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991.
Résumè Cet article a pour objet la recherche, à partir de la théorie des polynômes orthogonaux, de conditions permettant l'obtention de formules de quadrature numérique sur des domaines de n, avec fonction poids, à nombre minimal de noeuds et exactes sur les espacesQ
k de polynômes de degré k par rapport à chacune de leurn variables. Ces résultats, complétés par des exemples numériques originaux dans 2, adaptent à ces espacesQ
k ceux démontréq par H.J. Schmid [14] dans le cadre des espacesP
k de polynômes.
About Cubature formulas with a minimal number of knots
Summary In this paper we search, from the orthogonal polynomial theory, for conditions which allow to obtain cubature formulas on sets of n, with weight function. which have a minimal number of knots and which are exact on the spaceQ k of all polynomials of degree k with respect to each variablex i, 1in.These results, completed by original numerical examples in 2, adapt to the spacesQ k those proved by H.J. Schmid [14] in the case of polynomial spacesP k.相似文献
992.
Summary We examine the problem:u+a(x)u–b(x)u=f(x) for 0<x<1,a(x)>0,b(x)>,
2
= 4>0,a, b andf inC
2 [0, 1], in (0, 1],u(0) andu(1) given. Using finite elements and a discretized Green's function, we show that the El-Mistikawy and Werle difference scheme on an equidistant mesh of widthh is uniformly second order accurate for this problem (i.e., the nodal errors are bounded byCh
2, whereC is independent ofh and ). With a natural choice of trial functions, uniform first order accuracy is obtained in theL
(0, 1) norm. On choosing piecewise linear trial functions (hat functions), uniform first order accuracy is obtained in theL
1 (0, 1) norm. 相似文献
993.
E. Hairer 《Numerische Mathematik》1986,48(4):383-389
Summary Using a special representation of Runge-Kutta methods (W-transformation), simple characterizations ofA-stability andB-stability have been obtained in [9, 8, 7]. In this article we will make this representation and their conclusions more transparent by considering the exact Runge-Kutta method. Finally we demonstrate by a numerical example that for difficult problemsB-stable methods are superior to methods which are onlyA-stable.Talk, presented at the conference on the occasion of the 25th anniversary of the founding ofNumerische Mathematik, TU Munich, March 19–21, 1984 相似文献
994.
Summary Runge-Kutta methods have been generalized to procedures with higher derivatives of the right side ofy=f(t,y) e.g. by Fehlberg 1964 and Kastlunger and Wanner 1972. In the present work some sufficient conditions for the order of consistence are derived for these methods using partially the degree of the corresponding numerical integration formulas. In particular, methods of Gauß, Radau, and Lobatto type are generalized to methods with higher derivatives and their maximum order property is proved. The applied technique was developed by Crouzeix 1975 for classical Runge-Kutta methods. Examples of simple explicit and semi-implicit methods are given up to order 7 and 6 respectively. 相似文献
995.
On the multi-level splitting of finite element spaces 总被引:13,自引:0,他引:13
Harry Yserentant 《Numerische Mathematik》1986,49(4):379-412
Summary In this paper we analyze the condition number of the stiffness matrices arising in the discretization of selfadjoint and positive definite plane elliptic boundary value problems of second order by finite element methods when using hierarchical bases of the finite element spaces instead of the usual nodal bases. We show that the condition number of such a stiffness matrix behaves like O((log )2) where is the condition number of the stiffness matrix with respect to a nodal basis. In the case of a triangulation with uniform mesh sizeh this means that the stiffness matrix with respect to a hierarchical basis of the finite element space has a condition number behaving like
instead of
for a nodal basis. The proofs of our theorems do not need any regularity properties of neither the continuous problem nor its discretization. Especially we do not need the quasiuniformity of the employed triangulations. As the representation of a finite element function with respect to a hierarchical basis can be converted very easily and quickly to its representation with respect to a nodal basis, our results mean that the method of conjugate gradients needs onlyO(log n) steps andO(n log n) computer operations to reduce the energy norm of the error by a given factor if one uses hierarchical bases or related preconditioning procedures. Heren denotes the dimension of the finite element space and of the discrete linear problem to be solved. 相似文献
996.
Klaus Bartke 《Numerische Mathematik》1984,43(3):379-388
Summary We determine the connected components of the set of normal elements of the family
m
n
[a,b] of rational functions. Numerical difficulties occuring with the computation of the Chebyshev approximation via the Remez algorithm can be caused by its disconnectedness. In order to illustrate this we give numerical examples.
Gefördert von der DFG unter Nr. Be 808/2 相似文献
997.
Summary The existence of optimal nodes with preassigned multiplicities is proved for the Hardy spacesH
p
(1<p<). This is then used to show that the exact order of convergence for the optimal qudrature formula withN nodes (including multiplicity) is
where 1/p+1/q=1 and 1p. 相似文献
998.
Rudolf Wegmann 《Numerische Mathematik》1984,44(3):435-461
Summary The iterative method as introduced in [8] and [9] for the determination of the conformal mapping of the unit disc onto a domainG is here described explicitly in terms of the operatorK, which assigns to a periodic functionu its periodic conjugate functionK u. It is shown that whenever the boundary curve ofG is parametrized by a function with Lipschitz continuous derivative
then the method converges locally in the Sobolev spaceW of 2-periodic absolutely continuous functions with square integrable derivative. If is in a Hölder classC
2+, the order of convergence is at least 1+. If is inC
l+1+ withl1, 0<<1, then the iteration converges inC
l+. For analytic boundary curves the convergence takes place in a space of analytic functions.For the numerical implementation of the method the operatorK can be approximated by Wittich's method, which can be applied very effectively using fast Fourier transform. The Sobolev norm of the numerical error can be estimated in terms of the numberN of grid points. It isO(N
1–l–) if is inC
l+1+, andO (exp (–N/2)) if is an analytic curve. The number in the latter formula is bounded by logR, whereR is the radius of the largest circle into which can be extended analytically such that'(z)0 for |z|<R. The results of some test calculations are reported. 相似文献
999.
Summary A Gauss-Seidel procedure for accelerating the convergence of the generalized method of the root iterations type of the (k+2)-th order (kN) for finding polynomial complex zeros, given in [7], is considered in this paper. It is shown that theR-order of convergence of the accelerated method is at leastk+1+
n
(k), where
n
(k)>1 is the unique positive root of the equation
n
--k-1 = 0 andn is the degree of the polynomial. The examples of algebraic equations in ordinary and circular arithmetic are given. 相似文献
1000.
Rolf Rannacher 《Numerische Mathematik》1984,43(2):309-327
Summary Diffusion problems occuring in practice often involve irregularities in the initial or boundary data resulting in a local break-down of the solution's regularity. This may drastically reduce the accuracy of discretization schemes over the whole interval of integration, unless certain precautions are taken. The diagonal Padé schemes of order 2, combined with a standard finite element discretization, usually require an unnatural step size restriction in order to achieve even locally optimal accuracy. It is shown here that this restriction can be avoided by means of a sample damping procedure which preserves the order of the discretization and, in the case =1, does not increase the costs. 相似文献