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Abstract. In this paper we study finite element methods of least-squares type for the stationary, incompressible Navier-Stokes equations in 2 and 3 dimensions. We consider methods based on velocity-vorticity-pressure form of the Navier-Stokes equations augmented with several nonstandard boundary conditions. Least-squares minimization principles for these boundary value problems are developed with the aid of Agmon-Douglis-Nirenberg elliptic theory. Among the main results of this paper are optimal error estimates for conforming finite element approximations, and analysis of some nonstandard boundary conditions. Results of several computational experiments with least-squares methods which illustrate, among other things, the optimal convergence rates are also reported. Key words. Navier-Stokes equations, least-squares principle, finite element methods, velocityvorticity-pressure equations. AMS subject classifications. 76D05, 76D07, 65F10, 65F30 1. Introduction. In

this paper we will derive a 3-G type theorem as in (1) but with G 1;n replaced by the Green function G m;n for the m-polyharmonic operator with Dirichlet boundary conditions and with

Abstract. The main result of this paper is the existence of asymptotically hyperbolic Einstein metrics with prescribed conformal infinity sufficiently close to that of a given asymptotically hyperbolic Einstein metric with nonpositive curvature. If the conformal infinities are sufficiently smooth, the resulting Einstein metrics have optimal Hölder regularity at the boundary. The proof is based on sharp Fredholm theorems for self-adjoint geometric linear elliptic operators on asymptotically hyperbolic manifolds. 1.

Abstract. We consider the application of least-squares variational principles to the numerical solution of partial differential equations. Our main focus is on the development of least-squares finite element methods for elliptic boundary value problems arising in fields such as fluid flows, linear elasticity, and convection-diffusion. For many of these problems, least-squares principles offer numerous theoretical and computational advantages in the algorithmic design and implementation of corresponding finite element methods, that are not present in standard Galerkin discretizations. Most notably, the use of least-squares principles leads to symmetric and positive definite algebraic problems and allows one to circumvent stability conditions such as the inf-sup condition arising in mixed methods for the Stokes and Navier-Stokes equations. As a result, application of least-squares principles has led to the development of robust and efficient finite element methods for a large class of problems of practical importance. Key words. least-squares finite element methods, elliptic equations AMS subject classifications. 65N30 1. Introduction. The

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In this paper we derive global W 1, ∞ and piecewise C 1,α estimates for solutions to divergence form elliptic equations with piecewise Hölder continuous coefficients. The novelty of these estimates is that, even though they depend on the shape and on the size of the surfaces of discontinuity of the coefficients, they are independent of the distance between these surfaces. 1.

. Existence and uniqueness of classical solutions for the multi-dimensional expanding Hele-Shaw problem is proved. Key words. Classical solutions, Hele-Shaw model, moving boundary problem, maximal regularity. AMS subject classifications. 35R35, 35K55, 35S30, 76D99. 1. The problem. We are concerned with a class of moving boundary problems for bounded domains in R n , which comprise in particular the so called single phase Hele-Shaw problem. In order to describe precisely the involved geometry, let\Omega be a bounded domain in R n and assume that its boundary @\Omega is of class C 1 . Moreover, assume that @\Omega consists of two disjoint non-empty components J and \Gamma. Later on, we will model over the exterior component \Gamma a moving interface, whereas the interior component J describes a fixed portion of the boundary. Let denote the outer unit normal field over \Gamma and fix ff 2 (0; 1). Given a ? 0, set U := fae 2 C 2+ff (\Gamma) ; kaek C 1 (\Gamma) ! ag: Fo...

This paper is concerned with least squares methods for the numerical solution of operator equations. Our primary focus is the discussion of the following conceptual issues: the selection of appropriate least squares functionals, their numerical evaluation in the special light of recent developments of wavelet methods and a natural way of preconditioning the resulting systems of linear equations. We describe first a general format of variational problems that are well posed in a certain natural topology. In order to illustrate the scope of these problems we identify several special cases such as second order elliptic boundary value problems, their formulation as a first order system, transmission problems, the system of Stokes equations or more general saddle point problems. Particular emphasis is placed on the separate treatment of essential non-homogeneous boundary conditions. We propose a unified treatment based on wavelet expansions. In particular, we exploit the fact that weighted s...

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