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122
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
Computational Differential Equations
, 1996
"... Introduction This first part has two main purposes. The first is to review some mathematical prerequisites needed for the numerical solution of differential equations, including material from calculus, linear algebra, numerical linear algebra, and approximation of functions by (piecewise) polynomial ..."
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Cited by 44 (4 self)
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Introduction This first part has two main purposes. The first is to review some mathematical prerequisites needed for the numerical solution of differential equations, including material from calculus, linear algebra, numerical linear algebra, and approximation of functions by (piecewise) polynomials. The second purpose is to introduce the basic issues in the numerical solution of differential equations by discussing some concrete examples. We start by proving the Fundamental Theorem of Calculus by proving the convergence of a numerical method for computing an integral. We then introduce Galerkin's method for the numerical solution of differential equations in the context of two basic model problems from population dynamics and stationary heat conduction.
Instability and stability of rolls in the SwiftHohenberg equation
 COMM. MATH. PHYS
, 1997
"... We develop a method for the stability analysis of bifurcating spatially periodic patterns under general nonperiodic perturbations. In particular, it enables us to detect sideband instabilities. We treat in all detail the stability question of roll solutions in the twodimensional SwiftHohenberg equ ..."
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Cited by 25 (4 self)
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We develop a method for the stability analysis of bifurcating spatially periodic patterns under general nonperiodic perturbations. In particular, it enables us to detect sideband instabilities. We treat in all detail the stability question of roll solutions in the twodimensional SwiftHohenberg equation and derive a condition on the amplitude and the wave number of the rolls which is necessary and sufficent for stability. Moreover, we characterize the set of those wave vectors oe 2 R² which give rise to unstable perturbations.
Adaptive wavelet scheme for nonlinear variational problems with convergence rates
 SIAM J Numer Anal
"... Abstract. We develop and analyze wavelet based adaptive schemes for nonlinear variational problems. We derive estimates for convergence rates and corresponding work counts that turn out to be asymptotically optimal. Our approach is based on a new paradigm that has been put forward recently for a cla ..."
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Cited by 21 (10 self)
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Abstract. We develop and analyze wavelet based adaptive schemes for nonlinear variational problems. We derive estimates for convergence rates and corresponding work counts that turn out to be asymptotically optimal. Our approach is based on a new paradigm that has been put forward recently for a class of linear problems. The original problem is transformed first into an equivalent one which is well posed in the Euclidean metric ℓ2. Then conceptually one seeks iteration schemes for the infinite dimensional problem that exhibits at least a fixed error reduction per step. This iteration is then realized approximately through an adaptive application of the involved operators with suitable dynamically updated accuracy tolerances. The main conceptual ingredients center around nonlinear tree approximation and the sparse evaluation of nonlinear mappings of wavelet expansions. We prove asymptotically optimal complexity for adaptive realizations of first order iterations and of Newton’s method. Key words. variational problems, wavelet representations, semilinear equations, mapping properties, gradient iteration, convergence rates, adaptive application of operators, sparse evaluation of nonlinear mappings of wavelet expansions, tree approximation, Newton’s scheme
An Elliptic Problem Arising From the Unsteady Transonic Small Disturbance Equation
, 1995
"... We prove a theorem on existence of a weak solution of the Dirichlet problem for a quasilinear elliptic equation with a degeneracy on one part of the boundary. The degeneracy is of a type ("Keldysh type") associated with singular behavior  blowup of a derivative  at the boundary. We define an ..."
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Cited by 20 (8 self)
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We prove a theorem on existence of a weak solution of the Dirichlet problem for a quasilinear elliptic equation with a degeneracy on one part of the boundary. The degeneracy is of a type ("Keldysh type") associated with singular behavior  blowup of a derivative  at the boundary. We define an associated operator which is continuous, pseudomonotone and coercive and show that a weak solution displaying singular behavior at the boundary exists. Contents 1
An operator splitting method for nonlinear convectiondi usion equations
 Numer. Math
, 1997
"... Abstract. We present a semidiscrete method for constructing approximate solutions to the initial value problem for the mdimensional convectiondi usion equation ut + r f(u) = " u. The method is based on the use of operator splitting to isolate the convection part and the di usion part of the equat ..."
