Victor Nistor

Bernd Ammann, Catarina Carvalho, Victor Nistor

We prove a regularity result in weighted Sobolev spaces (or Babuska--Kondratiev spaces) for the eigenfunctions of a Schrödinger operator. More precisely, let K_{a}^{m}(\mathbb{R}^{3N}) be the weighted Sobolev space obtained by blowing up the set of singular points of the Coulomb type potential V(x) = \sum_{1 \le j \le N} \frac{b_j}{|x_j|} + \sum_{1 \le i < j \le N} \frac{c_{ij}}{|x_i-x_j|}, x in \mathbb{R}^{3N}, b_j, c_{ij} in \mathbb{R}. If u in L^2(\mathbb{R}^{3N}) satisfies (-Δ+ V) u = λu in distribution sense, then u belongs to K_{a}^{m} for all m \in \mathbb{Z}_+ and all a \le 0. Our result extends to the case when b_j and c_{ij} are suitable bounded functions on the blown-up space. In the single-electron, multi-nuclei case, we obtain the same result for all a<3/2.

Constantin Bacuta, Hengguang Li, Victor Nistor

We study families of strongly elliptic, second order differential operators with singular coefficients on domains with conical points. We obtain uniform estimates on their inverses and on the regularity of the solutions to the associated Poisson problem with mixed boundary conditions. The coefficients and the solutions belong to (suitable) weighted Sobolev spaces. The space of coefficients is a Banach space that contains, in particular, the space of smooth functions. Hence, our results extend classical well-posedness results for strongly elliptic equations in domains with conical points to problems with singular coefficients. We furthermore provide precise uniform estimates on the norms of the solution operators.

Eugenie Hunsicker, Hengguang Li, Victor Nistor et al.

Let $V$ be a potential on $\RR^3$ that is smooth everywhere except at a discrete set $\maS$ of points, where it has singularities of the form $Z/ρ^2$, with $ρ(x) = |x - p|$ for $x$ close to $p$ and $Z$ continuous on $\RR^3$ with $Z(p) > -1/4$ for $p \in \maS$. Also assume that $ρ$ and $Z$ are smooth outside $\maS$ and $Z$ is smooth in polar coordinates around each singular point. We either assume that $V$ is periodic or that the set $\maS$ is finite and $V$ extends to a smooth function on the radial compactification of $\RR^3$ that is bounded outside a compact set containing $\maS$. In the periodic case, we let $Λ$ be the periodicity lattice and define $\TT := \RR^3/ Λ$. We obtain regularity results in weighted Sobolev space for the eigenfunctions of the Schrödinger-type operator $H = -Δ+ V$ acting on $L^2(\TT)$, as well as for the induced $\vt k$--Hamiltonians $\Hk$ obtained by restricting the action of $H$ to Bloch waves. Under some additional assumptions, we extend these regularity and solvability results to the non-periodic case. We sketch some applications to approximation of eigenfunctions and eigenvalues that will be studied in more detail in a second paper.

Hengguang Li, Victor Nistor, Yu Qiao

Let $Ω\subset \RR^d$, $d \geqslant 1$, be a bounded domain with piecewise smooth boundary $\partial Ω$ and let $U$ be an open subset of a Banach space $Y$. Motivated by questions in "Uncertainty Quantification," we consider a parametric family $P = (P_y)_{y \in U}$ of uniformly strongly elliptic, second order partial differential operators $P_y$ on $Ω$. We allow jump discontinuities in the coefficients. We establish a regularity result for the solution $u: Ω\times U \to \RR$ of the parametric, elliptic boundary value/transmission problem $P_y u_y = f_y$, $y \in U$, with mixed Dirichlet-Neumann boundary conditions in the case when the boundary and the interface are smooth and in the general case for $d=2$. Our regularity and well-posedness results are formulated in a scale of broken weighted Sobolev spaces $\hat\maK^{m+1}_{a+1}(Ω)$ of Babuška-Kondrat'ev type in $Ω$, possibly augmented by some locally constant functions. This implies that the parametric, elliptic PDEs $(P_y)_{y \in U}$ admit a shift theorem that is uniform in the parameter $y\in U$. In turn, this then leads to $h^m$-quasi-optimal rates of convergence (i.e. algebraic orders of convergence) for the Galerkin approximations of the solution $u$, where the approximation spaces are defined using the "polynomial chaos expansion" of $u$ with respect to a suitable family of tensorized Lagrange polynomials, following the method developed by Cohen, Devore, and Schwab (2010).

Eugenie Hunsicker, Hengguang Li, Victor Nistor et al.

Let $V$ be a {\em periodic} potential on $\RR^3$ that is smooth everywhere except at a discrete set $\maS$ of points, where it has singularities of the form $Z/ρ^2$, with $ρ(x) = |x - p|$ for $x$ close to $p$ and $Z$ is continuous, $Z(p) > -1/4$ for $p \in \maS$. We also assume that $ρ$ and $Z$ are smooth outside $\maS$ and $Z$ is smooth in polar coordinates around each singular point. Let us denote by $Λ$ the periodicity lattice and set $\TT := \RR^3/ Λ$. In the first paper of this series \cite{HLNU1}, we obtained regularity results in weighted Sobolev space for the eigenfunctions of the Schrödinger-type operator $H = -Δ+ V$ acting on $L^2(\TT)$, as well as for the induced $\vt k$--Hamiltonians $\Hk$ obtained by resticting the action of $H$ to Bloch waves. In this paper we present two related applications: one to the Finite Element approximation of the solution of $(L+\Hk) v = f$ and one to the numerical approximation of the eigenvalues, $λ$, and eigenfunctions, $u$, of $\Hk$. We give optimal, higher order convergence results for approximation spaces defined piecewise polynomials. Our numerical tests are in good agreement with the theoretical results.

