A mortar element method for hyperbolic problems

A non-conforming finite element method based on non-overlapping domain decomposition is extended to linear hyperbolic problems. The method is based on streamline-diffusion/discontinuous Galerkin methods and the mortar element method. A weak flux continuity condition at the inflow interface is enforced by means of Lagrange multipliers. This weak flux continuity condition replaces the usual mortar condition for elliptic problems, and allows non-matching grids at the subdomain interfaces. To cite this article: Y. Bourgault, A. El Boukili, C. R. Acad. Sci. Paris, Ser. I 338 (2004).     

The Nitsche mortar method for matching grids in a mixed finite element method

It is well-known that nonmatching grids are often used in finite element methods. Usually, grids are being matched along lines or surfaces that divide a domain into subdomains. Such lines or surfaces are called interfaces. The interface matching means the satisfaction of some continuity conditions when crossing the interface. The direct matching procedures fall into three groups: Methods that use Lagrange multipliers, mortar methods based on the Nitsche technique, and penalty methods.     

The Mortar Element Method with Lagrange Multipliers for Stokes Problem

In this paper,we propose a mortar element method with Lagrange multiplier for incompressible Stokes problem,i.e.,the matching constraints of velocity on mortar edges are expressed in terms of Lagrange multipliers.We also present P_1 noncon- forming element attached to the subdomains.By proving inf-sup condition,we derive optimal error estimates for velocity and pressure.Moreover,we obtain satisfactory approximation for normal derivatives of the velocity across the interfaces.     

Two-level primal-dual decomposition technique for large-scale nonconvex optimization problems with constraints

A two-level decomposition method for nonconvex separable optimization problems with additional local constraints of general inequality type is presented and thoroughly analyzed in the paper. The method is of primal-dual type, based on an augmentation of the Lagrange function. Previous methods of this type were in fact three-level, with adjustment of the Lagrange multipliers at one of the levels. This level is eliminated in the present approach by replacing the multipliers by a formula depending only on primal variables and Kuhn-Tucker multipliers for the local constraints. The primal variables and the Kuhn-Tucker multipliers are together the higher-level variables, which are updated simultaneously. Algorithms for this updating are proposed in the paper, together with their convergence analysis, which gives also indications on how to choose penalty coefficients of the augmented Lagrangian. Finally, numerical examples are presented.     

Error reduction of the adaptive conforming and nonconforming finite element methods with red–green refinement

In this paper, we analyze the convergence of the adaptive conforming and nonconforming $P_1$ finite element methods with red–green refinement based on standard Dörfler marking strategy. Since the mesh after refining is not nested into the one before, the usual Galerkin-orthogonality or quasi-orthogonality for newest vertex bisection does not hold for this case. To overcome such a difficulty, we develop some new quasi-orthogonality instead under certain condition on the initial mesh (Condition A). Consequently, we show convergence of the adaptive methods by establishing the reduction of some total errors. To weaken the condition on the initial mesh, we propose a modified red–green refinement and prove the convergence of the associated adaptive methods under a much weaker condition on the initial mesh (Condition B). Furthermore, we also develop an initial mesh generator which guarantee that all the interior triangles are equilateral triangles (satisfy Condition A) and the other triangles containing at least one vertex on the boundary satisfy Condition B.     

基于几何非协调分解的Lagrange乘子区域分解方法

本文考虑将Lagrange乘子区域分解方法应用于几何非协调分解的情况来求解二阶椭圆问题.由于采用几何非协调区域分解,每个局部乘子空间关联到多个界面,我们按照一定的规则选取合适的乘子面来定义乘子空间.利用局部正则化技巧,可以消去内部变量,得到关于Lagrange乘子的界面方程.采用一种经济的预条件迭代方法求解界面方程,且相关的预条件子是可扩展的.     

Parameter identification in periodic delay differential equations with distributed delay

In this study, a parameter identification approach for identifying the parameters of a periodic delayed system with distributed delay is introduced based on time series analysis and spectral element analysis. Using this approach the parameters of the distributed delayed system can be identified from the time series of the response of the system. The experimental or numerical data of the response is examined with Floquet theory and time series analysis techniques to estimate a reduced order dynamics, or truncated state space to identify the Floquet multipliers. Parameter identification is then completed using a dynamic map developed for the assumed model of the system which can relate the Floquet multipliers to the unknown parameters in the model. The parameter identification technique is validated numerically for first and second order delay differential equations with distributed delay.     