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Cited by 16 (8 self)
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Abstract. We present a semidiscrete method for constructing approximate solutions to the initial value problem for the mdimensional convectiondi usion equation ut + r f(u) = " u. The method is based on the use of operator splitting to isolate the convection part and the di usion part of the equation. In the case m>1, dimensional splitting is used to reduce the mdimensional convection problem to a series of onedimensional problems. We showthat the method produces a compact sequence of approximate solutions which converges to the exact solution. Finally, a fully discrete method is analyzed, and demonstrated in the case of one and two space dimensions. 1. Introduction. In
Analysis of a finite PML approximation for the three dimensional timeharmonic maxwell and acoustic scattering problems
 MATH. COMP
, 2006
"... We consider the approximation of the frequency domain threedimensional Maxwell scattering problem using a truncated domain perfectly matched layer (PML). We also treat the timeharmonic PML approximation to the acoustic scattering problem. Following work of Lassas and Somersalo in 1998, a transiti ..."
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Cited by 15 (7 self)
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We consider the approximation of the frequency domain threedimensional Maxwell scattering problem using a truncated domain perfectly matched layer (PML). We also treat the timeharmonic PML approximation to the acoustic scattering problem. Following work of Lassas and Somersalo in 1998, a transitional layer based on spherical geometry is defined, which results in a constant coefficient problem outside the transition. A truncated (computational) domain is then defined, which covers the transition region. The truncated domain need only have a minimally smooth outer boundary (e.g., Lipschitz continuous). We consider the truncated PML problem which results when a perfectly conducting boundary condition is imposed on the outer boundary of the truncated domain. The existence and uniqueness of solutions to the truncated PML problem will be shown provided that the truncated domain is sufficiently large, e.g., contains a sphere of radius Rt. Wealsoshow exponential (in the parameter Rt) convergence of the truncated PML solution to the solution of the original scattering problem inside the transition layer. Our results are important in that they are the first to show that the truncated PML problem can be posed on a domain with nonsmooth outer boundary. This allows the use of approximation based on polygonal meshes. In addition, even though the transition coefficients depend on spherical geometry, they can be made arbitrarily smooth and hence the resulting problems are amenable to numerical quadrature. Approximation schemes based on our analysis are the focus of future research.
Stability and convergence of efficient NavierStokes solvers via a commutator estimate 0
, 2005
"... For strong solutions of the incompressible NavierStokes equations in bounded domains with velocity specified at the boundary, we establish the unconditional stability and convergence of discretization schemes that decouple the updates of pressure and velocity through explicit timestepping for pres ..."
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Cited by 13 (6 self)
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For strong solutions of the incompressible NavierStokes equations in bounded domains with velocity specified at the boundary, we establish the unconditional stability and convergence of discretization schemes that decouple the updates of pressure and velocity through explicit timestepping for pressure. These schemes require no solution of stationary Stokes systems, nor any compatibility between velocity and pressure spaces to ensure an infsup condition, and are representative of a class of highly efficient computational methods that have recently emerged. The proofs are simple, based upon a new, sharp estimate for the commutator of the Laplacian and Helmholtz projection operators. This allows us to treat an unconstrained formulation of the NavierStokes equations as a perturbed diffusion equation. 1
Domain decomposition preconditioners for linear–quadratic elliptic optimal control problems
, 2004
"... ABSTRACT. We develop and analyze a class of overlapping domain decomposition (DD) preconditioners for linearquadratic elliptic optimal control problems. Our preconditioners utilize the structure of the optimal control problems. Their execution requires the parallel solution of subdomain linearquad ..."
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Cited by 12 (4 self)
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ABSTRACT. We develop and analyze a class of overlapping domain decomposition (DD) preconditioners for linearquadratic elliptic optimal control problems. Our preconditioners utilize the structure of the optimal control problems. Their execution requires the parallel solution of subdomain linearquadratic elliptic optimal control problems, which are essentially smaller subdomain copies of the original problem. This work extends to optimal control problems the application and analysis of overlapping DD preconditioners, which have been used successfully for the solution of single PDEs. We prove that for a class of problems the performance of the twolevel versions of our preconditioners is independent of the mesh size and of the subdomain size. 1.
A combinatorial approach to involution and δregularity II: Structure analysis of polynomial modules with Pommaret bases
, 2002
"... Abstract Involutive bases are a special form of nonreduced Gröbner bases with additional combinatorial properties. Their origin lies in the JanetRiquier theory of linear systems of partial differential equations. We study them for a rather general class of polynomial algebras including also nonco ..."
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Cited by 12 (3 self)
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Abstract Involutive bases are a special form of nonreduced Gröbner bases with additional combinatorial properties. Their origin lies in the JanetRiquier theory of linear systems of partial differential equations. We study them for a rather general class of polynomial algebras including also noncommutative algebras like those generated by linear differential and difference operators or universal enveloping algebras of (finitedimensional) Lie algebras. A number of basic properties are derived and we provide concrete algorithms for their construction. Furthermore, we develop a theory for involutive bases with respect to semigroup orders (as they appear in local computations) and over coefficient rings, respectively. In both cases it turns out that generally only weak involutive bases exist. 1