15,136 research outputs found
Energy-corrected FEM and explicit time-stepping for parabolic problems
The presence of corners in the computational domain, in general, reduces the
regularity of solutions of parabolic problems and diminishes the convergence
properties of the finite element approximation introducing a so-called
"pollution effect". Standard remedies based on mesh refinement around the
singular corner result in very restrictive stability requirements on the
time-step size when explicit time integration is applied. In this article, we
introduce and analyse the energy-corrected finite element method for parabolic
problems, which works on quasi-uniform meshes, and, based on it, create fast
explicit time discretisation. We illustrate these results with extensive
numerical investigations not only confirming the theoretical results but also
showing the flexibility of the method, which can be applied in the presence of
multiple singular corners and a three-dimensional setting. We also propose a
fast explicit time-stepping scheme based on a piecewise cubic energy-corrected
discretisation in space completed with mass-lumping techniques and numerically
verify its efficiency
Galerkin FEM for fractional order parabolic equations with initial data in
We investigate semi-discrete numerical schemes based on the standard Galerkin
and lumped mass Galerkin finite element methods for an initial-boundary value
problem for homogeneous fractional diffusion problems with non-smooth initial
data. We assume that , is a convex
polygonal (polyhedral) domain. We theoretically justify optimal order error
estimates in - and -norms for initial data in . We confirm our theoretical findings with a number of numerical tests
that include initial data being a Dirac -function supported on a
-dimensional manifold.Comment: 13 pages, 3 figure
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
Error Analysis of Semidiscrete Finite Element Methods for Inhomogeneous Time-Fractional Diffusion
We consider the initial boundary value problem for the inhomogeneous
time-fractional diffusion equation with a homogeneous Dirichlet boundary
condition and a nonsmooth right hand side data in a bounded convex polyhedral
domain. We analyze two semidiscrete schemes based on the standard Galerkin and
lumped mass finite element methods. Almost optimal error estimates are obtained
for right hand side data , , for both semidiscrete schemes. For lumped mass method, the optimal
-norm error estimate requires symmetric meshes. Finally, numerical
experiments for one- and two-dimensional examples are presented to verify our
theoretical results.Comment: 21 pages, 4 figure
The Galerkin Finite Element Method for A Multi-term Time-Fractional Diffusion equation
We consider the initial/boundary value problem for a diffusion equation
involving multiple time-fractional derivatives on a bounded convex polyhedral
domain. We analyze a space semidiscrete scheme based on the standard Galerkin
finite element method using continuous piecewise linear functions. Nearly
optimal error estimates for both cases of initial data and inhomogeneous term
are derived, which cover both smooth and nonsmooth data. Further we develop a
fully discrete scheme based on a finite difference discretization of the
time-fractional derivatives, and discuss its stability and error estimate.
Extensive numerical experiments for one and two-dimension problems confirm the
convergence rates of the theoretical results.Comment: 22 pages, 4 figure
Implicit-Explicit Runge-Kutta schemes for hyperbolic systems and kinetic equations in the diffusion limit
We consider Implicit-Explicit (IMEX) Runge-Kutta (R-K) schemes for hyperbolic
systems with stiff relaxation in the so-called diffusion limit. In such regime
the system relaxes towards a convection-diffusion equation. The first objective
of the paper is to show that traditional partitioned IMEX R-K schemes will
relax to an explicit scheme for the limit equation with no need of modification
of the original system. Of course the explicit scheme obtained in the limit
suffers from the classical parabolic stability restriction on the time step.
The main goal of the paper is to present an approach, based on IMEX R-K
schemes, that in the diffusion limit relaxes to an IMEX R-K scheme for the
convection-diffusion equation, in which the diffusion is treated implicitly.
This is achieved by an original reformulation of the problem, and subsequent
application of IMEX R-K schemes to it. An analysis on such schemes to the
reformulated problem shows that the schemes reduce to IMEX R-K schemes for the
limit equation, under the same conditions derived for hyperbolic relaxation.
Several numerical examples including neutron transport equations confirm the
theoretical analysis
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