Acceleration of convergence for finite element solutions of the Poisson equation

Summary Letu _h be the finite element solution to–u=f with zero boundary conditions in a convex polyhedral domain . Fromu _h we calculate for eachz and ||1 an approximationu _h ^– (z) toD u(z) with |D u(z)–u _h ^– (z)|=O(h ^2k–2) wherek is the order of the finite elements. The same superconvergence order estimates are obtained also for the boundary flux. We need not work on a regular mesh but we have to compute averages ofu _h where the diameter of the domain of integration must not depend onh.     

Some remarks on the elastically supported membrane

Zusammenfassung Diese Arbeit entwickelt ein Paar Maximumprinzipien für Funktionen der Form |gradu|² u ^–2 und |gradv|² u ^–2. Dabei istu die erste Eigenfunktion einer am Rande elastisch gebundenen Membran, undv ist die zweite Eigenfunktion einer freien Membran. Im zweiten Falle wird ein Vergleichstheorem zwischen den zugehörigen Eigenwerten ₁ und ₂ hergeleitet.

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Résumé Cet article présente des principes de maximum pour des fonctions de la forme |gradu|² u ^–2 et |gradv|² u ^–2 oùu est la première fonction propre d'une membrane vibrante élastiquement liée sur son contour, etv est la seconde fonction propre d'une membrane libre. Dans le second cas, on obtient un théoreme de comparaison entre les valeurs propres correspondantes ₁ et ₂.

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This research was supported by the Swiss Nationalfonds.     

A Note on the Optimal L 2-Estimate of the Finite Volume Element Method

In this note, the optimal L ²-error estimate of the finite volume element method (FVE) for elliptic boundary value problem is discussed. It is shown that u–u _{h 0}Ch ²|ln h|^1/2f_1,1 and u–u _{h 0}Ch ²f_1,p , p>1, where u is the solution of the variational problem of the second order elliptic partial differential equation, u _h is the solution of the FVE scheme for solving the problem, and f is the given function in the right-hand side of the equation.     

Quelques propriétés des solutions de l'équation de Ginzburg-Landau sur un ouvert de ∝2

For a solution u of –u=u(1–|u|²) on the whole plane, |u|<1 holds everywhere unless u=eⁱ for some ; the derivatives of order k have moduli a constant M _kdepending only on k. For a solution u on an open set ², the moduli of u and its derivatives have upper bounds depending only on the distance to ²\ therefore the set of solutions on a given is compact in C() for the topology of uniform convergence on compact subsets of . For a solution u such that |u|<1, 1–|u| satisfies an estimation similar to the classical Harnack inequality for positive harmonic functions.Finally, if is bounded and |u| has a lim supm at each boundary point, the |u|m in if m1, but if m<1 then |u| admits only a majorant S _m with values in ]m, 1[ and sufficient conditions are given for lim S _m =0 or S _m =O(m) as m0.

     

On Non-Linear Rayleigh quotients

The stability with respect top of the non-linear eigenvalue problem div(|u| ^p–2u)+|u| ^p–2 u=0 is studied.     

On the Choquet Charge of δ-superharmonic Functions

Let u v be positive superharmonic functions in a general potential-theoretic setting, where these functions have a Choquet-type integral representation by minimal such functions with Choquet charges (i.e. representing measures) and , respectively. We show that on the contact set {u – v = 0} of the -superharmonic function u – v, if this set is properly interpreted as the set of those minimal superharmonic functions s which satisfy lim sup _{T s} v/u = 1 for the co-fine neighborhood filter T _s associated with s. In the setting of classical potential theory for Laplace's equation this result improves on results obtained by Fuglede in 1992.     

