The authors establish error estimates for recently developed finite-element methods for incompressible viscous flow in domains with no-slip boundary conditions. The methods arise by discretization of a well-posed extended Navier-Stokes dynamics for which pressure is determined from current velocity and force fields. The methods use C1 elements for velocity and C0 elements for pressure. A stability estimate is proved for a related finite-element projection method close to classical time-splitting methods of Orszag, Israeli, DeVille and Karniadakis.

In this paper we introduce a definition of the local conservation property for numerical methods solving time dependent conservation laws, which generalizes the classical local conservation definition. The motivation of our definition is the Lax-Wendroff theorem, and thus we prove it for locally conservative numerical schemes per our definition in one and two space dimensions. Several numerical methods, including continuous Galerkin methods and compact schemes, which do not fit the classical local conservation definition, are given as examples of locally conservative methods under our generalized definition.

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In this paper we show by example that there is no uniform estimate for the L p-Minkowski problem. As a result we obtain the nonuniqueness of solutions to the problem.

Let <i>M</i> be a compact Khler manifold and <i>N</i> be a subvariety with codimension greater than or equal to 2. We show that there are no complete KhlerEinstein metrics on M - N . As an application, let <i>E</i> be an exceptional divisor of <i>M</i>. Then M - N cannot admit any complete KhlerEinstein metric if blow-down of <i>E</i> is a complex variety with only canonical or terminal singularities. A similar result is shown for pairs.