2,907 research outputs found
Numerical analysis of nonlinear subdiffusion equations
We present a general framework for the rigorous numerical analysis of
time-fractional nonlinear parabolic partial differential equations, with a
fractional derivative of order in time. The framework relies
on three technical tools: a fractional version of the discrete Gr\"onwall-type
inequality, discrete maximal regularity, and regularity theory of nonlinear
equations. We establish a general criterion for showing the fractional discrete
Gr\"onwall inequality, and verify it for the L1 scheme and convolution
quadrature generated by BDFs. Further, we provide a complete solution theory,
e.g., existence, uniqueness and regularity, for a time-fractional diffusion
equation with a Lipschitz nonlinear source term. Together with the known
results of discrete maximal regularity, we derive pointwise norm
error estimates for semidiscrete Galerkin finite element solutions and fully
discrete solutions, which are of order (up to a logarithmic factor)
and , respectively, without any extra regularity assumption on
the solution or compatibility condition on the problem data. The sharpness of
the convergence rates is supported by the numerical experiments
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
Discrete maximal regularity of time-stepping schemes for fractional evolution equations
In this work, we establish the maximal -regularity for several time
stepping schemes for a fractional evolution model, which involves a fractional
derivative of order , , in time. These schemes
include convolution quadratures generated by backward Euler method and
second-order backward difference formula, the L1 scheme, explicit Euler method
and a fractional variant of the Crank-Nicolson method. The main tools for the
analysis include operator-valued Fourier multiplier theorem due to Weis [48]
and its discrete analogue due to Blunck [10]. These results generalize the
corresponding results for parabolic problems
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