A counterexample to the conjecture of Aksionov and Mel'nikov on non-3-colorable planar graphs

Aksinov and Mel'nikov conjectured that every edge-critical non-3-colorable planar graph with triangles at distance at least one has connectivity 2. By constructing 3-connected edge-critical non-3-colorable planar graphs in which the distance between triangles is 2 or more, this conjecture is refuted.     

Triangle Contact Representations and Duality

A?contact representation by triangles of a graph is a set of triangles in the plane such that two triangles intersect on at most one point, each triangle represents a vertex of the graph and two triangles intersects if and only if their corresponding vertices are adjacent. De Fraysseix, Ossona de Mendez and Rosenstiehl proved that every planar graph admits a contact representation by triangles. We strengthen this in terms of a simultaneous contact representation by triangles of a planar map and of its dual. A?primal?Cdual contact representation by triangles of a planar map is a contact representation by triangles of the primal and a contact representation by triangles of the dual such that for every edge uv, bordering faces f and g, the intersection between the triangles corresponding to u and v is the same point as the intersection between the triangles corresponding to f and g. We prove that every 3-connected planar map admits a primal?Cdual contact representation by triangles. Moreover, the interiors of the triangles form a tiling of the triangle corresponding to the outer face and each contact point is a corner of exactly three triangles. Then we show that these representations are in one-to-one correspondence with generalized Schnyder woods defined by Felsner for 3-connected planar maps.     

On the Helly Number for Line Transversals to Parallel Triangles

We prove two results about the problem of finding the Helly number for line transversals to a family of parallel triangles in the plane: (1) If each three triangles of a family of parallel right triangles are intersected by an ascending (or a descending) line, then there is an ascending (or a descending) line that intersects all     

The superiority of “optimal control” over simulation in policy analysis

The simulation approach to policy analysis usually concentrates on policy multipliers as a measure of the thrust of economic policy. However, this measure is inadequate for one branch of economic policy, namely, fiscal policy. The reason is that the effectiveness of fiscal policy depends, via the government budget constraint, on the method of finance. It is argued in this paper that for this very reason the conventional way of calculating simulation-based dynamic multipliers introduces a bias towards the no-crowding-out thesis. This bias arises even in models of monetarist persuasion. Furthermore, it is shown that this bias can be removed by utilizing multipliers based on optimal control. We illustrate this proposition by providing numerical results using a large-scale U.K. econometric model of international monetarist persuasion (the London Business School model, LBS). Section 1 builds up a framework through which policy optimization can be compared and evaluated to policy simulations. In Section 2 we derive and compare policy multipliers obtained through policy simulations and optimal control. Section 3 provides a numerical example with the findings being summarized in Section 4.     

The 4-triangles longest-side partition of triangles and linear refinement algorithms

In this paper we study geometrical properties of the iterative 4-triangles longest-side partition of triangles (and of a 3-triangles partition), as well as practical algorithms based on these partitions, used both directly for the triangulation refinement problem, and as a basis for point insertion strategies in Delaunay refinement algorithms. The 4-triangles partition is obtained by joining the midpoint of the longest side with the opposite vertex and the midpoints of the two remaining sides. By means of simple geometrical arguments we show that the iterative partition of obtuse triangles systematically improves the triangles (while they remain obtuse) in the following sense: the sequence of smallest angles monotonically increases while the sequence of largest angles monotonically decreases in an amount (at least) equal to the smallest angle of each iteration. This allows us to improve the known bound on the smallest angle (without making use of previous results), and to obtain a better a priori bound on the number of similarly distinct triangles, as a function of the geometry of the initial triangle. Numerical evidence, showing that the practical behavior of the 4-triangles partition is in complete agreement with this theory, is included. A 4-triangles refinement algorithm is also discussed and illustrated. Furthermore, we show that the time cost of the algorithm is linear independently of the size of the triangulation.

     

Size Analysis of Nearly Regular Delaunay Triangulations

Consider a nearly regular point pattern in which a Delaunay triangulation is comprised of nearly equilateral triangles of the same size. We propose to model this set of points with Gaussian perturbations about a regular mean configuration. By investigating triangle subsets in detail we obtain various distributions of statistics based on size, or squared size of the triangles which is closely related to the mean (squared) distance to the six nearest neighbors. A scaleless test statistic, corresponding to a coefficient of variation for squared sizes, is proposed and its asymptotic properties described. The methodology is applied to an investigation of regularity in human muscle fiber cross-sections. We compare the approach with an alternative technique in a power study.     

