We continue the study initiated in Hughes et al. (2010) in search of optimal quadrature rules for tensor product and hierarchically refined splines in isogeometric analysis. These rules are optimal ...in the sense that there exists no other quadrature rule that can exactly integrate the elements of the given spline space with fewer quadrature points. We extend the algorithm presented in Hughes et al. (2010) with an improved starting guess, which combined with arbitrary precision arithmetic, results in the practical computation of quadrature rules for univariate non-uniform splines up to any precision. Explicit constructions are provided in sixteen digits of accuracy for some of the most commonly used uniform spline spaces defined by open knot vectors. We study the efficacy of the proposed rules in the context of full and reduced quadrature applied to two- and three-dimensional diffusion–reaction problems using tensor product and hierarchically refined splines, and prove a theorem rigorously establishing the stability and accuracy of the reduced rules.
Radau- and Lobatto-type averaged Gauss rules Reichel, Lothar; Spalević, Miodrag M.
Journal of computational and applied mathematics,
February 2024, 2024-02-00, Letnik:
437
Journal Article
Recenzirano
We describe numerical methods for the construction of interpolatory quadrature rules of Radau and Lobatto types. In particular, we are interested in deriving efficient algorithms for computing ...optimal averaged Gauss–Radau and Gauss–Lobatto type javascript:undefined;quadrature rules. These averaged rules allow us to estimate the quadrature error in Gauss–Radau and Gauss–Lobatto quadrature rules. This is important since the latter rules have higher algebraic degree of exactness than the corresponding Gauss rules, and this makes it possible to construct averaged quadrature rules of higher algebraic degree of exactness than the corresponding “standard” averaged Gauss rules available in the literature.
In this paper we consider Gaussian type quadrature rules for trigonometric polynomials where an even number of nodes is fixed in advance. For an integrable and nonnegative weight function w on the ...interval E=a,a+2π), a∈R, these quadrature rules have the following form∫Et(x)w(x)dx=∑i=12kait(yi)+∑i=12(n+γ)Ait(xi),t∈T2(n+γ)+k−1, where the nodes yi∈E, i=1,2,…,2k, are fixed and prescribed in advance, γ∈{0,1/2} and Tn={coskx,sinkx|k=0,1,…,n}, n∈N.
Also, for γ=1/2, i.e., for the case of quadrature rules for trigonometric polynomials with odd number of nodes, we consider the optimal sets of quadrature rules in the sense of Borges (see 1,13) for trigonometric polynomials with even number of fixed nodes. Let n=(n1,n2,…,nr), r∈N, be a multi-index and let W=(w1,w2,…,wr) be a system of weight functions on the interval E=a,a+2π), a∈R. The optimal set of quadrature rules with respect to (W,n), with even number of fixed nodes, have the form∫Ef(x)wm(x)dx≈∑i=12kam,if(yi)+∑i=12|n|+1Am,if(xi),m=1,2,…r, where |n|=n1+n2+⋯+nr and the nodes yi∈E, i=1,2,…,2k, are fixed and prescribed in advance. For r=1 the optimal set of quadrature rules reduces to Gaussian quadrature rule for trigonometric polynomials with odd number of nodes.
For all mentioned quadrature rules, in addition to the theoretical results, we will present the method for construction and give appropriate numerical examples.
Despite extensive research on symmetric polynomial quadrature rules for triangles, as well as approaches to their calculation, few studies have focused on non-polynomial functions, particularly on ...their integration using symmetric triangle rules. In this paper, we present two approaches to computing symmetric triangle rules for singular integrands by developing rules that can integrate arbitrary functions. The first approach is well suited for a moderate amount of points and retains much of the efficiency of polynomial quadrature rules. The second approach better addresses large amounts of points, though it is less efficient than the first approach. We demonstrate the effectiveness of both approaches on singular integrands, which can often yield relative errors two orders of magnitude less than those from polynomial quadrature rules.
