Beta-function B-spline smoothing on triangulations

On the numerical performance of FEM based on piecewise rational smooth resolutions of unity on triangulations

“…For more details on the formulations, consult [23] and its upgrade [22]. For conciseness of the presentation, rather than presenting a self-contained introduction into the definition and properties of the three considered GERBS-based FEM, we refer to [13,6] and [23] for the first two constructions, and to [12] for the third one. Moreover, in the computations in this paper, we shall use a variant of the construction in [12] where the underlying univariate ERBS [9] in the construction is replaced by a respective logistic ERBS (LERBS) [10].…”

“…For conciseness of the presentation, rather than presenting a self-contained introduction into the definition and properties of the three considered GERBS-based FEM, we refer to [13,6] and [23] for the first two constructions, and to [12] for the third one. Moreover, in the computations in this paper, we shall use a variant of the construction in [12] where the underlying univariate ERBS [9] in the construction is replaced by a respective logistic ERBS (LERBS) [10]. The only reason for this modification is that the LERBS is explicitly computable in closed form in terms of rational and exponential functions, without the need of numerical integration as is the case with ERBS.…”

“…In the current study we focus our investigation to GERBS-based constructions. Together with the two GERBS-based variants addressed in [23] (the triangular Beta function B-spline (BFBS) [13] and the general smooth GERBS [5,6,8]), we also include a novel family of rational basis functions on triangulations as introduced in [12].…”

“…The present work is a first study of the potential for applications of this new construction in the numerical analysis of boundary-value problems for PDEs. It also represents a continuation and complement of certain aspects of the study in [23] which was conducted before the introduction of the new construction in [12] and the introduction of the logistic expo-rational B-spline in [10]. …”

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This work is a comparative study of a number of recently introduced finite element methods based on generalized expo-rational B-splines [7,9].In [12] a new construction of piecewise-rational C m -smooth resolution of unity on triangulated domains in R 2 , m = 1, 2,..., has been introduced. The present work is a first study of the potential for applications of this new construction in the numerical analysis of boundary-value problems for PDEs.

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