Preconditioning cubic spline collocation discretizations of elliptic equations

Summary. This work considers the uniformly elliptic operator defined by in (the unit square) with boundary conditions: on and on and its discretization based on Hermite cubic spline spaces and collocation at the Gauss points. Using an interpolatory basis with support on the Gauss points one obtains the matrix . We discuss the condition numbers and the distribution of -singular values of the preconditioned matrices where is the stiffness matrix associated with the finite element discretization of the positive definite uniformly elliptic operator given by in with boundary conditions: on on . The finite element space is either the space of continuous functions which are bilinear on the rectangles determined by Gauss points or the space of continuous functions which are linear on the triangles of the triangulation of using the Gauss points. When we obtain results on the eigenvalues of . In the general case we obtain bounds and clustering results on the -singular values of . These results are related to the results of Manteuffel and Parter [MP], Parter and Wong [PW], and Wong [W] for finite element discretizations as well as the results of Parter and Rothman [PR] for discretizations based on Legendre Spectral Collocation. Received January 1, 1994 / Revised version received February 7, 1995     

Eigenvalue asymptotics for an elliptic boundary problem involving an indefinite weight

We are concerned here with the eigenvalue asymptotics for a non-selfadjoint elliptic boundary problem involving an indefinite weight function which vanishes on a set of positive measure. The asymptotic behaviour of the eigenvalues is well known for the case of second order operators. However for higher order operators, results have only been established under the restriction that the order of the operator exceeds the dimension of the underlying Euclidean space in which the problem is set. In this paper we establish the eigenvalue asymptotics for the case of higher order operators without any such restriction.Supported in part by the John Knopfmacher Centre for Applicable Analysis and Number Theory, University of the Witwatersrand.     

Virtual Eigenvalues of the High Order Schrödinger Operator I

We consider the Schr?dinger operator H_γ = ( − Δ)^l + γ V(x)· acting in the space $$L_2 (\mathbb{R}^d ),$$ where 2l ≥ d, V (x) ≥ 0, V (x) is continuous and is not identically zero, and $$\lim _{|{\mathbf{x}}| \to \infty } V({\mathbf{x}}) = 0.$$ We obtain an asymptotic expansion as $$\gamma \uparrow 0$$of the bottom negative eigenvalue of H_γ, which is born at the moment γ = 0 from the lower bound λ = 0 of the spectrum σ(H₀) of the unperturbed operator H₀ = ( − Δ)^l (a virtual eigenvalue). To this end we develop a supplement to the Birman-Schwinger theory on the process of the birth of eigenvalues in the gap of the spectrum of the unperturbed operator H₀. Furthermore, we extract a finite-rank portion Φ(λ) from the Birman- Schwinger operator $$X_V (\lambda ) = V^{\frac{1} {2}} R_\lambda (H_0 )V^{\frac{1}{2}} ,$$ which yields the leading terms for the desired asymptotic expansion.     

Global solvability for first order real linear partial differential operators

F. Treves, in [17], using a notion of convexity of sets with respect to operators due to B. Malgrange and a theorem of C. Harvey, characterized globally solvable linear partial differential operators on C^∞(X), for an open subset X of Rn.Let P=L+c be a linear partial differential operator with real coefficients on a C^∞ manifold X, where L is a vector field and c is a function. If L has no critical points, J. Duistermaat and L. Hörmander, in [2], proved five equivalent conditions for global solvability of P on C^∞(X).Based on Harvey-Treves's result we prove sufficient conditions for the global solvability of P on C^∞(X), in the spirit of geometrical Duistermaat-Hörmander's characterizations, when L is zero at precisely one point. For this case, additional non-resonance type conditions on the value of c at the equilibrium point are necessary.     

Substochastic semigroups for transport equations with conservative boundary conditions

We consider the free streaming operator associated with conservative boundary conditions. It is known that this operator (with its usual domain) admits an extension A which generates a C₀-semigroup in L¹. With techniques borrowed from the additive perturbation theory of substochastic semigroups, we describe precisely its domain and provide necessary and sufficient conditions ensuring to be stochastic. We apply these results to examples from kinetic theory.     

