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1.
Hendrik Speleers Paul DierckxStefan Vandewalle 《Journal of Computational and Applied Mathematics》2007
Powell–Sabin splines are piecewise quadratic polynomials with a global C1-continuity, defined on conforming triangulations. Imposing boundary conditions on such a spline leads to a set of constraints on the spline coefficients. First, we discuss boundary conditions defined on a polygonal domain, before we treat boundary conditions on a general curved domain boundary. We consider Dirichlet and Neumann conditions, and we show that a particular choice of the PS-triangles at the boundary can greatly simplify the corresponding constraints. Finally, we consider an application where the techniques developed in this paper are used: the numerical solution of a partial differential equation by the Galerkin and collocation method. 相似文献
2.
Jan Grošelj 《BIT Numerical Mathematics》2016,56(4):1257-1280
In the paper, a family of bivariate super spline spaces of arbitrary degree defined on a triangulation with Powell–Sabin refinement is introduced. It includes known spaces of arbitrary smoothness r and degree \(3r-1\) but provides also other choices of spline degree for the same r which, in particular, generalize a known space of \(\mathscr {C}^{1}\) cubic super splines. Minimal determining sets of the proposed super spline spaces of arbitrary degree are presented, and the interpolation problems that uniquely specify their elements are provided. Furthermore, a normalized representation of the discussed splines is considered. It is based on the definition of basis functions that have local supports, are nonnegative, and form a partition of unity. The basis functions share numerous similarities with classical univariate B-splines. 相似文献
3.
Sara Remogna 《Advances in Computational Mathematics》2012,36(1):39-65
In this paper we construct discrete quasi-interpolants based on C
2 cubic multi-box splines on uniform Powell–Sabin triangulations of a rectangular domain. The main problem consists in finding
the coefficient functionals associated with boundary multi-box splines (i.e. multi-box splines whose supports overlap with
the domain) involving data points inside or on the boundary of the domain and giving the optimal approximation order. They
are obtained either by minimizing an upper bound for the infinity norm of the operator w.r.t. a finite number of free parameters,
or by inducing the superconvergence of the gradient of the quasi-interpolant at some specific points of the domain. Finally,
we give norm and error estimates and we provide some numerical examples illustrating the approximation properties of the proposed
operators. 相似文献
4.
Hendrik Speleers 《Constructive Approximation》2013,37(1):41-72
We construct a suitable B-spline representation for a family of bivariate spline functions with smoothness r≥1 and polynomial degree 3r?1. They are defined on a triangulation with Powell–Sabin refinement. The basis functions have a local support, they are nonnegative, and they form a partition of unity. The construction involves the determination of triangles that must contain a specific set of points. We further consider a number of CAGD applications. We show how to define control points and control polynomials (of degree 2r?1), and we provide an efficient and stable computation of the Bernstein–Bézier form of such splines. 相似文献
5.
《Journal of Computational and Applied Mathematics》2004,163(1):177-187
In this paper, we give a new proof of the famous Cayley–Bacharach theorem by means of interpolation, and deduce a general method of constructing properly posed set of nodes for bivariate Lagrange interpolation. As a result, we generalize the main results in Liang (On the interpolations and approximations in several variables, Jilin University, 1965), Liang and Lü (Approximation Theory IX, Vanderbilt University Press, 1988) and Liang et al. (Analysis, Combinatorics and Computing, Nova Science Publishers, Inc., New York, 2002) to the more extensive situations. 相似文献
6.
7.
H. Beirão da Veiga 《Annali dell'Universita di Ferrara》2014,60(1):23-34
These notes concern existence, and suitable formulation, of meaningful conditions on the direction of the vorticity which guarantee the regularity of the solutions to the evolution Navier–Stokes equations. A main concern here is to compare the different situations which appear in considering slip and no-slip boundary conditions. The paper reviews mainly results obtained in some of the references cited. 相似文献
8.
A finite element method of any order is applied on a Bakhvalov-type mesh to solve a singularly perturbed convection–diffusion equation in 2D, whose solution exhibits exponential boundary layers. A uniform convergence of (almost) optimal order is proved by means of a carefully defined interpolant. 相似文献
9.
Let * be the equilateral triangulation of the plane and let 1
* be the equilateral triangle formed by four triangles of *. We study the space of piecewise polynomial functions in C
k
(R
2) with support 1
*, having a sufficiently high degree n and which are invariant with respect to the group of symmetries of 1
*. Such splines are called 1
*-splines. We first compute the dimension of this space in function of n and k. Then, for any fixed k0, we prove the existence of 1
*-splines of class C
k
and minimal degree, but these splines are not unique. Finally, we describe an algorithm computing the Bernstein–Bézier coefficients of these splines. 相似文献
10.
