Discrete maximum principles for finite element solutions of nonlinear elliptic problems with mixed boundary conditions

Summary. One of the most important problems in numerical simulations is the preservation of qualitative properties of solutions of the mathematical models by computed approximations. For problems of elliptic type, one of the basic properties is the (continuous) maximum principle. In our work, we present several variants of the maximum principles and their discrete counterparts for (scalar) second-order nonlinear elliptic problems with mixed boundary conditions. The problems considered are numerically solved by the continuous piecewise linear finite element approximations built on simplicial meshes. Sufficient conditions providing the validity of the corresponding discrete maximum principles are presented. Geometrically, they mean that the employed meshes have to be of acute or nonobtuse type, depending of the type of the problem. Finally some examples of real-life problems, where the preservation of maximum principles plays an important role, are presented.The first author was supported by the Hungarian Research Fund OTKA under grant no. F034840The second author was supported by the Agora Center under Grant InBCT of TEKES, Finland, and by the Academy Research Fellowship no. 208628 from the Academy of FinlandMathematics Subject Classification (2000): 35B50, 35J65, 65N30, 65N50     

Simplicial finite elements in higher dimensions

Over the past fifty years, finite element methods for the approximation of solutions of partial differential equations (PDEs) have become a powerful and reliable tool. Theoretically, these methods are not restricted to PDEs formulated on physical domains up to dimension three. Although at present there does not seem to be a very high practical demand for finite element methods that use higher dimensional simplicial partitions, there are some advantages in studying the methods independent of the dimension. For instance, it provides additional insights into the structure and essence of proofs of results in one, two and three dimensions. In this survey paper we review some recent progress in this direction. The second author was supported by Grant No. 112444 of the Academy of Finland. The third author was supported by grant No. 201/04/1503 of the Grant Agency of the Czech Republic.     

Mean convergence of hermite interpolation revisited

The first author is supported by the Hungarian National Foundation for Scientific Research Grant No. 1910 and No. T7570, and the second author is supported by the National Science Foundation Grant No. 9302721. The work was done during the first author's visit in Eugene, Oregon in 1993, and was completed during the second author's visit to the Mathematisches Institut, University of Erlange-Nürnberg, supported by the Alexander von Humboldt Foundation.     

Local structure-preserving algorithms for partial differential equations

In this paper, we discuss the concept of local structure-preserving algorithms (SPAs) for partial differential equations, which are the natural generalization of the corresponding global SPAs. Local SPAs for the problems with proper boundary conditions are global SPAs, but the inverse is not necessarily valid. The concept of the local SPAs can explain the difference between different SPAs and provide a basic theory for analyzing and constructing high performance SPAs. Furthermore, it enlarges the applicable scopes of SPAs. We also discuss the application and the construction of local SPAs and derive several new SPAs for the nonlinear Klein-Gordon equation. This work was supported by the National Basic Research Program (Grant No. 2005CB321703). The first author was supported by the National Natural Science Foundation of China (Grant Nos. 40405019, 10471067) and the Major Research Projects of Jiangsu Province (Grant No. BK2006725); the second author was supported by the National Natural Science Foundation of China (Innovation Group) (Grant No. 40221503) and the third author was supported by the National Natural Science Foundation of China (Grant No. 10471145)     

On heat kernel estimates and parabolic Harnack inequality for jump processes on metric measure spaces

In this paper, we discuss necessary and sufficient conditions on jumping kernels for a class of jump-type Markov processes on metric measure spaces to have scale-invariant finite range parabolic Harnack inequality.     

On the congruence lattice of a Scott-domain

We prove that the congruence lattice of a Scott-domain can be characterized as a complete lattice.Presented by V. Trnkova.The research of the first author was supported by the NSERC of Canada.The research of the second author was supported by the Hungarian National Foundation for Scientific Research, under Grant No. 1903.     

Maximum principle of optimal control for functional differential systems

We show that the maximum principle holds for optimal periodic control problems governed by functional differential equations under a Lipschitz condition on the value functional. Generalizations to other boundary conditions are also considered.This research was partially supported by NSF Grant No. DMS-84-01719.The first author was partially supported by the Science Fund of the Chinese Academy of Sciences, Beijing, China.     

On critical orientations in the kedem-sharir motion planning algorithm

We discuss a technical problem arising in the motion planning algorithm of Kedem and Sharir [KS], and propose a way to overcome it without increasing the asymptotic complexity of the algorithm The first author was supported by the Eshkol Grant 04601-90 from the Israeli Ministry of Science and Technology. The second author was partly supported by the Fund for Basic Research administered by the Israeli Academy of Sciences, by National Science Foundation Grants CCR-91-22103 and CCR-93-11127, and by grants from the U.S.-Israeli Binational Science Foundation, and the G.I.F., the German-Israeli Foundation for Scientific Research and Development. The third author was partly supported by the Interdisciplinary Program at Tel-Aviv University. The third author’s current address is: Department of Computer Science, MIT, Boston, MA, USA.     

Logics with disjunction and proof by cases

This paper is a contribution to the general study of consequence relations which contain (definable) connective of “disjunction”. Our work is centered around the “proof by cases property”, we present several of its equivalent definitions, and show some interesting applications, namely in constructing axiomatic systems for intersections of logics and recognizing weakly implicative fuzzy logics among the weakly implicative ones. The work of the first author was supported by the National Foundation of Natural Sciences of China (Grant no. 60663002) and by the Grant Project of science and technology of The Education Department of Jiangxi Province under Grant no. 200618. The work of the second author was supported by grant A100300503 of the Grant Agency of the Academy of Sciences of the Czech Republic and by Institutional Research Plan AVOZ10300504.     