The need to evaluate Gauss quadrature rules arises in many applications in science and engineering. It often is important to be able to estimate the quadrature error when applying an ℓ-point Gauss ...rule, Gℓ(f), where f is an integrand of interest. Such an estimate often is furnished by applying another quadrature rule, Qk(f), with k>ℓ nodes, and using the difference Qk(f)−Gℓ(f) or its magnitude as an estimate for the quadrature error in Gℓ(f) or its magnitude. The classical approach to estimate the error in Gℓ(f) is to let Qk(f), with k=2ℓ+1, be the Gauss-Kronrod quadrature rule associated with Gℓ(f). However, it is well known that the Gauss-Kronrod rule associated with a Gauss rule Gℓ(f) might not exist for certain measures that determine the Gauss rule and for certain numbers of nodes. This prompted M. M. Spalević 1 to develop generalized averaged Gauss rules, Gˆ2ℓ+1, with 2ℓ+1 nodes for estimating the error in Gℓ(f). Similarly as for (2ℓ+1)-node Gauss-Kronrod rules, ℓ nodes of the rule Gˆ2ℓ+1 agree with the nodes of Gℓ. However, generalized averaged Gauss rules are not internal for some measures. They therefore may not be applicable when the integrand only is defined on the convex hull of the support of the measure. This paper describes a new kind of quadrature rules that may be internal also when generalized averaged quadrature rules are not. The construction of the new quadrature rules is based on theory developed by Peherstorfer 2. Their application is particularly attractive when the rule Gˆ2ℓ+1 is not internal, the integrand cannot be evaluated at all its nodes, and the integrand is inexpensive to evaluate at the quadrature points. Computed examples that illustrate the performance of the new quadrature rules introduced in this paper are presented.
We explore the use of various element-based reduced quadrature strategies for bivariate and trivariate quadratic and cubic spline elements used in isogeometric analysis. The rules studied encompass ...tensor-product Gauss and Gauss–Lobatto rules, and certain so-called monomial rules that do no possess a tensor-product structure. The objective of the study is to determine quadrature strategies, which enjoy the same accuracy and stability behavior as full Gauss quadrature, but with significantly fewer quadrature points. Several cases emerge that satisfy this objective and also demonstrate superior efficiency compared with standard C0-continuous finite elements of the same order.
•We test various element-based reduced quadrature rules for quadratic and cubic spline elements.•They encompass tensor-product Gauss and Gauss–Lobatto rules, and monomial rules.•Some rules enjoy the same accuracy and stability as full Gauss quadrature, but with significantly fewer quadrature points.•They can substantially reduce the formation and assembly effort in isogeometric analysis.
Recently, Zayernouri and Karniadakis in (2013) 78 investigated two classes of fractional Sturm–Liouville eigenvalue problems on compact interval a,b in more detail. They were the first authors who ...not only obtained some explicit forms for the eigensolutions of these problems but also derived some useful spectral properties of the obtained eigensolutions. Until now, to the best of our knowledge, fractional Sturm–Liouville eigenvalue problems on non-compact interval, such as 0,+∞) are not analyzed. So, our aim in this paper is to study these problems in detail. To do so, we study at first fractional Sturm–Liouville operators (FSLOs) of the confluent hypergeometric differential equations of the first kind and then two special cases of FSLOs: FSLOs-1 and FSLOs-2 are considered. After this, we obtain the analytical eigenfunctions for the cases and investigate the spectral properties of eigenfunctions and their corresponding eigenvalues. Also, we derive two fractional types of the associated Laguerre differential equations. Due to the non-polynomial nature of the eigenfunctions obtained from the two fractional associated Laguerre differential equations, they are defined as generalized associated Laguerre functions of the first and second kinds, GALFs-1 and GALFs-2. Furthermore, we prove that these fractional Sturm–Liouville operators are self-adjoint and the obtained eigenvalues are all real, the corresponding eigenfunctions are orthogonal with respect to the weight function associated to FSLOs-1 and FSLOs-2 and form two sets of non-polynomial bases. At the end, two new quadrature rules and L2-orthogonal projections with respect to and based on GALFs-1 and GALFs-2 are introduced. The upper bounds of the truncation errors of these new orthogonal projections according to some prescribed norm are proved and then verified numerically with some test examples. Finally, some fractional differential equations are provided and analyzed numerically.
PDF
