Functional A Posteriori Error Estimates for Parabolic Time-Periodic Boundary Value Problems

Abstract

Abstract:The paper is concerned with parabolic time-periodic boundary value problems which are of theoretical interest and arise in di erent practical applications. The multiharmonic nite element method is well adapted to this class of parabolic problems. We study properties of multiharmonic approximations and derive guaranteed and fully computable bounds of approximation errors. For this purpose, we use the functional a posteriori error estimation techniques earlier introduced by S. Repin. Numerical tests con rm the e ciency of the a posteriori error bounds derived.

Functional A Posteriori Error Estimates for Time-Periodic Parabolic Optimal Control Problems

“…(div τ c k (x) cos(kωt) + div τ s k (x) sin(kωt)) , and the L 2 (Q T )-norms of the functions R 1 , R 2 , R 3 and R 4 defined in (27) can be represented in the form, which exposes each mode separately. More precisely, we have…”

“…We wish to deduce majorants for the cost functional J (y(u), u) of the exact control u and corresponding state y(u) by using some of the results presented in [27], which are obtained for the time-periodic boundary value problem (2). In [27], the following functional a posteriori error estimate for problem (2) has been proved:…”

“…This can be done by various techniques. In [27], we have used Raviart-Thomas elements of the lowest order (see also [31,8,34]), in order to regularize the fluxes by a post-processing operator, which maps the L 2 -functions into H(div, Ω). Collecting all the fluxes corresponding to the modes together yields the reconstructed flux τ N h = R flux h (ν∇y N h ).…”

“…In particular, our functional a posteriori error analysis uses the techniques close to those suggested in [32], but the analysis contains essential changes due to the MhFEM setting. In [27], the authors already derived functional a posteriori error estimates for MhFE approximations to parabolic time-periodic boundary value problems. Similar results are now obtained for the MhFE approximations to the state and co-state, which are the unique solutions of the reduced optimality system.…”

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

“…Results for functional a posteriori error estimates of parabolic time-periodic boundary value problems using the same discretization and flux reconstruction techniques can be found in [19]. In order to solve the saddle point systems (46), we use the AMLI preconditioner proposed by Kraus in [18] for an inexact realization of the block-diagonal preconditioner (48) in the MINRES method.…”