A nonconforming finite element method for a two-dimensional curl–curl and grad-div problem

A numerical method for a two-dimensional curl–curl and grad-div problem is studied in this paper. It is based on a discretization using weakly continuous P ₁ vector fields and includes two consistency terms involving the jumps of the vector fields across element boundaries. Optimal convergence rates (up to an arbitrary positive ) in both the energy norm and the L ₂ norm are established on graded meshes. The theoretical results are confirmed by numerical experiments. The work of the first author was supported in part by the National Science Foundation under Grant No. DMS-03-11790 and by the Humboldt Foundation through her Humboldt Research Award. The work of the third author was supported in part by the National Science Foundation under Grant No. DMS-06-52481.     

Sharp Minima for Multiobjective Optimization in Banach Spaces

We study sharp minima for multiobjective optimization problems. In terms of the Mordukhovich coderivative and the normal cone, we present sufficient and or necessary conditions for existence of such sharp minima, some of which are new even in the single objective setting.This research was supported by a Central Research Grant of The Hong Kong Polytechnic University (Grant No. G-T 507). Research of the first author was also supported by the National Natural Science Foundation of PR China (Grant No. 10361008) and the Natural Science Foundation of Yunnan Province, China (Grant No. 2003A002M).     

A note on some analytic center cutting plane methods for convex feasibility and minimization problems

Recently Goffin, Luo and Ye have analyzed the complexity of an analytic center algorithm for convex feasibility problems defined by a separation oracle. The oracle is called at the (possibly approximate) analytic center of the set given by the linear inequalities which are the previous answers of the oracle. We discuss oracles for the problem of minimizing a convex (possibly nondifferentiable) function subject to box constraints, and give corresponding complexity estimates.The research of the first author is supported by the Polish Academy of Sciences; the research of the second author is supported by the State Committee for Scientific Research under Grant 8S50502206.     

On topological reflexivity of the groups of*-automorphisms and surjective isometrics of ∇ (H)

This research was partially carried out while the first author was a visitor to Abteilung Viathematik V, UniversitÄt Ulm, Germany. The second author's research was partially supported by the Hungarian National Research Foundation (OTKA), Grant No. F-019322     

A second-order method for the general nonlinear programming problem

This paper presents a multiplier-type method for nonlinear programming problems with both equality and inequality constraints. Slack variables are used for the inequalities. The penalty coefficient is adjusted automatically, and the method converges quadratically to points satisfying second-order conditions.The work of the first author was supported by NSF RANN and JSEP Contract No. F44620-71-C-0087; the work of the second author was supported by the National Science Foundation Grant No. ENG73-08214A01 and US Army Research Office Durham Contract No. DAHC04-73-C-0025.     

Some Geometrical Aspects of Efficient Points in Vector Optimization

We study efficient point sets in terms of extreme points, positive support points and strongly positive exposed points. In the case when the ordering cone has a bounded base, we prove that the efficient point set of a weakly compact convex set is contained in the closed convex hull of its strongly positive exposed points, thereby extending the Phelps theorem. We study also the density of positive proper efficient point sets. This research was supported by a Central Research Grant of Hong Kong Polytechnic University, Grant G-T 507. Research of the first author was also supported by the National Natural Science Foundation of P.R. China, Grant 10361008, and the Natural Science Foundation of Yunnan Province, China, Grant 2003A002M. Research of the second author was also supported by the Natural Science Foundation of Chongqing. Research of the third author was supported by a research grant from Australian Research Counsil.     

Complete solution of a family of quartic Thue equations

This paper evolved from a visit of the first author to the University of Debrecen, Hungary, which was supported by the Austrian-Hungarian Science Cooperation project Nr. 10-U-3. Research of the second author was partially supported by Hungarian National Foundation for Scientific Research Grant 1641/90.     

Blow-up Results for Some Nonlinear Delay Differential Equations

In this work we study the blow up phenomena for some scalar delay differential equations. In particular, we make connection with the blow up of ordinary differential equations that are related to the delay differential equations. The first author is supported by a Grant from TWAS under contract No: 03-030 RG/MATHS/AF/AC. The second author is supported by a grant from the Lebanese National Council for Scientific Research.     

Applications of a new space-partitioning technique

We present several applications of a recent space-partitioning technique of Chazelle, Sharir, and Welzl (Proceedings of the 6th Annual ACM Symposium on Computational Geometry, 1990, pp. 23–33). Our results include efficient algorithms for output-sensitive hidden surface removal, for ray shooting in two and three dimensions, and for constructing spanning trees with low stabbing number.Work on this paper has been supported by DIMACS, an NSF Science and Technology Center, under Grant STC-88-09684. The second author has been supported by Office of Naval Research Grants N00014-89-J-3042 and N00014-90-J-1284, by National Science Foundation Grant CCR-89-01484, and by grants from the U.S.-Israeli Binational Science Foundation, the Fund for Basic Research administered by the Israeli Academy of Sciences, and the G.I.F., the German-Israeli Foundation for Scientific Research and Development.     

Error bounds and convergence analysis of feasible descent methods: a general approach

We survey and extend a general approach to analyzing the convergence and the rate of convergence of feasible descent methods that does not require any nondegeneracy assumption on the problem. This approach is based on a certain error bound for estimating the distance to the solution set and is applicable to a broad class of methods.The research of the first author is supported by the Natural Sciences and Engineering Research Council of Canada, Grant No. OPG0090391, and the research of the second author is supported by the National Science Foundation, Grant No. CCR-9103804.