We apply the continuous interior penalty method to the three fields Stokes problem. We prove an inf-sup condition for the proposed method leading to optimal a priori error estimates for smooth exact solutions. Moreover we propose an iterative algorithm for the separate solution of the velocities and the pressures on the one hand and the extra-stress on the other. The stability of the iterative ...

For a general class of saddle point problems sharp estimates for Babuška’s inf-sup stability constants are derived in terms of the constants in Brezzi’s theory. In the finite-dimensional Hermitian case more detailed spectral properties of preconditioned saddle point matrices are presented, which are helpful for the convergence analysis of common Krylov subspace methods. The theoretical results ...

We investigate the possible limit distributions of zeros and poles associated with ray sequences of rational functions that are asymptotically optimal for weighted Zolotarev problems. For disjoint compacta E1 , E2 in the complex plane, the Zolotarev problem entails minimizing the ratio of the sup over E1 of the modulus of a weighted rational to its inf over E2 . Potential theoretic tools are ut...

The Galerkin discretization of the Electric Field Integral Equation is reinvestigated. We prove quasi-optimal convergence estimates at nonresonant frequencies, using orthogonal splittings of the Galerkin space. At resonant frequencies we show that the spurious electric currents radiate only weakly in the exterior domain. This is achieved through the study of some finitely degenerated problems i...

We consider the Stokes problem on a plane polygonal domain Ω ⊂ R2 . We propose a finite element method for overlapping or nonmatching grids for the Stokes problem based on the partition of unity method. We prove that the discrete inf-sup condition holds with a constant independent of the overlapping size of the subdomains. The results are valid for multiple subdomains and any spatial dimension.

A free boundary problem arising from the optimal reinforcement of a membrane or from the reduction of traffic congestion is considered; it is of the form sup ∫ D θ dx=m inf u∈H1 0 (D) ∫ D (1 + θ 2 |∇u| − fu ) dx. We prove the existence of an optimal reinforcement θ and that it has some higher integrability properties. We also provide some numerical computations for θ and u.

Small ball estimates for Brownian motion and the Brownian sheet are obtained when the balls are given by certain H61der norms. As an application, we establish some lira inf results for Gaussian samples of these processes, as well as Chung's LIL in this setting, but now we introduce some notation. Let I = [0, 1 ], with Co(I) the continuous real-valued function f ( . ) on I such that f ( 0 ) = 0....

In this work we present a stable proper orthogonal decomposition (POD)-Galerkin approximation for parametrized steady Navier-Stokes equations. The stabilization is guaranteed by the use of supremizers solutions that enrich the reduced velocity space. Numerical results show that an equivalent inf-sup condition is fulfilled, yielding stability for both velocity and pressure. Our stability analysi...

A time-dependent system modeling the interaction between a Stokes fluid and an elastic structure is studied. A divergence-free weak formulation is introduced which does not involve the fluid pressure field. The existence and uniqueness of a weak solution is proved. Strong energy estimates are derived under additional assumptions on the data. The existence of an L2 integrable pressure field is e...

Abstract. We extend the results on minimal stabilization of Burman and Stamm [J. Sci. Comp., 33 (2007), pp. 183-208] to the case of the local discontinuous Galerkin methods on mixed form. The penalization term on the faces is relaxed to act only on a part of the polynomial spectrum. Stability in the form of a discrete inf-sup condition is proved and optimal convergence follows. Some numerical e...