11.
In this paper, the optimal replacement problem is investigated for a system with two types of failures. One type of failure is repairable, which is conducted by a repairman when it occurs, and the other is unrepairable, which leads to a replacement of the system at once. The repair of the system is not “as good as new”. The consecutive operating times of the system after repair form a decreasing geometric process, while the repair times after failure are assumed to be independent and identically distributed. Replacement policy N is adopted, where N is the number of repairable failures. The system will be replaced at the Nth repairable failure or at the unrepairable failure, whichever occurs first. Two replacement models are considered, one is based on the limiting availability and the other based on the long-run average cost rate of the system. We give the explicit expressions for the limiting availability and the long-run average cost rate of the system under policy N, respectively. By maximizing the limiting availability A(N) and minimizing the long-run average cost rate C(N), we theoretically obtain the optimal replacement policies N∗ in both cases. Finally, some numerical simulations are presented to verify the theoretical results. 相似文献
12.
This article is concerned with the asymptotical behavior of solutions for the three-dimensional damped Navier–Stokes equations with additive noise. Due to the shortage of the existence proof of the existence of random absorbing sets in a more regular phase space, we cannot obtain some kind of compactness of the cocycle associated with the three-dimensional damped Navier–Stokes equations with additive noise by the Sobolev compactness embedding theorem. In this paper, we prove the existence of a random attractor for the three-dimensional damped Navier–Stokes equations with additive noise by verifying the pullback flattening property. 相似文献
13.
14.
Wei Wang Li-Ping Pang Zun-Quan Xia 《应用数学学报(英文版)》2007,23(1):29-38
In this paper,the UV-theory and P-differential calculus are employed to study second-order ex-pansion of a class of D.C.functions and minimization problems.Under certain conditions,some properties ofthe U-Lagrangian,the second-order expansion of this class of functions along some trajectories are formulated.Some first and second order optimality conditions for the class of D.C.optimization problems are given. 相似文献
15.
We investigate blow-up properties for the initial-boundary value problem of a Keller–Segel model with consumption of chemoattractant when the spatial dimension is three. Through a kinetic reformulation of the Keller–Segel system, we first derive some higher-order estimates and obtain certain blow-up criteria for the local classical solutions. These blow-up criteria generalize the results in [4], [5] from the whole space to the case of bounded smooth domain . Lower global blow-up estimate on is also obtained based on our higher-order estimates. Moreover, we prove local non-degeneracy for blow-up points. 相似文献
16.
This paper studies the solutions of a reaction–diffusion system with nonlinearities that generalize the Lengyel–Epstein and FitzHugh–Nagumo nonlinearities. Sufficient conditions are derived for the global asymptotic stability of the solutions. Furthermore, we present some numerical examples. 相似文献
17.
A. A. Koshelev 《Russian Mathematics (Iz VUZ)》2016,60(2):25-32
In this paper we solve the problem of maximizing the value of the Laplace operator at the origin for functions such that the second degree of the Laplace operator belongs to the space L∞ on the unit ball of the Euclidean space. The problem is solved under restrictions on the uniform norm of a function and the L∞-norm of the second degree of the Laplace operator of this function. 相似文献
18.
In this paper, we consider the least l 2-norm solution for a possibly inconsistent system of nonlinear inequalities. The objective function of the problem is only first-order continuously differentiable. By introducing a new smoothing function, the problem is approximated by a family of parameterized optimization problems with twice continuously differentiable objective functions. Then a Levenberg–Marquardt algorithm is proposed to solve the parameterized smooth optimization problems. It is proved that the algorithm either terminates finitely at a solution of the original inequality problem or generates an infinite sequence. In the latter case, the infinite sequence converges to a least l 2-norm solution of the inequality problem. The local quadratic convergence of the algorithm was produced under some conditions. 相似文献
19.
A diffusive predator–prey model with predator competition is considered under Dirichlet boundary conditions. Some existence and non-existence results are firstly obtained. Then by investigating the bifurcation of positive solutions, the multiplicity of positive solutions is established for suitably large m. Furthermore, by meticulously analyzing the asymptotic behaviors of positive solutions when k goes to ∞, we find that there is at most a positive solution for any c∈R when k is sufficiently large. At last, some numerical simulations are presented to supplement the analytic results in one dimension. 相似文献
20.
G. V. Ravindra 《Mathematische Annalen》2009,345(3):731-748
We prove that any arithmetically Gorenstein curve on a smooth, general hypersurface of degree at least 6, is a complete intersection. This gives a characterisation of complete intersection curves on general
type hypersurfaces in . We also verify that certain 1-cycles on a general quintic hypersurface are non-trivial elements of the Griffiths group. 相似文献