We present and analyze a stable space-time multi-patch discontinuous Galerkin Isogeometric Analysis (dG-IGA) scheme for the numerical solution of parabolic evolution equations in moving space-time computational domains. Following Langer et. al, 2016, we use a time-upwind test function and apply multi-patch discontinuous Galerkin IGA methodology for discretizing the evolution problem both in space and in time. This yields a discrete bilinear form which is elliptic on the IGA space with respect to a space-time dG norm. This property together with a corresponding boundedness property, consistency and approximation results for the IGA spaces yields an a priori discretization error estimate with respect to the space-time dG norm. The theoretical results are confirmed by several numerical experiments with low- and high-order IGA spaces.
In this talk we consider conservation laws with discontinuous flux. Such equations have a
wide range of applications including vehicle traffic flow in the presence of abruptly
varying road conditions and two-phase flow through heterogeneous porous media.
We consider the class of adapted entropy solutions where the spatial flux dependency is
piecewise constant. This setting allows the flux to change abruptly across finitely many
points in space. We prove that adapted entropy solutions are L1 stable with respect to
changes in the initial datum, the flux function, and the spatial dependency parameter.
This result allows us to derive a convergence rate estimate for the front tracking method -
a numerical method which is widely used in the field of conservation laws with
discontinuous flux. To the best of our knowledge, both of these results are the first of
their kind in the literature on conservation laws with discontinuous flux.
We briefly
present numerical experiments motivated by uncertainty quantification for two-phase
reservoir simulations for reservoirs with varying geological properties.
It is well known that if the singular values of a matrix A are distinct, then the best
rank-n approximation to A is uniquely determined in the Frobenius norm and given by the
truncated singular value decomposition. On the other hand, this uniqueness is in general
not true for best rank-n approximations in the spectral norm.
In this talk we relate the
problem of finding best rank-n approximations in the spectral norm to Kolmogorov n-widths
and corresponding optimal spaces. By providing new criteria for optimality of spaces with
respect to the n-width, we describe a large family of best rank-n approximations to A.
This results in a variety of solutions to the best low-rank approximation problem and
provides alternatives to the truncated singular value decomposition. This variety can be
exploited to obtain best low-rank approximations with problem-oriented properties.
Special attention is paid to the case of rank-1 approximation.
This talk is based on
joint work with Michael S. Floater, Carla Manni and Hendrik Speleers.