presented to compute the effect of the non-discretized part of the domain on the error starting from a coarse mesh. This especially implies that parts of the domain, where the measured error is small, stay non-discretized. The presented algorithm includes a stable path following to supply a sufficient polygonal approximation of the boundary, the reliable computation of the a posteriori estimates and a mesh adaptation based on Delaunay techniques. Numerical examples illustrate that errors outside the initial discretization will be detected.