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Buyang Li

Publications and source records attributed to Buyang Li.

At least 73 records · Page 4Linked to original sources
arXiv2018

Sharp convergence rates of time discretization for stochastic time-fractional PDEs subject to additive space-time white noise

The stochastic time-fractional equation $\partial_t ψ-Δ\partial_t^{1-α} ψ= f + \dot W$ with space-time white noise $\dot W$ is discretized in time by a backward-Euler convolution quadrature for which the sharp-order error estimate \[ {\mathbb E}\|ψ(\cdot,t_n)-ψ_n\|_{L^2(\mathcal{O})}^2=O(τ^{1-αd/2}) \] is established for $α\in(0,2/d)$, where $d$ denotes the spatial dimension, $ψ_n$ the approximate solution at the $n^{\rm th}$ time step, and $\mathbb{E}$ the expectation operator. In particular, the result indicates optimal convergence rates of numerical solutions for both stochastic subdiffusion and diffusion-wave problems in one spatial dimension. Numerical examples are presented to illustrate the theoretical analysis.

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arXiv2017

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 $α\in(0,1)$ 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 $h$ and time stepsize $τ$, we establish the following order of convergence for the numerical solutions of the optimal control problem: $O(τ^{\min({1}/{2}+α-ε,1)}+h^2)$ in the discrete $L^2(0,T;L^2(Ω))$ norm and $O(τ^{α-ε}+\ell_h^2h^2)$ in the discrete $L^\infty(0,T;L^2(Ω))$ norm, with any small $ε>0$ and $\ell_h=\ln(2+1/h)$. The analysis relies essentially on the maximal $L^p$-regularity and its discrete analogue for the subdiffusion problem. Numerical experiments are provided to support the theoretical results.

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arXiv2017

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 $α\in(0,1)$ in time. The framework relies on three technical tools: a fractional version of the discrete Grönwall-type inequality, discrete maximal regularity, and regularity theory of nonlinear equations. We establish a general criterion for showing the fractional discrete Grönwall 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 $L^2(Ω)$ norm error estimates for semidiscrete Galerkin finite element solutions and fully discrete solutions, which are of order $O(h^2)$ (up to a logarithmic factor) and $O(τ^α)$, 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.

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arXiv2017

Convergence of finite element solutions of stochastic partial integro-differential equations driven by white noise

Numerical approximation of a stochastic partial integro-differential equation driven by a space- time white noise is studied by truncating a series representation of the noise, with finite element method for spatial discretization and convolution quadrature for time discretization. Sharp-order convergence of the numerical solutions is proved up to a logarithmic factor. Numerical examples are provided to support the theoretical analysis.

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arXiv2017

Analyticity, maximal regularity and maximum-norm stability of semi-discrete finite element solutions of parabolic equations in nonconvex polyhedra

In general polygons and polyhedra, possibly nonconvex, the analyticity of the finite element heat semigroup in the $L^q$ norm, $1\leq q\leq\infty$, and the maximal $L^p$-regularity of semi-discrete finite element solutions of parabolic equations are proved. By using these results, the problem of maximum-norm stability of the finite element parabolic projection is reduced to the maximum-norm stability of the Ritz projection, which currently is known to hold for general polygonal domains and convex polyhedral domains.

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arXiv2017

Discrete maximal regularity of time-stepping schemes for fractional evolution equations

In this work, we establish the maximal $\ell^p$-regularity for several time stepping schemes for a fractional evolution model, which involves a fractional derivative of order $α\in(0,2)$, $α\neq 1$, 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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arXiv2017

Correction of high-order BDF convolution quadrature for fractional evolution equations

We develop proper correction formulas at the starting $k-1$ steps to restore the desired $k^{\rm th}$-order convergence rate of the $k$-step BDF convolution quadrature for discretizing evolution equations involving a fractional-order derivative in time. The desired $k^{\rm th}$-order convergence rate can be achieved even if the source term is not compatible with the initial data, which is allowed to be nonsmooth. We provide complete error estimates for the subdiffusion case $α\in (0,1)$, and sketch the proof for the diffusion-wave case $α\in(1,2)$. Extensive numerical examples are provided to illustrate the effectiveness of the proposed scheme.

