Asymptotically compatible schemes for the approximation of fractional Laplacian and related nonlocal diffusion problems on bounded domains

Approximations of solutions of fractional Laplacian equations on bounded domains are considered. Such equations allow global interactions between points separated by arbitrarily large distances. Two approximations are introduced. First, interactions are localized so that only points less than some specified distance, referred to as the interaction radius, are allowed to interact. The resulting truncated problem is a special case of a more general nonlocal diffusion problem. The second approximation is the spatial discretization of the related nonlocal diffusion problem. A recently developed abstract framework for asymptotically compatible schemes is applied to prove convergence results for solutions of the truncated and discretized problem to the solutions of the fractional Laplacian problems. Intermediate results also provide new convergence results for the nonlocal diffusion problem. Special attention is paid to limiting behaviors as the interaction radius increases and the spatial grid size decreases, regardless of how these parameters may or may not be dependent. In particular, we show that conforming Galerkin finite element approximations of the nonlocal diffusion equation are always asymptotically compatible schemes for the corresponding fractional Laplacian model as the interaction radius increases and the grid size decreases. The results are developed with minimal regularity assumptions on the solution and are applicable to general domains and general geometric meshes with no restriction on the space dimension and with data that are only required to be square integrable. Furthermore, our results also solve an open conjecture given in the literature about the convergence of numerical solutions on a fixed mesh as the interaction radius increases.     

Multiplicative perturbations of constant-domain evolution equations

I prove a variation-of-constants formula and an existence theorem for multiplicative perturbations of nonautonomous linear equations, in the constant-domain, nonparabolic case (CD-systems).We use the properties of the evolution process generated by a CD-system: in particular an estimate of the integral product of the process with the perturbation term, taken in the constant Favard class of the CD-system. Using the extrapolation spaces and an extension of U(t, s) we are able to define a mild solution and to prove a corresponding existence and regularity theorem.As application I treat a size-structured population equation. (This paper was written with the financial support of the CNR (Italy).)     

Time-dependent gradient perturbations of fractional Laplacian

We construct a fundamental solution of the equation ${\partial_t - \Delta^{\alpha/2} - b(\cdot, \cdot) \cdot\nabla_{x} = 0}We construct a fundamental solution of the equation ?_t - D^a/2 - b(·, ·) ·?_x = 0{\partial_t - \Delta^{\alpha/2} - b(\cdot, \cdot) \cdot\nabla_{x} = 0} for a ? (1, 2){\alpha \in (1, 2)} and b satisfying a certain integral space-time condition. We also show it has α-stable upper and lower bounds.     

TheL pDirichlet problem for small perturbations of the Laplacian

We show that Dahlberg's vanishing trace condition measuring the disagreement between the coefficients of two elliptic operators preserves harmonic measures whose logarithm belongs to VMO.     

Smooth approximation of singular perturbations

We consider a certain subclass of self-adjoint extensions of the symmetric operator

     

Sharp spectral bounds for complex perturbations of the indefinite Laplacian

We derive quantitative bounds for eigenvalues of complex perturbations of the indefinite Laplacian on the real line. Our results substantially improve existing results even for real potentials. For $L^1$-potentials, we obtain optimal spectral enclosures which accommodate also embedded eigenvalues, while our result for $L^p$-potentials yield sharp spectral bounds on the imaginary parts of eigenvalues of the perturbed operator for all $p\in [1,\infty )$. The sharpness of the results are demonstrated by means of explicit examples.     

Linear self-adjoint multipoint boundary value problems and related approximation schemes

Summary Linear self-adjoint multipoint boundary value problems are investigated. The case of the homogeneous equation is shown to lead to spline solutions, which are then utilized to construct a Green's function for the case of homogeneous boundary conditions. An approximation scheme is described in terms of the eigen-functions of the inverse of the Green's operator and is shown to be optimal in the sense of then-widths of Kolmogorov. Convergence rates are given and generalizations to more general boundary value problems are discussed.     

