Victor Nistor

Improving the rate of convergence of ‘high order finite elements’ on polygons and domains with cusps

Summary.Let u and uV ∈ V be the solution and, respectively, the discrete solution of the non-homogeneous Dirichlet problem Δu=f on ℙ, u|∂ℙ=0. For any m ∈ ℕ and any bounded polygonal domain ℙ, we provide a construction of a new sequence of finite dimensional subspaces Vn such that where f ∈ Hm−1(ℙ) is arbitrary and C is a constant that depends only on ℙ and not on n (we do not assume u ∈ Hm+1(ℙ)). The existence of such a sequence of subspaces was first proved in a ground–breaking paper by Babuška [8]. Our method is different; it is based on the homogeneity properties of Sobolev spaces with weights and the well–posedness of non-homogeneous Dirichlet problem in suitable Sobolev spaces with weights, for which we provide a new proof, and which is a substitute of the usual “shift theorems” for boundary value problems in domains with smooth boundary. Our results extended right away to domains whose boundaries have conical points. We also indicate some of the changes necessary to deal with domains with cusps. Our numerical computation are in agreement with our theoretical results.

Analysis of geometric operators on open manifolds: A groupoid approach

The first five sections of this paper are a survey of algebras of pseudodifferential operators on groupoids. We thus review differentiable groupoids, the definition of pseudodifferential operators on groupoids, and some of their properties. We use then this background material to establish a few new results on these algebras, results that are useful for the analysis of geometric operators on non-compact manifolds and singular spaces. The first step is to establish that the geometric operators on groupoids are in our algebras. This then leads to criteria for the Fredholmness of geometric operators on suitable non-compact manifolds, as well as to an inductive procedure to study their essential spectra. As an application, we answer a question of Melrose on the essential spectrum of the Laplace operator on manifolds with multi-cylindrical ends.

Improving the Rate of Convergence of High-Order Finite Elements on Polyhedra I: A Priori Estimates

ABSTRACT Let 𝒯 k be a sequence of triangulations of a polyhedron Ω ⊂ ℝ n and let S k be the associated finite element space of continuous, piecewise polynomials of degree m. Let u k ∈ S k be the finite element approximation of the solution u of a second-order, strongly elliptic system Pu = f with zero Dirichlet boundary conditions. We show that a weak approximation property of the sequence S k ensures optimal rates of convergence for the sequence u k . The method relies on certain a priori estimates in weighted Sobolev spaces for the system Pu = 0 that we establish. The weight is the distance to the set of singular boundary points. We obtain similar results for the Poisson problem with mixed Dirichlet–Neumann boundary conditions on a polygon.

ON THE GEOMETRY OF RIEMANNIAN MANIFOLDS WITH A LIE STRUCTURE AT INFINITY

We study a generalization of the geodesic spray and give conditions for noncomapct manifolds with a Lie structure at infinity to have positive injectivity radius. We also prove that the geometric operators are generated by the given Lie algebra of vector fields. This is the first one in a series of papers devoted to the study of the analysis of geometric differential operators on manifolds with Lie structure at infinity.

Groupoids and an index theorem for conical pseudo-manifolds

Abstract We define an analytical index map and a topological index map for conical pseudomanifolds. These constructions generalize the analogous constructions used by Atiyah and Singer in the proof of their topological index theorem for a smooth, compact manifold M. A main new ingredient in our proof is a non-commutative algebra that plays in our setting the role of 𝒞0(T*M). We prove a Thom isomorphism between non-commutative algebras which gives a new example of wrong way functoriality in K-theory. We then give a new proof of the Atiyah-Singer Index Theorem using deformation groupoids and show how it generalizes to conical pseudomanifolds. We thus prove a topological index theorem for conical pseudomanifolds.