Constantin Bacuta, Anna L. Mazzucato, Victor Nistor

We consider the model Poisson problem $-Δu = f \in Ω$, $u = g$ on $\pa Ω$, where $ Ω$ is a bounded polyhedral domain in $\RR^n$. The objective of the paper is twofold. The first objective is to review the well posedness and the regularity of our model problem using appropriate weighted spaces for the data and the solution. We use these results to derive the domain of the Laplace operator with zero boundary conditions on a concave domain, which seems not to have been fully investigated before. We also mention some extensions of our results to interface problems for the Elasticity equation. The second objective is to illustrate how anisotropic weighted regularity results for the Laplace operator in 3D are used in designing efficient finite element discretizations of elliptic boundary value problems, with the focus on the efficient discretization of the Poisson problem on polyhedral domains in $\RR^3$, following {\em Numer. Funct. Anal. Optim.}, 28(7-8):775--824, 2007. The anisotropic weighted regularity results described and used in the second part of the paper are a consequence of the well-posedness results in (isotropically) weighted Sobolev spaces described in the first part of the paper. The paper is based on the talk by the last named author at the Congress of Romanian Mathematicians, Brasov 2011, and is largely a survey paper.

Olesya Grishchenko, Xiao Han, Victor Nistor

We propose a general, very fast method to quickly approximate the solution of a parabolic Partial Differential Equation (PDEs) with explicit formulas. Our method also provides equaly fast approximations of the derivatives of the solution, which is a challenge for many other methods. Our approach is based on a computable series expansion in terms of a "small" parameter. As an example, we treat in detail the important case of the SABR PDE for $β= 1$, namely $\partial_τu = σ^2 \big [ \frac{1}{2} (\partial^2_xu - \partial_xu) + νρ\partial_x\partial_σu + \frac{1}{2} ν^2 \partial^2_σu \, \big ] + κ(θ- σ) \partial_σ$, by choosing $ν$ as small parameter. This yields $u = u_0 + νu_1 + ν^2 u_2 + \ldots$, with $u_j$ independent of $ν$. The terms $u_j$ are explicitly computable, which is also a challenge for many other, related methods. Truncating this expansion leads to computable approximations of $u$ that are in "closed form," and hence can be evaluated very quickly. Most of the other related methods use the "time" $τ$ as a small parameter. The advantage of our method is that it leads to shorter and hence easier to determine and to generalize formulas. We obtain also an explicit expansion for the implied volatility in the SABR model in terms of $ν$, similar to Hagan's formula, but including also the {\em mean reverting term.} We provide several numerical tests that show the performance of our method. In particular, we compare our formula to the one due to Hagan. Our results also behave well when used for actual market data and show the mean reverting property of the volatility.

Constantin Bacuta, Victor Nistor, Ludmil Zikatanov

We announce a well-posedness result for the Laplace equation in weighted Sobolev spaces on polyhedral domains in $\RR^n$ with Dirichlet boundary conditions. The weight is the distance to the set of singular boundary points. We give a detailed sketch of the proof in three dimensions.

Ivo Babuska, Victor Nistor

We study the approximation properties of a harmonic function $u \in H\sp{1-k}(Ω)$, $k > 0$, on relatively compact sub-domain $A$ of $Ω$, using the Generalized Finite Element Method. For smooth, bounded domains $Ω$, we obtain that the GFEM--approximation $u_S$ satisfies $\|u - u_S\|_{H\sp{1}(A)} \le C h^γ\|u\|_{H\sp{1-k}(Ω)}$, where $h$ is the typical size of the ``elements'' defining the GFEM--space $S$ and $γ\ge 0 $ is such that the local approximation spaces contain all polynomials of degree $k + γ+ 1$. The main technical result is an extension of the classical super-approximation results of Nitsche and Schatz \cite{NitscheSchatz72} and, especially, \cite{NitscheSchatz74}. It turns out that, in addition to the usual ``energy'' Sobolev spaces $H^1$, one must use also the negative order Sobolev spaces $H\sp{-l}$, $l \ge 0$, which are defined by duality and contain the distributional boundary data.

Marius Mitrea, Victor Nistor

We extend the method of layer potentials to manifolds with boundary and cylindrical ends. To obtain this extension along the classical lines, we have to deal with several technical difficulties due to the non-compactness of the boundary, which prevents us from using the standard characterization of Fredholm and compact (pseudo-)differential operators between Sobolev spaces. Our approach, which involves the study of layer potentials depending on a parameter on compact manifolds as an intermediate step, yields the invertibility of the relevant boundary integral operators in the global, non-compact setting, which is rather unexpected. As an application, we prove the well-posedness of the non-homogeneous Dirichlet problem on manifolds with boundary and cylindrical ends. We also prove the existence of the Dirichlet-to-Neumann map, which we show to be a pseudodifferential operator in the calculus of pseudodifferential operators that are ``almost translation invariant at infinity,'' a calculus that is closely related to Melrose's b-calculus \cite{me81, meaps}, which we study in this paper. The proof of the convergence of the layer potentials and of the existence of the Dirichlet-to-Neumann map are based on a good understanding of resolvents of elliptic operators that are translation invariant at infinity.