Multiplicative symmetry and related functional equations

Summary Let (G, *) be a commutative monoid. Following J. G. Dhombres, we shall say that a functionf: G G is multiplicative symmetric on (G, *) if it satisfies the functional equationf(x * f(y)) = f(y * f(x)) for allx, y inG. (1)Equivalently, iff: G G satisfies a functional equation of the following type:f(x * f(y)) = F(x, y) (x, y G), whereF: G × G G is a symmetric function (possibly depending onf), thenf is multiplicative symmetric on (G, *).In Section I, we recall the results obtained for various monoidsG by J. G. Dhombres and others concerning the functional equation (1) and some functional equations of the formf(x * f(y)) = F(x, y) (x, y G), (E) whereF: G × G G may depend onf. We complete these results, in particular in the case whereG is the field of complex numbers, and we generalize also some results by considering more general functionsF. In Section II, we consider some functional equations of the formf(x * f(y)) + f(y * f(x)) = 2F(x, y) (x, y K), where (K, +, ·) is a commutative field of characteristic zero, * is either + or · andF: K × K K is some symmetric function which has already been considered in Section I for the functional equation (E). We investigate here the following problem: which conditions guarantee that all solutionsf: K K of such equations are multiplicative symmetric either on (K, +) or on (K, ·)? Under such conditions, these equations are equivalent to some functional equations of the form (E) for which the solutions have been given in Section I. This is a partial answer to a question asked by J. G. Dhombres in 1973.     

On the approximation of Laplacian eigenvalues in graph disaggregation

Graph disaggregation is a technique used to address the high cost of computation for power law graphs on parallel processors. The few high-degree vertices are broken into multiple small-degree vertices, in order to allow for more efficient computation in parallel. In particular, we consider computations involving the graph Laplacian, which has significant applications, including diffusion mapping and graph partitioning, among others. We prove results regarding the spectral approximation of the Laplacian of the original graph by the Laplacian of the disaggregated graph. In addition, we construct an alternate disaggregation operator whose eigenvalues interlace those of the original Laplacian. Using this alternate operator, we construct a uniform preconditioner for the original graph Laplacian.     

Multiplicative perturbations in semigroup theory with the (Z)-condition

In [4] the (Z)-condition was introduced by G. W. Desch and W. Schappacher. This condition on a Banach spaceZ and a generatorA inX ensures thatA(I+B) and(I+B)A generateC ₀-semigroups, ifB has its range inZ. In this paper we will consider how certain properties of the semigroup generated byA are inherited by the semigroups generated byA(I+B) and(I+B)A. We shall furthermore investigate a related condition, that simplifies certain sufficiency assumptions given previously.     

Some notes on the spectral perturbations of the signless Laplacian of a graph

Let G be a simple graph and let Q（G） be the signless Laplacian matrix of G. In this paper we obtain some results on the spectral perturbation of the matrix Q（G） under an edge addition or an edge contraction.     

Laplacian eigenvalues and partition problems in hypergraphs

We use the generalization of the Laplacian matrix to hypergraphs to obtain several spectral-like results on partition problems in hypergraphs which are computationally difficult to solve (NP-hard or NP-complete). Therefore it is very important to obtain nontrivial bounds. More precisely, the following parameters are bounded in the paper: bipartition width, averaged minimal cut, isoperimetric number, max-cut, independence number and domination number.     

Singular perturbations of elliptic problems

Summary The paper treates applications of singular perturbations of variational inequalities, to differential problems. Some informations on the boundary layer phenomenon are obtained. Entrata in Redazione il 20 ottobre 1971.     

Discontinuous Galerkin approximation with discrete variational principle for the nonlinear Laplacian

A discontinuous Galerkin method is analyzed to approximate the nonlinear Laplacian model problem. The salient feature of the proposed scheme is that it is endowed with a discrete variational principle. The convergence of the discrete approximations to the exact solution is proven. To cite this article: E. Burman, A. Ern, C. R. Acad. Sci. Paris, Ser. I 346 (2008).     

Complexity of approximation problems

We consider some aspects of optimal encoding and renewal related to the problem of complexity of the ε-definition of functions posed by Kolmogorov in 1962. We present some estimates for the ε-complexity of the problem of renewal of functions in the uniform metric and Hausdorff metric.