21 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
Fully Discrete Pointwise Smoothing Error Estimates for Measure Valued Initial Data
In this paper we analyze a homogeneous parabolic problem with initial data in
the space of regular Borel measures. The problem is discretized in time with a
discontinuous Galerkin scheme of arbitrary degree and in space with continuous
finite elements of orders one or two. We show parabolic smoothing results for
the continuous, semidiscrete and fully discrete problems. Our main results are
interior error estimates for the evaluation at the endtime, in cases
where the initial data is supported in a subdomain. In order to obtain these,
we additionally show interior error estimates for initial data
and quadratic finite elements, which extends the corresponding result
previously established by the authors for linear finite elements
Local energy estimates for the fractional Laplacian
The integral fractional Laplacian of order is a nonlocal
operator. It is known that solutions to the Dirichlet problem involving such an
operator exhibit an algebraic boundary singularity regardless of the domain
regularity. This, in turn, deteriorates the global regularity of solutions and
as a result the global convergence rate of the numerical solutions. For finite
element discretizations, we derive local error estimates in the -seminorm
and show optimal convergence rates in the interior of the domain by only
assuming meshes to be shape-regular. These estimates quantify the fact that the
reduced approximation error is concentrated near the boundary of the domain. We
illustrate our theoretical results with several numerical examples
Planar Double Bubbles on Flat Walls
In this paper, we investigate some properties of planar soap bubbles on a straight wall with a single corner. We show that when a wall has a single corner and a bubble consists of two connected regions, the perimeter minimizing bubble must be one of the types, two concentric circular arcs or a truncated standard double bubble, depending on the angle of the corner and areas of the regions