412 citations · 980 across the 11 of their papers we have counts for
13 papers
Construction of polynomial preserving cochain extensions by blending
Richard S. Falk, Ragnar Winther
A classical technique to construct polynomial preserving extensions of scalar functions defined on the boundary of an simplex to the interior is to use so-called rational blend…
Hidden Positivity and a New Approach to Numerical Computation of Hausdorff Dimension: Higher Order Methods
Richard S. Falk, Roger D. Nussbaum
In [14], the authors developed a new approach to the computation of the Hausdorff dimension of the invariant set of an iterated function system or IFS. In this paper, we extend thi…
Eigenfunctions of Perron-Frobenius Operators and a New Approach to Numerical Computation of Hausdorff Dimension: Applications in
Richard S. Falk, Roger D. Nussbaum
We develop a new approach to the computation of the Hausdorff dimension of the invariant set of an iterated function system or IFS. In the one dimensional case that we consider her…
A New Approach to Numerical Computation of Hausdorff Dimension of Iterated Function Systems: Applications to Complex Continued Fractions
Richard S. Falk, Roger D. Nussbaum
In a previous paper, dealing with "Applications in ," the authors developed a new approach to the computation of the Hausdorff dimension of the invariant set of an it…
C^m Eigenfunctions of Perron-Frobenius Operators and a New Approach to Numerical Computation of Hausdorff Dimension
Richard S. Falk, Roger D. Nussbaum
We develop a new approach to the computation of the Hausdorff dimension of the invariant set of an iterated function system or IFS. In the one dimensional case, our methods require…
The bubble transform: A new tool for analysis of finite element methods
Richard S. Falk, Ragnar Winther
The purpose of this paper is to discuss the construction of a linear operator, referred to as the bubble transform, which maps scalar functions defined on a bounded domain in $…
On the consistency of the combinatorial codifferential
Douglas N. Arnold, Richard S. Falk, Johnny Guzmán +1
In 1976, Dodziuk and Patodi employed Whitney forms to define a combinatorial codifferential operator on cochains, and they raised the question whether it is consistent in the sense…
Local bounded cochain projection
Richard S. Falk, Ragnar Winther
We construct projections from the space of differential k-forms which belong to L2 and whose exterior derivative also belongs to L2, to finite dimensional subspaces of piecewise po…
Mixed finite element approximation of the vector Laplacian with Dirichlet boundary conditions
Douglas N. Arnold, Richard S. Falk, Jay Gopalakrishnan
We consider the finite element solution of the vector Laplace equation on a domain in two dimensions. For various choices of boundary conditions, it is known that a mixed finite el…
Finite element exterior calculus: from Hodge theory to numerical stability
Douglas N. Arnold, Richard S. Falk, Ragnar Winther
This article reports on the confluence of two streams of research, one emanating from the fields of numerical analysis and scientific computation, the other from topology and geome…
Geometric decompositions and local bases for spaces of finite element differential forms
Douglas N. Arnold, Richard S. Falk, Ragnar Winther
We study the two primary families of spaces of finite element differential forms with respect to a simplicial mesh in any number of space dimensions. These spaces are generalizatio…
Mixed finite element methods for linear elasticity with weakly imposed symmetry
Douglas N. Arnold, Richard S. Falk, Ragnar Winther
In this paper, we construct new finite element methods for the approximation of the equations of linear elasticity in three space dimensions that produce direct approximations to b…
Approximation by quadrilateral finite elements
Douglas N. Arnold, Daniele Boffi, Richard S. Falk
We consider the approximation properties of finite element spaces on quadrilateral meshes. The finite element spaces are constructed starting with a given finite dimensional space…