Buyang Li

Georgios Akrivis, Buyang Li, Christian Lubich

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.

Max Gunzburger, Buyang Li, Jilu Wang

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.

Balázs Kovács, Buyang Li, Christian Lubich

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.

Bangti Jin, Buyang Li, Zhi Zhou

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.

Bangti Jin, Buyang Li, Zhi Zhou

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.

Balázs Kovács, Buyang Li, Christian Lubich et al.

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.

Max Gunzburger, Buyang Li, Jilu Wang

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.

Buyang Li, Weiwei Sun

In this paper, we study the unconditional convergence and error estimates of a Galerkin-mixed FEM with the linearized semi-implicit Euler time-discrete scheme for the equations of incompressible miscible flow in porous media. We prove that the optimal $L^2$ error estimates hold without any time-step (convergence) condition, while all previous works require certain time-step condition. Our theoretical results provide a new understanding on commonly-used linearized schemes for nonlinear parabolic equations. The proof is based on a splitting of the error function into two parts: the error from the time discretization of the PDEs and the error from the finite element discretization of corresponding time-discrete PDEs. The approach used in this paper is applicable for more general nonlinear parabolic systems and many other linearized (semi)-implicit time discretizations.

Peer C. Kunstmann, Buyang Li, Christian Lubich

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.

Bangti Jin, Buyang Li, Zhi Zhou

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.

Buyang Li, Weiwei Sun

This paper is concerned with the time-step condition of commonly-used linearized semi-implicit schemes for nonlinear parabolic PDEs with Galerkin finite element approximations. In particular, we study the time-dependent nonlinear Joule heating equations. We present optimal error estimates of the semi-implicit Euler scheme in both the $L^2$ norm and the $H^1$ norm without any time-step restriction. Theoretical analysis is based on a new splitting of the error and precise analysis of a corresponding time-discrete system. The method used in this paper can be applied to more general nonlinear parabolic systems and many other linearized (semi)-implicit time discretizations for which previous works often require certain restriction on the time-step size $τ$.

Bangti Jin, Buyang Li, Zhi Zhou

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.

Buyang Li, Weiwei Sun

This paper focuses on unconditionally optimal error analysis of an uncoupled and linearized Crank--Nicolson Galerkin finite element method for the time-dependent nonlinear thermistor equations in $d$-dimensional space, $d=2,3$. We split the error function into two parts, one from the spatial discretization and one from the temporal discretization, by introducing a corresponding time-discrete (elliptic) system. We present a rigorous analysis for the regularity of the solution of the time-discrete system and error estimates of the time discretization. With these estimates and the proved regularity, optimal error estimates of the fully discrete Crank--Nicolson Galerkin method are obtained unconditionally. Numerical results confirm our analysis and show the efficiency of the method.

Bangti Jin, Buyang Li, Zhi Zhou

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.

Buyang Li, Weiwei Sun

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.

Buyang Li

In this paper, we study the semi-discrete Galerkin finite element method for parabolic equations with Lipschitz continuous coefficients. We prove the maximum-norm stability of the semigroup generated by the corresponding elliptic finite element operator, and prove the space-time stability of the parabolic projection onto the finite element space in $L^\infty(Ω_T)$ and $L^p((0,T);L^q(Ω))$, $1<p,q<\infty$. The maximal $L^p$ regularity of the parabolic finite element equation is also established.

Georgios Akrivis, Buyang Li

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.

Buyang Li

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.

Buyang Li

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.

Buyang Li, Jilu Wang, Liwei Xu

A new fully discrete linearized $H^1$-conforming Lagrange finite element method is proposed for solving the two-dimensional magneto-hydrodynamics equations based on a magnetic potential formulation. The proposed method yields numerical solutions that converge in general domains that may be nonconvex, nonsmooth and multi-connected. The convergence of subsequences of the numerical solutions is proved only based on the regularity of the initial conditions and source terms, without extra assumptions on the regularity of the solution. Strong convergence in $L^2(0,T;{\bf L}^2(Ω))$ was proved for the numerical solutions of both ${\bm u}$ and ${\bm H}$ without any mesh restriction.

