26,956 research outputs found
Virtual Enriching Operators
We construct bounded linear operators that map conforming Lagrange
finite element spaces to conforming virtual element spaces in two and
three dimensions. These operators are useful for the analysis of nonstandard
finite element methods
Constructions of some minimal finite element systems
Within the framework of finite element systems, we show how spaces of
differential forms may be constructed, in such a way that they are equipped
with commuting interpolators and contain prescribed functions, and are minimal
under these constraints. We show how various known mixed finite element spaces
fulfill such a design principle, including trimmed polynomial differential
forms, serendipity elements and TNT elements. We also comment on virtual
element methods and provide a dimension formula for minimal compatible finite
element systems containing polynomials of a given degree on hypercubes.Comment: Various minor changes, based on suggestions of paper referee
Hot new directions for quasi-Monte Carlo research in step with applications
This article provides an overview of some interfaces between the theory of
quasi-Monte Carlo (QMC) methods and applications. We summarize three QMC
theoretical settings: first order QMC methods in the unit cube and in
, and higher order QMC methods in the unit cube. One important
feature is that their error bounds can be independent of the dimension
under appropriate conditions on the function spaces. Another important feature
is that good parameters for these QMC methods can be obtained by fast efficient
algorithms even when is large. We outline three different applications and
explain how they can tap into the different QMC theory. We also discuss three
cost saving strategies that can be combined with QMC in these applications.
Many of these recent QMC theory and methods are developed not in isolation, but
in close connection with applications
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