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Sickel: Optimal approximation of elliptic problems by linear and nonlinear mappings III
 Triebel, Function Spaces, Entropy Numbers, Differential Operators
, 1996
"... We study the optimal approximation of the solution of an operator equation A(u) = f by four types of mappings: a) linear mappings of rank n; b) nterm approximation with respect to a Riesz basis; c) approximation based on linear information about the right hand side f; d) continuous mappings. We co ..."
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Cited by 134 (28 self)
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We study the optimal approximation of the solution of an operator equation A(u) = f by four types of mappings: a) linear mappings of rank n; b) nterm approximation with respect to a Riesz basis; c) approximation based on linear information about the right hand side f; d) continuous mappings. We consider worst case errors, where f is an element of the unit ball of a Sobolev or Besov space Br q(Lp(Ω)) and Ω ⊂ Rd is a bounded Lipschitz domain; the error is always measured in the Hsnorm. The respective widths are the linear widths (or approximation numbers), the nonlinear widths, the Gelfand widths, and the manifold widths. As a technical tool, we also study the Bernstein numbers. Our main results are the following. If p ≥ 2 then the order of convergence is the same for all four classes of approximations. In particular, the best linear approximations are of the same order as the best nonlinear ones. The best linear approximation can be quite difficult to realize as a numerical algorithm since the optimal Galerkin space usually depends on the operator and of the shape of the domain Ω. For p < 2 there is a difference, nonlinear approximations are better than linear ones. However, in this case, it turns out that linear information about the right hand side f is again optimal. Our main theoretical tool is the best nterm approximation with respect to an optimal Riesz basis and related nonlinear widths. These general results are used to study the Poisson equation in a polygonal domain. It turns out that best nterm wavelet approximation is (almost) optimal. The main results of
An equation of MongeAmpère type in conformal geometry, and fourmanifolds of positive Ricci curvature
, 2004
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Infinitedimensional linear systems with unbounded control and observation: A functional analytic approach
 Transactions of the American Mathematical Society
, 1987
"... ABSTRACT. The object of this paper is to develop a unifying framework for the functional analytic representation of infinite dimensional linear systems with unbounded input and output operators. On the basis of the general approach new results are derived on the wellposedness of feedback systems and ..."
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Cited by 82 (1 self)
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ABSTRACT. The object of this paper is to develop a unifying framework for the functional analytic representation of infinite dimensional linear systems with unbounded input and output operators. On the basis of the general approach new results are derived on the wellposedness of feedback systems and on the linear quadratic control problem. The implications of the theory for large classes of functional and partial differential equations are discussed in detail. 1. Introduction. For
Gradient estimates for solutions to divergence form elliptic equations with discontinuous coefficients
 Arch. Rational Mech. Anal
, 2000
"... 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 ..."
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Cited by 68 (5 self)
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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.
A leastsquares finite element method for the NavierStokes equations
 Appl. Math. Lett
, 1993
"... Abstract. In this paper we study finite element methods of leastsquares type for the stationary, incompressible NavierStokes equations in 2 and 3 dimensions. We consider methods based on velocityvorticitypressure form of the NavierStokes equations augmented with several nonstandard boundary con ..."
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Cited by 66 (19 self)
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Abstract. In this paper we study finite element methods of leastsquares type for the stationary, incompressible NavierStokes equations in 2 and 3 dimensions. We consider methods based on velocityvorticitypressure form of the NavierStokes equations augmented with several nonstandard boundary conditions. Leastsquares minimization principles for these boundary value problems are developed with the aid of AgmonDouglisNirenberg 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 leastsquares methods which illustrate, among other things, the optimal convergence rates are also reported. Key words. NavierStokes equations, leastsquares principle, finite element methods, velocityvorticitypressure equations. AMS subject classifications. 76D05, 76D07, 65F10, 65F30 1. Introduction. In
Fredholm operators and Einstein metrics on conformally compact manifolds
"... 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, t ..."
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Cited by 59 (3 self)
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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 selfadjoint geometric linear elliptic operators on asymptotically hyperbolic manifolds. 1.
Finite element methods of leastsquares type
, 1998
"... We consider the application of leastsquares variational principles to the numerical solution of partial differential equations. Our main focus is on the development of leastsquares finite element methods for elliptic boundary value problems arising in fields such as fluid flows, linear elasticit ..."
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Cited by 50 (4 self)
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We consider the application of leastsquares variational principles to the numerical solution of partial differential equations. Our main focus is on the development of leastsquares finite element methods for elliptic boundary value problems arising in fields such as fluid flows, linear elasticity, and convectiondiffusion. For many of these problems, leastsquares 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 leastsquares principles leads to symmetric and positive definite algebraic problems and allows one to circumvent stability conditions such as the infsup condition arising in mixed methods for the Stokes and NavierStokes equations. As a result, application of leastsquares principles has led to the development of robust and efficient finite element methods for a large class of problems of practical importance.
The SchrödingerPoisson equation under the effect of a nonlinear local term
 J. Funct. Anal
"... of a nonlinear local term ..."
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Positivity for equations involving polyharmonic operators with Dirichlet boundary conditions
 MATHEMATISCHE ANNALEN
, 1995
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Graded Mesh Refinement and Error Estimates for Finite Element Solutions of Elliptic Boundary Value Problems in NonSmooth Domains
 BICOM Institute of Computational Mathematics
, 1993
"... . This paper is concerned with the effective numerical treatment of elliptic boundary value problems when the solutions contain singularities. The paper deals first with the theory of problems of this type in the context of weighted Sobolev spaces and covers problems in domains with conical vertices ..."
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Cited by 41 (4 self)
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. This paper is concerned with the effective numerical treatment of elliptic boundary value problems when the solutions contain singularities. The paper deals first with the theory of problems of this type in the context of weighted Sobolev spaces and covers problems in domains with conical vertices and nonintersecting edges, as well as polyhedral domains with Lipschitz boundaries. Finite element schemes on graded meshes for second order problems in polygonal/polyhedral domains are then proposed for problems with the above singularities. These schemes exhibit optimal convergence rates with decreasing mesh size. Finally, we describe numerical experiments which demonstrate the efficiency of our technique in terms of "actual" errors for specific (finite) mesh sizes in addition to the asymptotic rates of convergence. Key Words. Elliptic boundary value problem, singularities, finite element method, mesh grading. AMS(MOS) subject classification. 65N30 1 Introduction This paper is concern...