arXiv subjects

Sergey I. Repin

Publications and source records attributed to Sergey I. Repin.

6 recordsLinked to original sources

arXiv2021

Functional a posteriori error estimates for parabolic obstacle problems

The paper is concerned with functional type a posteriori estimates for the initial boundary value problem for a parabolic partial differential equation with an obstacle. We deduce a guaranteed and computable bound of the distance between the exact minimizer and any function from the admissible (energy) class of functions. Applications to the analysis of modeling errors caused by data implification are discussed. An important case of time incremental approximations is specially studied. Numerical examples presented in the last section show how the estimates work in practice.

math.AP↗

arXiv2020

Biharmonic obstacle problem: guaranteed and computable error bounds for approximate solutions

The paper is concerned with a free boundary problem generated by the biharmonic operator and an obstacle. The main goal is to deduce a fully guaranteed upper bound of the difference between the exact minimizer u and any function (approximation) from the corresponding energy class (which consists of the functions in $H^2$ satisfying the prescribed boundary conditions and the restrictions stipulated by the obstacle). For this purpose we use the duality method of the calculus of variations and general type error identities earlier derived for a wide class of convex variational problems. By this method, we define a combined primal--dual measure of error. It contains four terms of different nature. Two of them are the norms of the difference between the exact solutions (of the direct and dual variational problems) and corresponding approximations. Two others are nonlinear measures, related to approximation of the coincidence set (they vanish if the coincidence set defined by means of the approximate solution coincides with the exact one). The measure satisfies the error identity, which right hand side depends on approximate solutions only and, therefore, is fully computable. Thus, the identity provides direct estimation of the primal--dual errors. However, it contains a certain restriction on the form of the dual approximation. In the second part of the paper, we present a way to skip the restriction. As a result, we obtain a fully guaranteed and directly computable error majorant valid for a wide class of approximations regardless of the method used for their construction. The estimates are verified in a series of tests with different approximate solutions. Some of them are quite close to the exact solution and others are rather coarse and have coincidence sets that differ much from the exact one. The results show that the estimates are robust and effective in all the cases.

math.AP↗

arXiv2018

Thin obstacle problem: estimates of the distance to the exact solution

We consider elliptic variational inequalities generated by obstacle type problems with thin obstacles. For this class of problems, we deduce estimates of the distance (measured in terms of the natural energy norm) between the exact solution and any function that satisfies the boundary condition and is admissible with respect to the obstacle condition (i.e., it is valid for any approximation regardless of the method by which it was found). Computation of the estimates does not require knowledge of the exact solution and uses only the problem data and an approximation. The estimates provide guaranteed upper bounds of the error (error majorants) and vanish if and only if the approximation coincides with the exact solution. In the last section, the efficiency of error majorants is confirmed by an example, where the exact solution is known.

math.AP↗

arXiv2015

A fast iteration method for solving elliptic problems with quasiperiodic coefficients

The paper suggests a preconditioning type method for fast solving of elliptic equations with oscillating quasiperiodic coefficients $A_\epsilon$ specified by the small parameter $\epsilon>0$. We use an iteration method generated by an elliptic operator, associated with a certain simplified (e.g., homogenized) problem. On each step of this procedure it is required to solve an auxiliary elliptic boundary value problem with non--oscillating coefficients $A_0$. All the information related to complicated coefficients of the original differential problem is encompasses in the linear functional, which forms the right hand side of the auxiliary problem. Therefore, explicit inversion of the original operator associated with oscillating coefficients is avoided. The only operation used instead is multiplication of the operator by a vector (vector function), which can be efficiently performed due to the low-rank QTT tensor operations with the rank parameter controlled by the given precision $\delta >0$ independent on the parameter $\epsilon$. We deduce two--sided a posteriori error estimates that do not use $A^{-1}_\epsilon$ and provide guaranteed two sided bounds of the distance to the exact solution of the original problem for any step of the iteration process. The second part is concerned with realisations of the iteration method. For a wide class of oscillating coefficients, we obtain sharp QTT rank estimates for the stiffness matrix in tensor representation. In practice, this leads to the logarithmic complexity scaling of the approximation and solution process in both the FEM grid-size, and $O(\vert\log\epsilon\vert)$ cost in terms of $\epsilon$. Numerical tests in 1D confirm the logarithmic complexity scaling of our method applied to a class of complicated quasiperiodic coefficients.

math.NA↗

arXiv2015

Exact constants in Poincare type inequalities for functions with zero mean boundary traces

In the paper, we investigate Poincare type inequalities for the functions having zero mean value on the whole boundary of a Lipschitz domain or on a measurable part of the boundary. We derive exact and easily computable constants for some basic domains (rectangles, cubes, and right triangles). In the last section, we derive an a estimate of the difference between the exact solutions of two boundary value problems. Constants in Poincare type inequalities enter these estimates, which provide guaranteed a posteriori error control.

math.AP↗

arXiv2010

A Posteriori Error Estimates for Nonconforming 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.

math.NA↗