Adaptive cubature over a collection of triangles using the d-transformation

We describe an automatic routine to integrate a function over a collection of triangles where on each triangle the function is well behaved, or has singularities of certain types at one or more vertices or edges. The underlying algorithm is globally adaptive and incorporates the d-transformation for extrapolation. Results from performance profile testing indicate that the routine is superior to other published routines when the singularities are located along edges of the triangles.     

Triangulation refinement by using edge subdivision

It is proved that any triangulation of a flat polygonal region can be refined by using repeated subdivisions of an edge so that: (1) the maximum diameter of the triangles would be less than any pre-assigned positive number, and (2) the minimum interior angle of the triangles of the triangulation obtained would be not less than the minimum interior angle of the triangles of the original triangulation divided by 9. The required triangulation refinement is constructed in two steps: first, the triangulation is refined so that the triangles of the triangulation obtained can be combined into pairs, and only boundary triangles may be left unpaired; at this step each triangle is split into at most 4 parts. Then the triangulation obtained is refined once again in order that the diameter of each triangle be less then a prescribed ?. At each of the steps, the minimum interior angle of triangles is reduced by at most 3 times. This is guaranteed by the lemma saying that the interior angles of the triangles into which the original triangle is divided by a median are at least as great as one-third of the minimum interior angle of the original triangle.     

Pairwise reactive sor algorithm for quadratic programming of net import spatial equilibrium models

An iterative method based on the successive overrelaxation (SOR) is proposed to solve quadratic programming of net important spatial equilibrium models. The algorithm solves the problem by updating the variables pairwise at each iteration. The principal feature of the algorithm is that the lagrange multipliers corresponding to the constraints do not have to be calculated at each iteration as is the case in SOR based algorithms. Yet the Lagrange multipliers can easily be extracted from the solution values.This research was partially supported by grants from the James F. Kember Foundation and the School of Business, Loyola University of Chicago.     

On Reduced Triangles in Normed Planes

A convex body is reduced if it does not properly contain a convex body of the same minimal width. In this paper we present new results on reduced triangles in normed (or Minkowski) planes, clearly showing how basic seemingly elementary notions from Euclidean geometry (like that of the regular triangle) spread when we extend them to arbitrary normed planes. Via the concept of anti-norms, we study the rich geometry of reduced triangles for arbitrary norms giving bounds on their side-lengths and on their vertex norms. We derive results on the existence and uniqueness of reduced triangles, and also we obtain characterizations of the Euclidean norm by means of reduced triangles. In the introductory part we discuss different topics from Banach Space Theory, Discrete Geometry, and Location Science which, unexpectedly, benefit from results on reduced triangles.     

Triangles in Euclidean Arrangements

The number of triangles in arrangements of lines and pseudolines has been the object of some research. Most results, however, concern arrangements in the projective plane. In this article we add results for the number of triangles in Euclidean arrangements of pseudolines. Though the change in the embedding space from projective to Euclidean may seem small there are interesting changes both in the results and in the techniques required for the proofs. In 1926 Levi proved that a nontrivial arrangement—simple or not—of n pseudolines in the projective plane contains at least n triangles. To show the corresponding result for the Euclidean plane, namely, that a simple arrangement of n pseudolines contains at least n-2 triangles, we had to find a completely different proof. On the other hand a nonsimple arrangement of n pseudolines in the Euclidean plane can have as few as 2n/3 triangles and this bound is best possible. We also discuss the maximal possible number of triangles and some extensions. Received February 12, 1998, and in revised form April 7, 1998.     

On fat partitioning, fat covering and the union size of polygons

The complexity of the contour of the union of simple polygons with n vertices in total can be O(n²) in general. A notion of fatness for simple polygons is introduced that extends most of the existing fatness definitions. It is proved that a set of fat polygons with n vertices in total has union complexity O(n log log n), which is a generalization of a similar result for fat triangles (Matou ek et al., 1994). Applications to several basic problems in computational geometry are given, such as efficient hidden surface removal, motion planning, injection molding, and more. The result is based on a new method to partition a fat simple polygon P with n vertices into O(n) fat convex quadrilaterals, and a method to cover (but not partition) a fat convex quadrilateral with O(l) fat triangles. The maximum overlap of the triangles at any point is two, which is optimal for any exact cover of a fat simple polygon by a linear number of fat triangles.