Hamiltonian Paths in Cartesian Powers of Directed Cycles

The vertex set of the k ^th cartesian power of a directed cycle of length m can be naturally identified with the abelian group ( _m ) ^k . For any two elements u=(u ₁,,u _k ) and v=(v ₁,,v _k ) of ( _m ) ^k , it is easy to see that if there is a hamiltonian path from u to v, then     

Number of no nisomorphic subgraphs in an n-point graph

LetG(n) be the set of all nonoriented graphs with n enumerated points without loops or multiple lines, and let v_k(G) be the number of mutually nonisomorphic k-point subgraphs of G G(n). It is proved that at least |G(n)| (1–1/n) graphs G G(n) possess the following properties: a) for any k [6log₂n], where c=–c log₂c–(1–c)×log₂(1–c) and c>1/2, we havev _k(G) > C _n ^k (1–1/n²); b) for any k [cn + 5 log₂n] we havev _k(G) = C _n ^k . Hence almost all graphs G G(n) containv(G) 2ⁿ pairwise nonisomorphic subgraphs.Translated from Matematicheskie Zametki, Vol. 9, No. 3, pp. 263–273, March, 1971.     

Quelques observations concernant les ensembles de Ditkin d'un groupe localement compact

We prove that every closed normal subgroupH of a locally compact amenable groupG is a Ditkin set with respect to the Herz-Figà-Talamanca algebraA _p (G) (p>1). Let be the canonical map ofG ontoG/H andF a closed subset ofG/H. We show thatF is a Ditkin set if and only if everyuA _p (G), which vanishes on ^–1, lies on the norm closure of the subspace ofA _p (G) generated by {u _h |hH, vA _p (G)C ₀₀(G)} whereu _h (x)=u(x h). As far as we know, this result seems to be new even forG abelian andp=2.     

Difference schemes for fourth-order one-dimensional equations with nonlinear boundary condition

A difference scheme is constructed for a boundary-value problem for a one-dimensional biharmonic equation with nonlinear boundary condition. Under the hypothesis that the exact solution of the problem belongs to the Sobolev space W ₂ ^k(),k [2, 4], in the lattice norm L ₂ (), an estimate is obtained of the precision of the difference scheme to O(h^k–1,5).Translated fromVychislitel'naya i Prikladnaya Matematika, No. 69, pp. 43–50, 1989.     

Nonlinear elliptic equations on compact Riemannian manifolds and asymptotics of Emden equations

We study the asymptotics and the global solutions of the following Emden equations: –u=e ^u in a 3-dim domain (>0) or –u=u ^q +|x|^–2 u (q>1) in anN-dim domain. Precise behaviour is obtained by the use of Simon's results on analytic geometric functionals. In the case of the first equation, or the second equation with =0 andq=(N+1)/(N–3) (N>3), we point out how the asymptotics are described via the Moebius group onS ^N–1. For a conformally invariant equation –u=|u|^4/(N–2) u+|x|^–2 u(=±1) we prove the existence of a new type of solution of the formu(x)=|x|(^2–N)/2((Ln|x|)(x/|x|)) where is defined onS ^N–1 and C (;O(N)). Finnally, we extend and simplify the results of Gidas and Spruck on semilinear elliptic equations on compact Riemannian manifolds by a systematic use of the Bochner-Licherowicz-Weitzenböck formula.     

Graph Connectivity After Path Removal

Let G be a graph and u, v be two distinct vertices of G. A u—v path P is called nonseparating if G—V(P) is connected. The purpose of this paper is to study the number of nonseparating u—v path for two arbitrary vertices u and v of a given graph. For a positive integer k, we will show that there is a minimum integer (k) so that if G is an (k)-connected graph and u and v are two arbitrary vertices in G, then there exist k vertex disjoint paths P ₁[u,v], P ₂[u,v], . . ., P _k [u,v], such that G—V (P _i [u,v]) is connected for every i (i = 1, 2, ..., k). In fact, we will prove that (k) 22k+2. It is known that (1) = 3.. A result of Tutte showed that (2) = 3. We show that (3) = 6. In addition, we prove that if G is a 5-connected graph, then for every pair of vertices u and v there exists a path P[u, v] such that G—V(P[u, v]) is 2-connected.* Supported by NSF grant No. DMS-0070059 Supported by ONR grant N00014-97-1-0499 Supported by NSF grant No. 9531824     

On the structure of singular sets of convex functions

Whenf is a convex function ofR ^h, andk is an integer with 0<k, then the set ^k (f)=x:dim(f(x)k may be covered by countably many manifolds of dimensionh–k and classC ² except an _h–k negligible subset.The author is supported by INdAM     

A Note on an Error Bound for the AOR Method

Suppose Ax = b is a system of linear equations where the matrix A is symmetric positive definite and consistently ordered. A bound for the norm of the errors _k = x– x ^k of the AOR method in terms of the norms of _k = x ^k–x ^k–1 and _k+1 = x ^k+1–x ^k and their inner product is derived.     