On the essential spectrum of the linearized Navier-Stokes operator

We determine the essential spectrum of the linearized Navier-Stokes operator with physical boundary conditions. In contrast to other approaches we do not make use of pseudo-differential operators. We establish a direct proof using only some fundamental results for matrix operators.Supported in part by a grant from the NRF of South Africa and by the John Knopfmacher Centre for Applicable Analysis and Number Theory, University of the Witwatersrand, Johannesburg     

Spectrum of the One-dimensional Schrödinger Operator With a Periodic Potential Subjected to a Local Dilative Perturbation

We study the spectrum of the one-dimensional Schr?dinger operator with a potential, whose periodicity is violated via a local dilation. We obtain conditions under which this violation preserves the essential spectrum of the Schr?dinger operator and an infinite number of isolated eigenvalues appear in a gap of the essential spectrum. We show that the considered perturbation of the periodic potential is not relative compact in general.     

Generalized Anti-Wick Operators with Symbols in Distributional Sobolev spaces

Generalized Anti-Wick operators are introduced as a class of pseudodifferential operators which depend on a symbol and two different window functions. Using symbols in Sobolev spaces with negative smoothness and windows in so-called modulation spaces, we derive new conditions for the boundedness on L² of such operators and for their membership in the Schatten classes. These results extend and refine results contained in [20], [10], [5], [4], and [14].     

Bounded imaginary powers and <Emphasis Type="Italic">H</Emphasis><Subscript>∞</Subscript>-calculus of the Stokes operator in two-dimensional exterior domains

The present contribution deals with the Stokes operator A_q on L^q_σ(Ω), 1<q<∞, where Ω is an exterior domain in ℝ² of class C². It is proved that A_q admits a bounded H_∞-calculus. This implies the existence of bounded imaginary powers of A_q, which has several important applications. – So far this property was only known for exterior domains in ℝⁿ, n≥3. – In particular, this shows that A_q has maximal regularity on L^q_σ(Ω). For the proof the resolvent (λ+A_q)⁻¹ has to be analyzed for |λ|→∞ and λ→0. For large λ this is done using an approximate resolvent based on the results of [3], which were obtained by applying the calculus of pseudodifferential boundary value problems. For small λ we analyze the representation of the resolvent developed in [11] by a potential theoretical method.     

On the Friedrichs extension of some block operator matrices

We give a matrix representation for the resolvent of the Friedrichs extension of some semibounded 2×2 operator matrices and study their essential spectrum.     

Regularity of flat free boundaries in two-phase problems for the p-Laplace operator

In this paper we continue the study in Lewis and Nyström (2010) [19], concerning the regularity of the free boundary in a general two-phase free boundary problem for the p-Laplace operator, by proving regularity of the free boundary assuming that the free boundary is close to a Lipschitz graph.     

On the resolvent arising in a parameter‐elliptic multi‐order boundary problem

In this paper we consider a boundary problem for a parameter‐elliptic, multi‐order system of differential equations defined over a bounded region in $\mathbb {R}^n$ and under limited smoothness assumptions as well as under boundary conditions which include those of Dirichlet. Information is then derived concerning the asymptotic behaviour of the trace of a power of the resolvent of the Hilbert space operator, in general non‐selfadjoint, induced by the boundary problem under null boundary conditions. This information will then be used in a subsequent work to derive various results pertaining to the asymptotic behaviour of the eigenvalues of this operator.     

Operator Matrices as Generators of Cosine Operator Functions

We introduce an abstract setting that allows to discuss wave equations with time-dependent boundary conditions by means of operator matrices. We show that such problems are well-posed if and only if certain perturbations of the same problems with homogeneous, time-independent boundary conditions are well-posed. As applications we discuss two wave equations in L^p(0, 1) and in L²(Ω) equipped with dynamical and acoustic-like boundary conditions, respectively.     

