3 research outputs found
Sharp asymptotics of the Lp approximation error for interpolation on block partitions
Adaptive approximation (or interpolation) takes into account local variations
in the behavior of the given function, adjusts the approximant depending on it,
and hence yields the smaller error of approximation. The question of
constructing optimal approximating spline for each function proved to be very
hard. In fact, no polynomial time algorithm of adaptive spline approximation
can be designed and no exact formula for the optimal error of approximation can
be given. Therefore, the next natural question would be to study the asymptotic
behavior of the error and construct asymptotically optimal sequences of
partitions. In this paper we provide sharp asymptotic estimates for the error
of interpolation by splines on block partitions in IRd. We consider various
projection operators to define the interpolant and provide the analysis of the
exact constant in the asymptotics as well as its explicit form in certain
cases.Comment: 20 pages, 1 figur
Optimally Adapted Meshes for Finite Elements of Arbitrary Order and W1p Norms
Given a function f defined on a bidimensional bounded domain and a positive
integer N, we study the properties of the triangulation that minimizes the
distance between f and its interpolation on the associated finite element
space, over all triangulations of at most N elements. The error is studied in
the W1p norm and we consider Lagrange finite elements of arbitrary polynomial
order m-1. We establish sharp asymptotic error estimates as N tends to infinity
when the optimal anisotropic triangulation is used. A similar problem has been
studied earlier, but with the error measured in the Lp norm. The extension of
this analysis to the W1p norm is crucial in order to match more closely the
needs of numerical PDE analysis, and it is not straightforward. In particular,
the meshes which satisfy the optimal error estimate are characterized by a
metric describing the local aspect ratio of each triangle and by a geometric
constraint on their maximal angle, a second feature that does not appear for
the Lp error norm. Our analysis also provides with practical strategies for
designing meshes such that the interpolation error satisfies the optimal
estimate up to a fixed multiplicative constant. We discuss the extension of our
results to finite elements on simplicial partitions of a domain of arbitrary
dimension, and we provide with some numerical illustration in two dimensions.Comment: 37 pages, 6 figure