1,585 research outputs found
Pointwise best approximation results for Galerkin finite element solutions of parabolic problems
In this paper we establish a best approximation property of fully discrete
Galerkin finite element solutions of second order parabolic problems on convex
polygonal and polyhedral domains in the norm. The discretization
method uses of continuous Lagrange finite elements in space and discontinuous
Galerkin methods in time of an arbitrary order. The method of proof differs
from the established fully discrete error estimate techniques and for the first
time allows to obtain such results in three space dimensions. It uses elliptic
results, discrete resolvent estimates in weighted norms, and the discrete
maximal parabolic regularity for discontinuous Galerkin methods established by
the authors in [16]. In addition, the proof does not require any relationship
between spatial mesh sizes and time steps. We also establish a local best
approximation property that shows a more local behavior of the error at a given
point
Numerical methods for time-fractional evolution equations with nonsmooth data: a concise overview
Over the past few decades, there has been substantial interest in evolution
equations that involving a fractional-order derivative of order
in time, due to their many successful applications in
engineering, physics, biology and finance. Thus, it is of paramount importance
to develop and to analyze efficient and accurate numerical methods for reliably
simulating such models, and the literature on the topic is vast and fast
growing. The present paper gives a concise overview on numerical schemes for
the subdiffusion model with nonsmooth problem data, which are important for the
numerical analysis of many problems arising in optimal control, inverse
problems and stochastic analysis. We focus on the following aspects of the
subdiffusion model: regularity theory, Galerkin finite element discretization
in space, time-stepping schemes (including convolution quadrature and L1 type
schemes), and space-time variational formulations, and compare the results with
that for standard parabolic problems. Further, these aspects are showcased with
illustrative numerical experiments and complemented with perspectives and
pointers to relevant literature.Comment: 24 pages, 3 figure
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