Strong convergence of viscosity approximation methods for finding zeros of accretive operators in Banach spaces

In this paper, we introduce a composite iterative scheme by viscosity approximation method for finding a zero of an accretive operator in Banach spaces. Then, we establish strong convergence theorems for the composite iterative scheme. The main theorems improve and generalize the recent corresponding results of Kim and Xu [T.H. Kim, H.K. Xu, Strong convergence of modified Mann iterations, Nonlinear Anal. 61 (2005) 51-60], Qin and Su [X. Qin, Y. Su, Approximation of a zero point of accretive operator in Banach spaces, J. Math. Anal. Appl. 329 (2007) 415-424] and Xu [H.K. Xu, Strong convergence of an iterative method for nonexpansive and accretive operators, J. Math. Anal. Appl. 314 (2006) 631-643] as well as Aoyama et al. [K. Aoyama, Y Kimura, W. Takahashi, M. Toyoda, Approximation of common fixed points of a countable family of nonexpansive mappings in Banach spaces, Nonlinear Anal. 67 (2007) 2350-2360], Benavides et al. [T.D. Benavides, G.L. Acedo, H.K. Xu, Iterative solutions for zeros of accretive operators, Math. Nachr. 248-249 (2003) 62-71], Chen and Zhu [R. Chen, Z. Zhu, Viscosity approximation fixed points for nonexpansive and m-accretive operators, Fixed Point Theory and Appl. 2006 (2006) 1-10] and Kamimura and Takahashi [S. Kamimura, W. Takahashi, Approximation solutions of maximal monotone operators in Hilberts spaces, J. Approx. Theory 106 (2000) 226-240].     

The Friedrichs Extension of a Pair of Operators

Abstract

An extension of a pair of linear unbounded operators which map from a Banach space X to a Hilbert space Y is constructed and studied. The purpose of the extension is to obtain a pair of jointly closed operators which will be the generating pair of a B-evolution similar to the classical Friedrichs extension of a single operator which generates a holomorphic semigroup. The construction is based on spectral methods.     

Propagation of low regularity for solutions of nonlinear PDEs on a Riemannian manifold with a sub-Laplacian structure

Following Bernicot (2012) [7], we introduce a notion of paraproducts associated to a semigroup. We do not use Fourier transform arguments and the background manifold is doubling, endowed with a sub-Laplacian structure. Our main result is a paralinearization theorem in a non-Euclidean framework, with an application to the propagation of regularity for some nonlinear PDEs.     

On a Generic Topological Structure of the Spectrum to One-dimensional Schrödinger Operators with Complex Limit-periodic Potentials

We prove that for a massive set of limit-periodic complex-valued potentials V(x) of Stepanov class, the spectra (H_v) of the corresponding one-dimensional Schrödinger operators H_v are 0-dimensional topological subspaces of the complex plane . This is a generalization of a similar result obtained by J. Avron and B. Simon for the case of real-valued limit-periodic potentials. Our technique is based upon results of V. Tkachenko concerning an inverse spectral problem for the Hill operator with a complex-valued potential. We also make use of some facts from Differential Topology.     

Virtual Eigenvalues of the High Order Schrödinger Operator II

We consider the Schr?dinger operator H_γ = ( − Δ)^l + γ V(x)· acting in the space where 2l ≥ d, V (x) ≥ 0, V (x) is continuous and is not identically zero, and We study the asymptotic behavior as of the non-bottom negative eigenvalues of H_γ, which are born at the moment γ = 0 from the lower bound λ = 0 of the spectrum σ(H₀) of the unperturbed operator H₀ = ( − Δ)^l (virtual eigenvalues). To this end we use the Puiseux-Newton diagram for a power expansion of eigenvalues of some class of polynomial matrix functions. For the groups of virtual eigenvalues, having the same rate of decay, we obtain asymptotic estimates of Lieb-Thirring type.     

Boussinesq–Cerruti problems for the hexagonally symmetric system of elasticity in n–dimensional half-space

The Poisson matrices of the analoga to the Boussinesq–Cerruti boundary value problems for the operator of transversely isotropic elastostatics in n–dimensional half-space are computed by Fourier transformation and given in explicit form. (Received: May 4, 2004; revised: January 30, 2006)     

Finite difference preconditioning cubic spline collocation method of elliptic equations

Summary. We discuss a finite difference preconditioner for the interpolatory cubic spline collocation method for a uniformly elliptic operator defined by in (the unit square) with homogeneous Dirichlet boundary conditions. Using the generalized field of values arguments, we discuss the eigenvalues of the preconditioned matrix where is the matrix of the collocation discretization operator corresponding to , and is the matrix of the finite difference operator corresponding to the uniformly elliptic operator given by in with homogeneous Dirichlet boundary conditions. Finally we mention a bound of -singular values of for a general elliptic operator in . Received December 11, 1995 / Revised version received June 20, 1996