Solution of one problem of nonlinear adsorption kinetics by the method of lines

The method of lines is constructed and proved for numerical solution of a nonlinear initial-boundary-value problem of parabolic type describing the adsorption of a substance from an aqueous solution of bounded volume by a spherical adsorbent. The method is developed under natural assumptions on the smoothness of the solution of the original problem. The rate of convergence of the method depends on the smoothness of the initial function and is of order O(h) if v₀(x) 0, O(h^1/2) if v₀(x) C¹[0, 1], and 0(|v ₀(x)|W ₂ ¹ (O,h)).Translated from Vychislitel'naya i Prikladnaya Matematika, No. 63, pp. 24–30, 1987.     

Higher order methods for a singularly perturbed problem

Summary A finite element method using piecewise polynomials of degree k is used to approximate the problem u+u=f, >0 a small parameter. A very irregular mesh is used. On this mesh error estimates of order0(h ^k+1) are obtained uniformly in ,h the maximum stepsize, fork2. The condition number of the system of linear equations one has to solve in order to get the approximation is estimated. Extension of the results to more complicated problems is briefly indicated. Finally, a numerical example is given.Work performed while visiting the IBM Thomas J. Watson Research Center, Yorktown Heights, N.Y.     

A Class of Iterative Algorithms and Solvability of Nonlinear Variational Inequalities Involving Multivalued Mappings

The solvability of the following class of nonlinear variational inequality (NVI) problems based on a class of iterative procedures, which possess an equivalence to a class of projection formulas, is presented.Determine an element x ^* K and u ^* T(x ^*) such that u ^*, x – x ^* 0 for all x K where T: K P(H) is a multivalued mapping from a real Hilbert space H into P(H), the power set of H, and K is a nonempty closed convex subset of H. The iterative procedure adopted here is represented by a nonlinear variational inequality: for arbitrarily chosen initial points x ⁰, y ⁰ K, u ⁰ T(y ⁰) and v ⁰ T(x ⁰), we have u ^k + x ^k+1 – y ^k , x – x ^k+1 0, x K, for u ^k T(y ^k ) and for k 0where v ^k + y ^k – x ^k , x – y ^k 0, x K and for v ^k T(x ^k ).     

Some Asymptotic Formulae for q-Shifted Factorials

The q-shifted factorial defined by (a : q^k) _n = (1 – a) (1 – aq^k)(1 – aq^2k)... (1 – aq^{(n – 1)k}) appears in the terms of basic hypergeometric series. Complete asymptotic expansions as q 1 of some q-shifted factorials are given in terms of polylogarithms and Bernoulli polynomials.     

A property of nonlinear elliptic equations when the right-hand side is a measure

LetA(u)=–diva(x, u, Du) be a Leray-Lions operator defined onW ₀ ^1,p () and be a bounded Radon measure. For anyu SOLA (Solution Obtained as Limit of Approximations) ofA(u)= in ,u=0 on , we prove that the truncationsT _k(u) at heightk satisfyA(T _k(u)) A(u) in the weak * topology of measures whenk + .

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Résumé SoitA(u)=–diva(x, u, Du) un opérateur de Leray-Lions défini surW ₀ ^1,p () et une mesure de Radon bornée. Pour toutu SOLA (Solution Obtenue comme Limite d'Approximations) deA(u)= dans ,u=0 sur , nous démontrons que les troncaturesT _k(u) à la hauteurk vérifientA(T _k(u)) A(u) dans la topologie faible * des mesures quandk + .