We propose a construction of a trivariate C 1 macro-element over a special tetrahedral partition and compare our construction with known C 1 macro-elements which are summarized in this paper. Also, we propose an improvement of the Alfeld construction of a C 1 quintic macro-element such that the new scheme is able to reproduce all polynomials of total degree 5. x1. Introduction The study of triv...

In 1988, Worsey and Piper constructed a trivariate macro-element based on C1 quadratic splines defined over a split of a tetrahedron into 24 subtetrahedra. However, this local element can only be used to construct a corresponding macro-element spline space over tetrahedral partitions that satisfy some very restrictive geometric constraints. We show that by further refining their split, it is po...

In 1988, Worsey and Piper constructed a trivariate macro-element based on C quadratic splines defined over a split of a tetrahedron into 24 subtetrahedra. However, this local element can only be used to construct a corresponding macro-element spline space over tetrahedral partitions that satisfy some very restrictive geometric constraints. We show that by further refining their split, it is pos...

We show that a nested sequence of C macro-element spline spaces on quasi-uniform triangulations gives rise to hierarchical Riesz bases of Sobolev spaces H(Ω), 1 < s < r+ 3 2 , and H 0(Ω), 1 < s < σ+ 3 2 , s / ∈ Z+ 1 2 , as soon as there is a nested sequence of Lagrange interpolation sets with uniformly local and bounded basis functions, and, in case of H 0(Ω), the nodal interpolation operators ...

A variant of the classical C Powell-Sabin-12 macro-element is defined that has the same approximation power, but has fewer degrees of freedom and only requires vertex data. The same idea is then applied to a related C trivariate macro-element. §

In this paper, we consider eecient multilevel based iterative solvers and ee-cient and reliable a posteriori error estimators for mixed hybrid and macro-hybrid nite element discretizations of elliptic boundary value problems. We give an overview concerning the state-of-the-art techniques for these nonconforming approaches and illustrate the performance of the adaptivity concepts realized by som...

Trivariate C macro-elements defined in terms of polynomials of degree 8r + 1 on tetrahedra are analyzed. For r = 1, 2, these spaces reduce to well-known macro-element spaces used in data fitting and in the finite-element method. We determine the dimension of these spaces, and describe stable local minimal determining sets and nodal minimal determining sets. We also show that the spaces approxim...

A finite element heterogeneous multiscale method (FEHMM) is proposed for the time dependent wave equation with highly oscillatory, albeit not necessarily periodic, coefficients. It is based on a finite element discretization of an effective wave equation at the macro scale, whose a priori unknown effective coefficients are computed “on the fly” on sampling domains within each macro finite eleme...