We consider finite element approximations of a second order elliptic problem on a bounded polytopic domain in with . The constant appearing in Céa’s lemma and coming from its standard proof can be very large when the coefficients of an elliptic operator attain considerably different values. We restrict ourselves to regular families of uniform partitions and linear simplicial elements. Using a lower bound of the interpolation error and the supercloseness between the finite element solution and...
By means of eigenvalue error expansion and integral expansion techniques, we propose and analyze the stream function-vorticity-pressure method for the eigenvalue problem associated with the Stokes equations on the unit square. We obtain an optimal order of convergence for eigenvalues and eigenfuctions. Furthermore, for the bilinear finite element space, we derive asymptotic expansions of the eigenvalue error, an efficient extrapolation and an a posteriori error estimate for the eigenvalue. Finally,...
In this paper we study some special residuated lattices, namely, idempotent residuated chains. After giving some properties of Green’s relation on the monoid reduct of an idempotent residuated chain, we establish a structure theorem for idempotent residuated chains. As an application, we give necessary and sufficient conditions for a band with an identity to be the monoid reduct of some idempotent residuated chain. Finally, based on the structure theorem for idempotent residuated chains, we obtain...
In a case-cohort design, covariate histories are measured only on cases and a subcohort that is randomly selected from the entire cohort. This design has been widely used in large epidemiologic studies, especially when the exposures of interest are expensive to assemble for all the subjects. In this paper, we propose statistical procedures for analyzing case-cohort sampled current status data under the additive hazards model. Asymptotical properties of the proposed estimator are described and we...
We study a spin system with both mixed even-spin Sherrington–Kirkpatrick (SK) couplings and Curie–Weiss (CW) interaction. Our main results are: (i) The thermodynamic limit of the free energy is given by a variational formula involving the free energy of the SK model with a change in the external field. (ii) In the presence of a centered Gaussian external field, the positivity of the overlap and the extended Ghirlanda–Guerra identities hold on a dense subset of the temperature parameters. (iii) We...
We derive an optimal lower bound of the interpolation error for linear finite elements on a bounded two-dimensional domain. Using the supercloseness between the linear interpolant of the true solution of an elliptic problem and its finite element solution on uniform partitions, we further obtain two-sided bounds of the discretization error by means of the interpolation error. Two-sided bounds for bilinear finite elements are given as well. Numerical tests illustrate our theoretical analysis.
Integrated design of observer based Fault Detection (FD) for a class of uncertain nonlinear systems with Lipschitz nonlinearities is studied. In the context of norm based residual evaluation, the residual generator and evaluator are designed together in an integrated form, and, based on it, a trade-off FD system is finally achieved in the sense that, for a given Fault Detection Rate (FDR), the False Alarm Rate (FAR) is minimized. A numerical example is given to illustrate the effectiveness of the...
For a double complex , we show that if it satisfies the -lemma and the spectral sequence induced by does not degenerate at , then it degenerates at . We apply this result to prove the degeneration at of a Hodge-de Rham spectral sequence on compact bi-generalized Hermitian manifolds that satisfy a version of -lemma.
We derive an optimal lower bound of the
interpolation error for linear finite elements on a bounded two-dimensional
domain. Using the supercloseness between the linear interpolant
of the true solution of an elliptic problem and its finite element
solution on uniform partitions, we further
obtain two-sided bounds of the discretization error by means of the
interpolation error. Two-sided bounds for bilinear finite elements
are given as well. Numerical tests illustrate our theoretical
analysis.
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