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54
FirstOrder System Least Squares For SecondOrder Partial Differential Equations: Part I
, 1994
"... . This paper develops ellipticity estimates and discretization error bounds for elliptic equations (with lower order terms) that are reformulated as a leastsquares problem for an equivalent firstorder system. The main result is the proof of ellipticity, which is used in a companion paper to esta ..."
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Cited by 61 (14 self)
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. This paper develops ellipticity estimates and discretization error bounds for elliptic equations (with lower order terms) that are reformulated as a leastsquares problem for an equivalent firstorder system. The main result is the proof of ellipticity, which is used in a companion paper to establish optimal convergence of multiplicative and additive solvers of the discrete systems. Key words. leastsquares discretization, secondorder elliptic problems, RayleighRitz, finite elements 1. Introduction. The purpose of this paper is to analyse the leastsquares finite element method for secondorder convectiondiffusion equations written as a firstorder system. In general, the standard Galerkin finite element methods applied to nonselfadjoint elliptic equations with significant convection terms exhibit a variety of deficiencies, including oscillations or nonmonotonocity of the solution and poor approximation of its derivatives. A variety of stabilization techniques, such as upwin...
Adaptive Wavelet Methods II  Beyond the Elliptic Case
 FOUND. COMPUT. MATH
, 2000
"... This paper is concerned with the design and analysis of adaptive wavelet methods for systems of operator equations. Its main accomplishment is to extend the range of applicability of the adaptive wavelet based method developed in [CDD] for symmetric positive denite problems to indefinite or unsymmet ..."
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Cited by 47 (14 self)
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This paper is concerned with the design and analysis of adaptive wavelet methods for systems of operator equations. Its main accomplishment is to extend the range of applicability of the adaptive wavelet based method developed in [CDD] for symmetric positive denite problems to indefinite or unsymmetric systems of operator equations. This is accomplished by first introducing techniques (such as the least squares formulation developed in [DKS]) that transform the original (continuous) problem into an equivalent infinite system of equations which is now wellposed in the Euclidean metric. It is then shown how to utilize adaptive techniques to solve the resulting infinite system of equations. This second step requires a signicant modication of the ideas from [CDD]. The main departure from [CDD] is to develop an iterative scheme that directly applies to the innite dimensional problem rather than nite subproblems derived from the infinite problem. This rests on an adaptive application of the innite dimensional operator to finite vectors representing elements from finite dimensional trial spaces. It is shown that for a wide range of problems, this new adaptive method performs with asymptotically optimal complexity, i.e., it recovers an approximate solution with desired accuracy at a computational expense that stays proportional to the number of terms in a corresponding waveletbest Nterm approximation. An important advantage of this adaptive approach is that it automatically stabilizes the numerical procedure so that, for instance, compatibility constraints on the choice of trial spaces like the LBB condition no longer arise.
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 46 (16 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
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 31 (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.
Edge stabilization for Galerkin approximations of convectiondiffusionreaction problems
 Comp. Methods Appl. Mech. Engrg
"... Abstract. We analyze a nonlinear shockcapturing scheme for H 1conforming, piecewiseaffine finite element approximations of linear elliptic problems. The meshes are assumed to satisfy two standard conditions: a local quasiuniformity property and the Xu–Zikatanov condition ensuring that the stiffne ..."
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Cited by 23 (4 self)
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Abstract. We analyze a nonlinear shockcapturing scheme for H 1conforming, piecewiseaffine finite element approximations of linear elliptic problems. The meshes are assumed to satisfy two standard conditions: a local quasiuniformity property and the Xu–Zikatanov condition ensuring that the stiffness matrix associated with the Poisson equation is an Mmatrix. A discrete maximum principle is rigorously established in any space dimension for convectiondiffusionreaction problems. We prove that the shockcapturing finite element solution converges to that without shockcapturing if the cell Péclet numbers are sufficiently small. Moreover, in the diffusiondominated regime, the difference between the two finite element solutions superconverges with respect to the actual approximation error. Numerical experiments on test problems with stiff layers confirm the sharpness of the a priori error estimates. 1.
Wavelet Least Square Methods For Boundary Value Problems
 SIAM J. Numer. Anal
, 1999
"... 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 ..."
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Cited by 17 (13 self)
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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 nonhomogeneous boundary conditions. We propose a unified treatment based on wavelet expansions. In particular, we exploit the fact that weighted s...
