Combined a posteriori modeling-discretization error estimate for elliptic problems with complicated interfaces

Abstract:We derive computable bounds of deviations from the exact solution of the stationary Oseen problem. They are applied to approximations generated by the Uzawa iteration method. Also, we derive an advanced form of the estimate, which takes into account approximation errors arising due to discretization of the boundary value problem, generated by the main step of the Uzawa method. Numerical tests con rm our theoretical results and show practical applicability of the estimates.

Section: Related To the Approximation Error And The Second Term Presementioning

confidence: 99%

A Posteriori Error Bounds for Approximations of the Oseen Problem and Applications to the Uzawa Iteration Algorithm

Nokka¹,

Repin²

2014

“…There are many different ways to obtain suitable reconstructions with minimal expenditures (concerning this point we refer to [21] where the reader will find a systematic discussion of computational aspects in the context of various boundary value problems). Error majorants of this type can be also used for the evaluation of modeling errors (see [27,28]). Usually, the cost of a good estimate (with the efficiency index between 1 and 2) is comparable with the cost of a numerical solution.…”

Section: Remark 43mentioning

confidence: 99%

Rank Structured Approximation Method for Quasi-Periodic Elliptic Problems

Khoromskij

Repin

2017

Abstract:We consider an iteration method for solving an elliptic type boundary value problem Au = f , where a positive definite operator A is generated by a quasi-periodic structure with rapidly changing coefficients (a typical period is characterized by a small parameter ϵ). The method is based on using a simpler operator A (inversion of A is much simpler than inversion of A), which can be viewed as a preconditioner for A. We prove contraction of the iteration method and establish explicit estimates of the contraction factor q. Certainly the value of q depends on the difference between A and A . For typical quasi-periodic structures, we establish simple relations that suggest an optimal A (in a selected set of "simple" structures) and compute the corresponding contraction factor. Further, this allows us to deduce fully computable two-sided a posteriori estimates able to control numerical solutions on any iteration. The method is especially efficient if the coefficients of A admit low-rank representations and if algebraic operations are performed in tensor structured formats. Under moderate assumptions the storage and solution complexity of our approach depends only weakly (merely linear-logarithmically) on the frequency parameter ϵ .

“…It is computable, since the modeling error, which arises due to the replacement of the original problem with a simplified one, is also explicitly estimated. Estimates of this type for stationary diffusion problems with variable coefficients, which may sharply change values and have a complex behavior in the domain, have already been provided and discussed in [10].…”

Section: Tatiana Samrowskimentioning

confidence: 99%

“…Our analysis of the deviation from the exact solution is based upon the so-called functionaltype a posteriori error estimates (see [6][7][8][9][10] [9]) for the combined error norm…”

Section: Combined Error Estimatementioning

confidence: 99%

Combined Error Estimates in the Case of Dimension Reduction

Samrowski

2014

-We consider the stationary reaction-diffusion problem in a domain Ω ⊂ R 3 having the size along one coordinate direction essentially smaller than along the others. By an energy type argumentation, different simplified models of lower dimension can be deduced and solved numerically. For these models, we derive a guaranteed upper bound of the difference between the exact solution of the original problem and a three-dimensional reconstruction generated by the solution of a dimensionally reduced problem. This estimate of the total error is determined as the sum of discretization and modeling errors, which are both explicit and computable. The corresponding discretization errors are estimated by a posteriori estimates of the functional type. Modeling error majorants are also explicitly evaluated. Hence, a numerical strategy based on the balancing modeling and discretization errors can be derived in order to provide an economical way of getting an approximate solution with an a priori given accuracy. Numerical tests are presented and discussed.2010 Mathematical subject classification: 35J20, 65N15, 65N30.