16,104 research outputs found
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
An hp-version discontinuous Galerkin method for integro-differential equations of parabolic type
We study the numerical solution of a class of parabolic integro-differential equations with weakly singular kernels. We use an -version discontinuous Galerkin (DG) method for the discretization in time. We derive optimal -version error estimates and show that exponential rates of convergence can be achieved for solutions with singular (temporal) behavior near caused by the weakly singular kernel. Moreover, we prove that by using nonuniformly refined time steps, optimal algebraic convergence rates can be achieved for the -version DG method. We then combine the DG time-stepping method with a standard finite element discretization in space, and present an optimal error analysis of the resulting fully discrete scheme. Our theoretical results are numerically validated in a series of test problems
Analytic Regularity and GPC Approximation for Control Problems Constrained by Linear Parametric Elliptic and Parabolic PDEs
This paper deals with linear-quadratic optimal control problems constrained by a parametric or stochastic elliptic or parabolic PDE. We address the (difficult) case that the state equation depends on a countable number of parameters i.e., on with , and that the PDE operator may depend non-affinely on the parameters. We consider tracking-type functionals and distributed as well as boundary controls. Building on recent results in [CDS1, CDS2], we show that the state and the control are analytic as functions depending on these parameters . We
establish sparsity of generalized polynomial chaos (gpc) expansions of both, state and control, in terms of the stochastic coordinate sequence of the random inputs, and prove convergence rates of best -term truncations of these expansions. Such truncations are the key for subsequent computations since they do {\em not} assume that the stochastic input data has a finite expansion. In the follow-up paper [KS2], we explain two methods how such best -term truncations can practically be computed, by greedy-type algorithms
as in [SG, Gi1], or by multilevel Monte-Carlo methods as in
[KSS]. The sparsity result allows in conjunction with adaptive wavelet Galerkin schemes for sparse, adaptive tensor discretizations of control problems constrained by linear elliptic and parabolic PDEs developed in [DK, GK, K], see [KS2]
A Posteriori error control & adaptivity for Crank-Nicolson finite element approximations for the linear Schrodinger equation
We derive optimal order a posteriori error estimates for fully discrete
approximations of linear Schr\"odinger-type equations, in the
norm. For the discretization in time we use the Crank-Nicolson
method, while for the space discretization we use finite element spaces that
are allowed to change in time. The derivation of the estimators is based on a
novel elliptic reconstruction that leads to estimates which reflect the
physical properties of Schr\"odinger equations. The final estimates are
obtained using energy techniques and residual-type estimators. Various
numerical experiments for the one-dimensional linear Schr\"odinger equation in
the semiclassical regime, verify and complement our theoretical results. The
numerical implementations are performed with both uniform partitions and
adaptivity in time and space. For adaptivity, we further develop and analyze an
existing time-space adaptive algorithm to the cases of Schr\"odinger equations.
The adaptive algorithm reduces the computational cost substantially and
provides efficient error control for the solution and the observables of the
problem, especially for small values of the Planck constant
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