Wavelet Methods for PDEs  Some Recent Developments
 J. Comput. Appl. Math
, 1999
"... this article will be on recent developments in the last area. Rather than trying to give an exhaustive account of the state of the art I would like to bring out some mechanisms which are in my opinion important for the application of wavelets to operator equations. To accomplish this I found it nece ..."
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Cited by 13 (4 self)
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this article will be on recent developments in the last area. Rather than trying to give an exhaustive account of the state of the art I would like to bring out some mechanisms which are in my opinion important for the application of wavelets to operator equations. To accomplish this I found it necessary to address some of the pivotal issues in more detail than others. Nevertheless, such a selected `zoom in' supported by an extensive list of references should provide a sound footing for conveying also a good idea about many other related branches that will only be briey touched upon. Of course, the selection of material is biased by my personal experience and therefore is not meant to reect any objective measure of importance. The paper is organized around two essential issues namely adaptivity and the development of concepts for coping with a major obstruction in this context namely practically relevant domain geometries
Firstorder system least squares for velocityvorticitypressure form of the Stokes equations, with application to linear elasticity
 ETNA
, 1995
"... In this paper, we study the leastsquares method for the generalized Stokes equations (including linear elasticity) based on the velocityvorticitypressure formulation in d =2or3 dimensions. The leastsquares functional is defined in terms of the sum of the L2andH−1norms of the residual equation ..."
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Cited by 13 (7 self)
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In this paper, we study the leastsquares method for the generalized Stokes equations (including linear elasticity) based on the velocityvorticitypressure formulation in d =2or3 dimensions. The leastsquares functional is defined in terms of the sum of the L2andH−1norms of the residual equations, which is similar to that in [7], but weighted appropriately by the Reynolds number (Poisson ratio). Our approach for establishing ellipticity of the functional does not use ADN theory, but is founded more on basic principles. We also analyze the case where the H−1norm in the functional is replaced by a discrete functional to make the computation feasible. We show that the resulting algebraic equations can be preconditioned by wellknown techniques uniformly well in the Reynolds number (Poisson ratio).
FirstOrder System Least Squares (FOSLS) for Planar Linear Elasticity: Pure Traction Problem
 SIAM J. Numer. Anal
, 1998
"... This paper develops two firstorder system leastsquares (FOSLS) approaches for the solution of the pure traction problem in planar linear elasticity. Both are twostage algorithms that first solve for the gradients of displacement (which immediately yield deformation and stress), then for the displ ..."
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Cited by 10 (4 self)
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This paper develops two firstorder system leastsquares (FOSLS) approaches for the solution of the pure traction problem in planar linear elasticity. Both are twostage algorithms that first solve for the gradients of displacement (which immediately yield deformation and stress), then for the displacement itself (if desired). One approach, which uses L 2 norms to define the FOSLS functional, is shown under certain H 2 regularity assumptions to admit optimal H 1 like performance for standard finite element discretization and standard multigrid solution methods that is uniform in the Poisson ratio for all variables. The second approach, which is based on H 1 norms, is shown under general assumptions to admit optimal uniform performance for displacement flux in an L 2 norm and for displacement in an H 1 norm. These methods do not degrade as other methods generally do when the material properties approach the incompressible limit.
Adaptive wavelet techniques in numerical simulation
 Encyclopedia of Computational Mechanics
, 2004
"... This chapter highlights recent developments concerning adaptive wavelet methods for time dependent and stationary problems. The first problem class focusses on hyperbolic conservation laws where wavelet concepts exploit sparse representations of the conserved variables. Regarding the second problem ..."
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Cited by 10 (0 self)
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This chapter highlights recent developments concerning adaptive wavelet methods for time dependent and stationary problems. The first problem class focusses on hyperbolic conservation laws where wavelet concepts exploit sparse representations of the conserved variables. Regarding the second problem class, we begin with matrix compression in the context of boundary integral equations where the key issue is now to obtain sparse representations of (global) operators like singular integral operators in wavelet coordinates. In the remainder of the chapter a new fully adaptive algorithmic paradigm along with some analysis concepts are outlined which, in particular, works for nonlinear problems and where the sparsity of both, functions and operators, is exploited. key words: Conservation laws, boundary integral equations, elliptic problems, saddle point problems, mixed formulations, nonlinear problems, matrix compression, adaptive application of operators, best Nterm approximation, tree approximation, convergence rates, complexity estimates 1.