On the stability of Scott-Zhang type operators and application to multilevel preconditioning in fractional diffusion

We provide an endpoint stability result for Scott-Zhang type operators in Besov spaces. For globally continuous piecewise polynomials these are bounded from $H^{3/2}$ into $B^{3/2}_{2,\infty}$; for elementwise polynomials these are bounded from $H^{1/2}$ into $B^{1/2}_{2,\infty}$. As an application, we obtain a multilevel decomposition based on Scott-Zhang operators on a hierarchy of meshes generated by newest vertex bisection with equivalent norms up to (but excluding) the endpoint case. A local multilevel diagonal preconditioner for the fractional Laplacian on locally refined meshes with optimal eigenvalue bounds is presented.

On a Right-Inverse for the Divergence Operator in Spaces of Continuous Piecewise Polynomials

This paper is concerned with the norm of a right-inverse for the divergence operator between spaces of piecewise polynomials on triangular elements. More specifically, one tries to construct a right-inverse acting from the space W of continuous piecewise polynomials of degree p into the space V of R2-valued piecewise polynomials of degree p+1. Our results are as follows. Assume that we are dealing with a quasiuniform family of triangulations {Mh} satisfying two additional hypotheses: Mh can be transformed into a quadriangular mesh by grouping its element two by two and Mh has no boundary vertex shared by only one or exactly three elements. In that context, one proves that the divergence has a right-inverse with an operator norm growing at most like [Formula: see text] when both W and V are equipped with the H1-norm and at most like [Formula: see text] if W is equipped with the L2-norm only. An application of the first of these two results is the approximation of the thermoelasticity equations by the p-version of the finite element methods. One also shows how the second result can be used in the context of higher-order Hood–Taylor method for Stokes problem.

A C0-Collocation-like method for elliptic equations on rectangular regions

We describe a C0-collocation-like method for solving two-dimensional elliptic Dirichlet problems on rectangular regions, using tensor products of continuous piecewise polynomials. Nodes of the Lobatto quadrature formula are taken as the points of collocation. We show that the method is stable and convergent with order hr(r ≥ 1) in the H1–norm and hr+1(r ≥ 2) in the L2–norm, if the collocation solution js a piecewise polynomial of degree not greater than r with respect to each variable. The method has an advantage over the Galerkin procedure for the same space in that no integrals need be evaluated or approximated.