108 research outputs found
Functional a posteriori error estimates for parabolic time-periodic boundary value problems
The paper is concerned with parabolic time-periodic boundary value problems
which are of theoretical interest and arise in different practical
applications. The multiharmonic finite 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
confirm the efficiency of the a posteriori error bounds derived
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.Comment: A gap in the proof of Theorem 3.2 is fixed; 19 pages, 3 figure
Functional a posteriori error estimates for time-periodic parabolic optimal control problems
This paper is devoted to the a posteriori error analysis of multiharmonic
finite element approximations to distributed optimal control problems with
time-periodic state equations of parabolic type. We derive a posteriori
estimates of functional type, which are easily computable and provide
guaranteed upper bounds for the state and co-state errors as well as for the
cost functional. These theoretical results are confirmed by several numerical
tests that show high efficiency of the a posteriori error bounds
A posteriori error estimates of functional type for variational problems related to generalized Newtonian fluids
The paper is focused on functional type a posteriori estimates of the difference between the exact solution of a variational problem modeling certain types of generalized Newtonian fluids and any function from the admissible energy class. In contrast to the a posteriori estimates obtained for example by the finite element method our estimates do not contain any local (mesh dependent) constants, and therefore they can be used regardless of the way in which an approximation has been constructed
Estimates of the deviations from the exact solutions for variational inequalities describing the stationary flow of certain viscous incompressible fluids
This paper is concerned with computable and guaranteed upper bounds of the difference between exact solutions of variational inequalities arising in the theory of viscous fluids and arbitrary approximations in the corresponding energy space. Such estimates (also called error majorants of functional type) have been derived for the considered class of nonlinear boundary value problems in [11] with the help of variational methods based on duality theory from convex analysis. In the present paper it is shown that error majorants can be derived in a different way by certain transformations of the variational inequalities that define generalized solutions. The error bounds derived by this techniques for the velocity function differ from those obtained by the variational method. These estimates involve only global constants coming from Korn and Friedrichs type inequalities, which are not difficult to evaluate in case of Dirichlet boundary conditions. For the case of mixed boundary conditions, we also derive another form of the estimate which contains only one constant coming from the following assertion: the L^{2} norm of a vector valued function from H^{1}(\Omega) in the factor-space generated by the equivalence with respect to rigid motions is bounded by the L^{2} norm of the symmetric part of the gradient tensor. Since for some ”simple” domains like squares or cubes, the constants in this inequality can be found analytically (or numerically), we obtain a unified form of an error majorant for any domain that admits a decomposition into such subdomains
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