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arXiv2017

An Analysis of the Crank-Nicolson Method for Subdiffusion

In this work, we analyze a Crank-Nicolson type time stepping scheme for the subdiffusion equation, which involves a Caputo fractional derivative of order $α\in (0,1)$ in time. It hybridizes the backward Euler convolution quadrature with a $θ$-type method, with the parameter $θ$ dependent on the fractional order $α$ by $θ=α/2$, and naturally generalizes the classical Crank-Nicolson method. We develop essential initial corrections at the starting two steps for the Crank-Nicolson scheme, and together with the Galerkin finite element method in space, obtain a fully discrete scheme. The overall scheme is easy to implement, and robust with respect to data regularity. A complete error analysis of the fully discrete scheme is provided, and a second-order accuracy in time is established for both smooth and nonsmooth problem data. Extensive numerical experiments are provided to illustrate its accuracy, efficiency and robustness, and a comparative study also indicates its competitive with existing schemes.

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arXiv2017

Boundary problems for the fractional and tempered fractional operators

For characterizing the Brownian motion in a bounded domain: $Ω$, it is well-known that the boundary conditions of the classical diffusion equation just rely on the given information of the solution along the boundary of a domain; on the contrary, for the Lévy flights or tempered Lévy flights in a bounded domain, it involves the information of a solution in the complementary set of $Ω$, i.e., $\mathbb{R}^n\backslash Ω$, with the potential reason that paths of the corresponding stochastic process are discontinuous. Guided by probability intuitions and the stochastic perspectives of anomalous diffusion, we show the reasonable ways, ensuring the clear physical meaning and well-posedness of the partial differential equations (PDEs), of specifying `boundary' conditions for space fractional PDEs modeling the anomalous diffusion. Some properties of the operators are discussed, and the well-posednesses of the PDEs with generalized boundary conditions are proved.

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arXiv2017

Convergence of finite elements on an evolving surface driven by diffusion on the surface

For a parabolic surface partial differential equation coupled to surface evolution, convergence of the spatial semidiscretization is studied in this paper. The velocity of the evolving surface is not given explicitly, but depends on the solution of the parabolic equation on the surface. Various velocity laws are considered: elliptic regularization of a direct pointwise coupling, a regularized mean curvature flow and a dynamic velocity law. A novel stability and convergence analysis for evolving surface finite elements for the coupled problem of surface diffusion and surface evolution is developed. The stability analysis works with the matrix-vector formulation of the method and does not use geometric arguments. The geometry enters only into the consistency estimates. Numerical experiments complement the theoretical results.

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arXiv2016

Combining maximal regularity and energy estimates for time discretizations of quasilinear parabolic equations

We analyze fully implicit and linearly implicit backward difference formula (BDF) methods for quasilinear parabolic equations, without making any assumptions on the growth or decay of the coefficient functions. We combine maximal parabolic regularity and energy estimates to derive optimal-order error bounds for the time-discrete approximation to the solution and its gradient in the maximum norm and energy norm.

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arXiv2016

Runge-Kutta time discretization of nonlinear parabolic equations studied via discrete maximal parabolic regularity

For a large class of fully nonlinear parabolic equations, which include gradient flows for energy functionals that depend on the solution gradient, the semidiscretization in time by implicit Runge-Kutta methods such as the Radau IIA methods of arbitrary order is studied. Error bounds are obtained in the $W^{1,\infty}$ norm uniformly on bounded time intervals and, with an improved approximation order, in the parabolic energy norm. The proofs rely on discrete maximal parabolic regularity. This is used to obtain $W^{1,\infty}$ estimates, which are the key to the numerical analysis of these problems.

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arXiv2016

Maximum norm analysis of implicit-explicit backward difference formulas for nonlinear parabolic equations

We establish optimal order a priori error estimates for implicit-explicit BDF methods for abstract semilinear parabolic equations with time-dependent operators in a complex Banach space settings, under a sharp condition on the non-self-adjointness of the linear operator. Our approach relies on the discrete maximal parabolic regularity of implicit BDF schemes for autonomous linear parabolic equations, recently established in [20], and on ideas from [7]. We illustrate the applicability of our results to four initial and boundary value problems, namely two for second order, one for fractional order, and one for fourth order, namely the Cahn-Hilliard, parabolic equations.