Improving the rate of convergence of `high order finite elements' on polyhedra II: Mesh refinements and interpolation

We construct a sequence of meshes 𝒯 k ′ that provides quasi-optimal rates of convergence for the solution of the Poisson equation on a bounded polyhedral domain with right-hand side in H m−1, m ≥ 2. More precisely, let Ω ⊂ ℝ3 be a bounded polyhedral domain and let u ∈ H 1(Ω) be the solution of the Poisson problem − Δ u = f ∈ H m−1(Ω), m ≥ 2, u = 0 on ∂ Ω. Also, let S k be the finite element space of continuous, piecewise polynomials of degree m ≥ 2 on 𝒯 k ′ and let u k ∈ S k be the finite element approximation of u, then ‖u − u k ‖ H 1(Ω) ≤ C dim(S k )−m/3 ‖f‖ H m−1(Ω), with C independent of k and f. Our method relies on the a priori estimate ‖u‖𝒟 ≤ C ‖f‖ H m−1(Ω) in certain anisotropic weighted Sobolev spaces , with a > 0 small, determined only by Ω. The weight is the distance to the set of singular boundary points (i.e., edges). The main feature of our mesh refinement is that a segment AB in 𝒯 k ′ will be divided into two segments AC and CB in 𝒯 k+1′ as follows: |AC| = |CB| if A and B are equally singular and |AC| = κ |AB| if A is more singular than B. We can choose κ ≤ 2−m/a . This allows us to use a uniform refinement of the tetrahedra that are away from the edges to construct 𝒯 k ′.

Cyclic cohomology of crossed products by algebraic groups

4 Reduction to the Maximal Compact Subgroup 19 4.1 The ring C∞ inv(G) . . . . . . . . . . . . . . . . . . . . . . . . . 19 4.2 Localization at x and at Gx . . . . . . . . . . . . . . . . . . . 20 4.3 Definition of X (j) and δ . . . . . . . . . . . . . . . . . . . . . . 22 4.4 The definition of σ . . . . . . . . . . . . . . . . . . . . . . . . 23 4.5 The definition of η and the equation ∇ = σδ + δσ . . . . . . . 23 4.6 The condition of the Main Lemma are satisfied . . . . . . . . . 25 4.7 The Main Theorem . . . . . . . . . . . . . . . . . . . . . . . . 27 4.8 Some nonalgebraic groups . . . . . . . . . . . . . . . . . . . . 28

COMPLEX POWERS AND NON-COMPACT MANIFOLDS

Abstract We study the complex powers A z of an elliptic, strictly positive pseudodifferential operator A using an axiomatic method that combines the approaches of Guillemin and Seeley. In particular, we introduce a class of algebras, called “Guillemin algebras, ” whose definition was inspired by Guillemin [Guillemin, V. (1985). A new proof of Weyls formula on the asymptotic distribution of eigenvalues. Adv. in Math. 55:131–160]. A Guillemin algebra can be thought of as an algebra of “abstract pseudodifferential operators.” Most algebras of pseudodifferential operators belong to this class. Several results typical for algebras of pseudodifferential operators (asymptotic completeness, construction of Sobolev spaces, boundedness between appropriate Sobolev spaces,…) generalize to Guillemin algebras. Most important, this class of algebras provides a convenient framework to obtain precise estimates at infinity for A z , when A > 0 is elliptic and defined on a non-compact manifold, provided that a suitable ideal of regularizing operators is specified (a submultiplicative Ψ*-algebra). We shall use these results in a forthcoming paper to study pseudodifferential operators and Sobolev spaces on manifolds with a Lie structure at infinity (a certain class of non-compact manifolds that has emerged from Melroses work on geometric scattering theory [Melrose, R. B. (1995). Geometric Scattering Theory. Stanford Lectures. Cambridge: Cambridge University Press]).

Weighted Sobolev spaces and regularity for polyhedral domains

We prove a regularity result for the Poisson problem -Δu=f, u|∂P=g on a polyhedral domain P⊂R3 using the Babuska–Kondratiev spaces Kam(P). These are weighted Sobolev spaces in which the weight is given by the distance to the set of edges [4,33]. In particular, we show that there is no loss of Kam—regularity for solutions of strongly elliptic systems with smooth coefficients. We also establish a “trace theorem” for the restriction to the boundary of the functions in Kam(P).

SPECTRAL INVARIANCE FOR CERTAIN ALGEBRAS OF PSEUDODIFFERENTIAL OPERATORS

We construct algebras of pseudodifferential operators on a continuous family groupoid G that are closed under holomorphic functional calculus, contain the algebra of all pseudodifferential operators of order 0 on G as a dense subalgebra, and reflect the smooth structure of the groupoid G, when G is smooth. As an application, we get a better understanding on the structure of inverses of elliptic pseudodifferential operators on classes of non-compact manifolds. For the construction of these algebras closed under holomorphic functional calculus, we develop three methods: one using two-sided semi-ideals, one using commutators, and one based on Schwartz spaces on the groupoid.