1,950 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 Analysis of Sparse Initial Data Identification for Parabolic Problems
In this paper we consider a problem of initial data identification from the
final time observation for homogeneous parabolic problems. It is well-known
that such problems are exponentially ill-posed due to the strong smoothing
property of parabolic equations. We are interested in a situation when the
initial data we intend to recover is known to be sparse, i.e. its support has
Lebesgue measure zero. We formulate the problem as an optimal control problem
and incorporate the information on the sparsity of the unknown initial data
into the structure of the objective functional. In particular, we are looking
for the control variable in the space of regular Borel measures and use the
corresponding norm as a regularization term in the objective functional. This
leads to a convex but non-smooth optimization problem. For the discretization
we use continuous piecewise linear finite elements in space and discontinuous
Galerkin finite elements of arbitrary degree in time. For the general case we
establish error estimates for the state variable. Under a certain structural
assumption, we show that the control variable consists of a finite linear
combination of Dirac measures. For this case we obtain error estimates for the
locations of Dirac measures as well as for the corresponding coefficients. The
key to the numerical analysis are the sharp smoothing type pointwise finite
element error estimates for homogeneous parabolic problems, which are of
independent interest. Moreover, we discuss an efficient algorithmic approach to
the problem and show several numerical experiments illustrating our theoretical
results.Comment: 43 pages, 10 figure
Pointwise-in-time error estimates for an optimal control problem with subdiffusion constraint
In this work, we present numerical analysis for a distributed optimal control
problem, with box constraint on the control, governed by a subdiffusion
equation which involves a fractional derivative of order in
time. The fully discrete scheme is obtained by applying the conforming linear
Galerkin finite element method in space, L1 scheme/backward Euler convolution
quadrature in time, and the control variable by a variational type
discretization. With a space mesh size and time stepsize , we
establish the following order of convergence for the numerical solutions of the
optimal control problem: in the
discrete norm and
in the discrete
norm, with any small and
. The analysis relies essentially on the maximal
-regularity and its discrete analogue for the subdiffusion problem.
Numerical experiments are provided to support the theoretical results.Comment: 20 pages, 6 figure
- …