275 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
A linear Galerkin numerical method for a quasilinear subdiffusion equation
We couple the L1 discretization for Caputo derivative in time with spectral
Galerkin method in space to devise a scheme that solves quasilinear
subdiffusion equations. Both the diffusivity and the source are allowed to be
nonlinear functions of the solution. We prove method's stability and
convergence with spectral accuracy in space. The temporal order depends on
solution's regularity in time. Further, we support our results with numerical
simulations that utilize parallelism for spatial discretization. Moreover, as a
side result we find asymptotic exact values of error constants along with their
remainders for discretizations of Caputo derivative and fractional integrals.
These constants are the smallest possible which improves the previously
established results from the literature.Comment: This is the accepted version of the manuscript published in Applied
Numerical Mathematic
Mathematical properties and numerical approximation of pseudo-parabolic systems
The paper is concerned with the mathematical theory and numerical
approximation of systems of partial differential equations (pde) of hyperbolic,
pseudo-parabolic type. Some mathematical properties of the
initial-boundary-value problem (ibvp) with Dirichlet boundary conditions are
first studied. They include the weak formulation, well-posedness and existence
of traveling wave solutions connecting two states, when the equations are
considered as a variant of a conservation law. Then, the numerical
approximation consists of a spectral approximation in space based on Legendre
polynomials along with a temporal discretization with strong stability
preserving (SSP) property. The convergence of the semidiscrete approximation is
proved under suitable regularity conditions on the data. The choice of the
temporal discretization is justified in order to guarantee the stability of the
full discretization when dealing with nonsmooth initial conditions. A
computational study explores the performance of the fully discrete scheme with
regular and nonregular data
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
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]
- ā¦