Sergey Repin

Mixed Finite Element Method on Polygonal and Polyhedral Meshes

A new mixed finite element method for the diffusion equations on general polygonal and polyhedral meshes is presented. The basis vector functions in macrocells are designed by solving the local mixed finite element problems with the lowest order Raviart-Thomas elements. Numerical results for the Poisson equation on distorted prismatic meshes are given.

A posteriori error estimation for the Poisson equation with mixed Dirichlet/Neumann boundary conditions

The present work is devoted to the a posteriori error estimation for the Poisson equation with mixed Dirichlet/Neumann boundary conditions. Using the duality technique we derive a reliable and efficient a posteriori error estimator that measures the error in the energy norm. The estimator can be used in assessing the error of any approximate solution which belongs to the Sobolev space H1, independently of the discretization method chosen. Only two global constants appear in the definition of the estimator; both constants depend solely on the domain geometry, and the estimator is quite nonsensitive to the error in the constants evaluation. It is also shown how accurately the estimator captures the local error distribution, thus, creating a base for a justified adaptivity of an approximation.

Two-Sided A Posteriori Error Estimates for Mixed Formulations of Elliptic Problems

The present work is devoted to the a posteriori error estimation for mixed approximations of linear self-adjoint elliptic problems. New guaranteed upper and lower bounds for the error measured in the natural product norm are derived, and individual sharp upper bounds are obtained for approximation errors in each of the physical variables. All estimates are reliable and valid for any approximate solution from the class of admissible functions. The estimates contain only global constants depending solely on the domain geometry and the given operators. Moreover, it is shown that, after an appropriate scaling of the coordinates and the equation, the ratio of the upper and lower bounds for the error in the product norm never exceeds 3. The possible methods of finding the approximate mixed solution in the class of admissible functions are discussed. The estimates are computationally very cheap and can also be used for the indication of the local error distribution. As applications, the diffusion problem as well as the problem of linear elasticity are considered.

DUALITY BASED A POSTERIORI ERROR ESTIMATES FOR HIGHER ORDER VARIATIONAL INEQUALITIES WITH POWER GROWTH FUNCTIONALS

We consider variational inequalities of higher order with p-growth potentials over a domain in the plane by the way including the obstacle problem for a plate with power hardening law. Using duality methods we prove a posteriori error estimates of functional type for the difference of the exact solution and any admissible comparision function.

A Posteriori Estimates for Cost Functionals of Optimal Control Problems

A posteriori analysis has become an inherent part of numerical mathematics. Methods of a posteriori error estimation for finite element approximations were actively developed in the last two decades (see, e.g., [1, 2, 3, 13] and references therein). For problems in the theory of optimization these methods started receiving attention much later. In particular, for optimal control problems governed by PDE’s the literature on this matter is rather scarce. In this work, we present a new approach to a class of optimal control problems associated with elliptic type partial differential equations. In the framework of this approach, we obtain directly computable upper bounds for the cost functionals of the respective optimal control problems. Let Ω ∈ R be a Lipschitz domain with boundary Γ := ∂Ω. Problem 1. Given ψ ∈ L∞(Ω), y ∈ L2(Ω), u ∈ L2(Ω), f ∈ L2(Ω), and a ∈ R+, consider the distributed control problem

A posteriori error estimates for approximations of evolutionary convection–diffusion problems

We derive computable upper bounds for the difference between an exact solution of the evolutionary convection-diffusion problem and an approximation of this solution. The estimates are obtained by certain transformations of the integral identity that defines the generalized solution. These estimates depend on neither special properties of the exact solution nor its approximation and involve only global constants coming from embedding inequalities. The estimates are first derived for functions in the corresponding energy space, and then possible extensions to classes of piecewise continuous approximations are discussed. Bibliography: 7 titles.

Estimates for the deviation from exact solutions of variational problems with power growth functionals

We study the nonlinear power growth variational problem J_{\alpha}[w]:=\int_{\Omega}\left[\frac{1}{\alpha}\left|\nabla w\right|^{\alpha}-fw\right]dx\rightarrow\textrm{min} and establish directly computable estimates for the deviation from exact solutions. In the case of superquadratic growth, these estimates are given in terms of the energy norm, in the subquadratic case we pass to estimates for the solution of the dual variational problem. Various boundary conditions are included in our considerations.

Computable majorants of the limit load in Hencky’s plasticity problems

Abstract We propose a new method for analyzing the limit (safe) load of elastoplastic media governed by the Hencky plasticity law and deduce fully computable bounds of this load. The main idea of the method is based on a combination of kinematic approach and new estimates of the distance to the set of divergence free fields. We show that two sided bounds of the limit load are sharp and the computational efficiency of the method is confirmed by numerical experiments.

Overview of Other Results and Open Problems

This chapter presents an overview of results related to error control methods, which were not considered in previous chapters. In the first part, we discuss possible extensions of the theory exposed in Chaps. 3 and 4 to nonconforming approximations and certain classes of nonlinear problems. Also, we shortly discuss some results related to explicit evaluation of modeling errors. The remaining part of the chapter is devoted to a posteriori estimates of errors in iteration methods. Certainly, the overview is not complete. A posteriori error estimation methods are far from having been fully explored and this subject contains many unsolved problems and open questions, some of which we formulate in the last section.

Errors Arising in Computer Simulation Methods

The goal of this introductory chapter is to discuss in general terms different classes of errors arising in computer simulation methods and to direct the reader to the chapters and sections of the book where these errors are analyzed. Moreover, we describe the error estimation methodology applied in this book.