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arXiv2016

Convergence of a decoupled mixed FEM for the dynamic Ginzburg--Landau equations in nonsmooth domains with incompatible initial data

In this paper, we propose a fully discrete mixed finite element method for solving the time-dependent Ginzburg--Landau equations, and prove the convergence of the finite element solutions in general curved polyhedra, possibly nonconvex and multi-connected, without assumptions on the regularity of the solution. Global existence and uniqueness of weak solutions for the PDE problem are also obtained in the meantime. A decoupled time-stepping scheme is introduced, which guarantees that the discrete solution has bounded discrete energy, and the finite element spaces are chosen to be compatible with the nonlinear structure of the equations. Based on the boundedness of the discrete energy, we prove the convergence of the finite element solutions by utilizing a uniform $L^{3+δ}$ regularity of the discrete harmonic vector fields, establishing a discrete Sobolev embedding inequality for the Nédélec finite element space, and introducing a $\ell^2(W^{1,3+δ})$ estimate for fully discrete solutions of parabolic equations. The numerical example shows that the constructed mixed finite element solution converges to the true solution of the PDE problem in a nonsmooth and multi-connected domain, while the standard Galerkin finite element solution does not converge.

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arXiv2016

Maximal $\bf L^p$ analysis of finite element solutions for parabolic equations with nonsmooth coefficients in convex polyhedra

The paper is concerned with Galerkin finite element solutions for parabolic equations in a convex polygon or polyhehron with a diffusion coefficient in $W^{1,N+ε}$ for some $ε>0$, where $N$ denotes the dimension of the domain. We prove the analyticity of the semigroup generated by the discrete elliptic operator, the discrete maximal $L^p$ regularity and the optimal $L^p$ error estimate of the finite element solution for the parabolic equation.

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arXiv2015

A-stable time discretizations preserve maximal parabolic regularity

It is shown that for a parabolic problem with maximal $L^p$-regularity (for $1<p<\infty$), the time discretization by a linear multistep method or Runge--Kutta method has maximal $\ell^p$-regularity uniformly in the stepsize if the method is A-stable (and satisfies minor additional conditions). In particular, the implicit Euler method, the Crank-Nicolson method, the second-order backward difference formula (BDF), and the Radau IIA and Gauss Runge--Kutta methods of all orders preserve maximal regularity. The proof uses Weis' characterization of maximal $L^p$-regularity in terms of $R$-boundedness of the resolvent, a discrete operator-valued Fourier multiplier theorem by Blunck, and generating function techniques that have been familiar in the stability analysis of time discretization methods since the work of Dahlquist. The A($α$)-stable higher-order BDF methods have maximal $\ell^p$-regularity under an $R$-boundedness condition in a larger sector. As an illustration of the use of maximal regularity in the error analysis of discretized nonlinear parabolic equations, it is shown how error bounds are obtained without using any growth condition on the nonlinearity or for nonlinearities having singularities.

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arXiv2014

Mathematical and numerical analysis of time-dependent Ginzburg--Landau equations in nonconvex polygons based on Hodge decomposition

We prove well-posedness of time-dependent Ginzburg--Landau system in a nonconvex polygonal domain, and decompose the solution as a regular part plus a singular part. We see that the magnetic potential is not in $H^1$ in general, and the finite element method (FEM) may give incorrect solutions. To remedy this situation, we reformulate the equations into an equivalent system of elliptic and parabolic equations based on the Hodge decomposition, which avoids direct calculation of the magnetic potential. The essential unknowns of the reformulated system admit $H^1$ solutions and can be solved correctly by the FEMs. We then propose a decoupled and linearized FEM to solve the reformulated equations and present error estimates based on proved regularity of the solution. Numerical examples are provided to support our theoretical analysis and show the efficiency of the method.

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