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Register a new userStability of the Stokes projection on weighted spaces and applications
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We show that, on convex polytopes and two or three dimensions, the finite element Stokes projection is stable on weighted spaces $mathbf{W}^{1,p}_0(omega,Omega) times L^p(omega,Omega)$, where the weight belongs to a certain Muckenhoupt class and the integrability index can be different from two. We show how this estimate can be applied to obtain error estimates for approximations of the solution to the Stokes problem with singular sources.
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We show stability of the $L^2$-projection onto Lagrange finite element spaces with respect to (weighted) $L^p$ and $W^{1,p}$-norms for any polynomial degree and for any space dimension under suitable conditions on the mesh grading. This includes $W^{1,2}$-stability in two space dimensions for any polynomial degree and meshes generated by newest vertex bisection. Under realistic but conjectured assumptions on the mesh grading in three dimensions we show $W^{1,2}$-stability for all polynomial degrees. We also propose a modified bisection strategy that leads to better $W^{1,p}$-stability. Moreover, we investigate the stability of the $L^2$-projection onto Crouzeix-Raviart elements.
The long-standing problem of minimal projections is addressed from a computational point of view. Techniques to determine bounds on the projection constants of univariate polynomial spaces are presented. The upper bound, produced by a linear program, and the lower bound, produced by a semidefinite program exploiting the method of moments, are often close enough to deduce the projection constant with reasonable accuracy. The implementation of these programs makes it possible to find the projection constant of several three-dimensional spaces with five digits of accuracy, as well as the projection constants of the spaces of cubic, quartic, and quintic polynomials with four digits of accuracy. Beliefs about uniqueness and shape-preservation of minimal projections are contested along the way.
In this paper we derive stability estimates in $L^{2}$- and $L^{infty}$- based Sobolev spaces for the $L^{2}$ projection and a family of quasiinterolants in the space of smooth, 1-periodic, polynomial splines defined on a uniform mesh in $[0,1]$. As a result of the assumed periodicity and the uniform mesh, cyclic matrix techniques and suitable decay estimates of the elements of the inverse of a Gram matrix associated with the standard basis of the space of splines, are used to establish the stability results.
In two dimensions, we show existence of solutions to the stationary Navier Stokes equations on weighted spaces $mathbf{H}^1_0(omega,Omega) times L^2(omega,Omega)$, where the weight belongs to the Muckenhoupt class $A_2$. We show how this theory can be applied to obtain a priori error estimates for approximations of the solution to the Navier Stokes problem with singular sources.
In this paper, we analyse a Vector Penalty Projection Scheme (see [1]) to treat the displacement of a moving body in incompressible viscous flows in the case where the interaction of the fluid on the body can be neglected. The presence of the obstacle inside the computational domain is treated with a penalization method introducing a parameter $eta$. We show the stability of the scheme and that the pressure and velocity converge towards a limit when the penalty parameter $epsilon$, which induces a small divergence and the time step $delta$t tend to zero with a proportionality constraint $epsilon$ = $lambda$$delta$t. Finally, when $eta$ goes to 0, we show that the problem admits a weak limit which is a weak solution of the Navier-Stokes equations with no-sleep condition on the solid boundary. R{e}sum{e} Dans ce travail nous analysons un sch{e}ma de projection vectorielle (voir [1]) pour traiter le d{e}placement dun corps solide dans un fluide visqueux incompressible dans le cas o` u linteraction du fluide sur le solide est n{e}gligeable. La pr{e}sence de lobstacle dans le domaine solide est mod{e}lis{e}e par une m{e}thode de p{e}nalisation. Nous montrons la stabilit{e} du sch{e}ma et la convergence des variables vitesse-pression vers une limite quand le param etre $epsilon$ qui assure une faible divergence et le pas de temps $delta$t tendent vers 0 avec une contrainte de proportionalit{e} $epsilon$ = $lambda$$delta$t. Finalement nous montrons que leprob{`i} eme converge au sens faible vers une solution des equations de Navier-Stokes avec une condition aux limites de non glissement sur lafront{`i} ere immerg{e}e quand le param etre de p{e}nalisation $eta$ tend vers 0.
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