Buyang Li, Weiwei Sun

The paper focuses on unconditionally optimal error analysis of the fully discrete Galerkin finite element methods for a general nonlinear parabolic system in $\R^d$ with $d=2,3$. In terms of a corresponding time-discrete system of PDEs as proposed in \cite{LS1}, we split the error function into two parts, one from the temporal discretization and one the spatial discretization. We prove that the latter is $τ$-independent and the numerical solution is bounded in the $L^{\infty}$ and $W^{1,\infty}$ norms by the inverse inequalities. With the boundedness of the numerical solution, optimal error estimates can be obtained unconditionally in a routine way. Several numerical examples in two and three dimensional spaces are given to support our theoretical analysis.

Guangwei Gao, Buyang Li, Rong Tang

We propose new formulations of geometric curvature flows -- referred to as \emph{dual formulations} -- that are equivalent to the original formulations but provide a novel framework for constructing linearly implicit and energy-stable schemes for curvature-driven surface evolution, including mean curvature flow, surface diffusion, and solid-state dewetting on a substrate with a moving contact line. The dual formulations are derived by introducing, at the continuous level, an additional unknown in the form of a dual multiplier. This augmentation does not alter the continuous dynamics but makes the underlying energy-dissipation structure explicit and, in turn, enables a systematic design of linearly implicit discretizations that inherit energy stability. A key feature of this framework is that it accommodates a broad class of artificial tangential motions which can be used to maintain good mesh quality of the computed surfaces. As an illustration, we combine the framework with the minimal-deformation-rate (MDR) tangential motion, leading to what we call the \emph{dual-MDR} scheme. The resulting method is linearly implicit and energy-stable, while retaining the MDR tangential motion to maintain good mesh quality. Extensive numerical experiments demonstrate the convergence of the proposed schemes, their structure-preserving properties, and advantages on representative benchmark problems.

Jiachuan Cao, Buyang Li, Hao Li

We introduce an entropy-compatible neural network method for scalar hyperbolic conservation laws based on a computable surrogate of the Kružkov entropy residual. The proposed loss is designed to enforce weak consistency and entropy admissibility while remaining amenable to rigorous analysis. For piecewise smooth entropy solutions, we construct explicit neural network competitors with provably small loss by combining shock-adapted continuous piecewise linear approximations with representability results for neural networks. These approximation bounds, together with entropy-based stability estimates, yield rigorous $L^1$ error estimates for minimizers of the resulting loss functional. In particular, when the network size scales comparably to the number of degrees of freedom of a space time mesh of size $h$, we recover the classical $O(h^{1/2})$ convergence rate in shock-dominated regimes. The analysis also covers solutions containing rarefaction waves and regular shock interactions; in this extended version, it further treats smooth initial data that develop a shock in finite time, for which an $L^1$ estimate of order $O(h^{1/2}|\ln h|)$ is obtained. Numerical experiments in one and two space dimensions illustrate the effectiveness of the method and its ability to resolve nonsmooth wave structures accurately.

Bangti Jin, Buyang Li, Zhi Zhou

In this work, a complete error analysis is presented for fully discrete solutions of the subdiffusion equation with a time-dependent diffusion coefficient, obtained by the Galerkin finite element method with conforming piecewise linear finite elements in space and backward Euler convolution quadrature in time. The regularity of the solutions of the subdiffusion model is proved for both nonsmooth initial data and incompatible source term. Optimal-order convergence of the numerical solutions is established using the proven solution regularity and a novel perturbation argument via freezing the diffusion coefficient at a fixed time. The analysis is supported by numerical experiments.

Wentao Cai, Buyang Li, Yanping Lin et al.

Fully discrete Galerkin finite element methods are studied for the equations of miscible displacement in porous media with the commonly-used Bear--Scheidegger diffusion-dispersion tensor: $$ D({\bf u}) = γd_m I + |{\bf u}|\bigg( α_T I + (α_L - α_T) \frac{{\bf u} \otimes {\bf u}}{|{\bf u}|^2}\bigg) \, . $$ Previous works on optimal-order $L^\infty(0,T;L^2)$-norm error estimate required the regularity assumption $\nabla_x\partial_tD({\bf u}(x,t)) \in L^\infty(0,T;L^\infty(Ω))$, while the Bear--Scheidegger diffusion-dispersion tensor is only Lipschitz continuous even for a smooth velocity field ${\bf u}$. In terms of the maximal $L^p$-regularity of fully discrete finite element solutions of parabolic equations, optimal error estimate in $L^p(0,T;L^q)$-norm and almost optimal error estimate in $L^\infty(0,T;L^q)$-norm are established under the assumption of $D({\bf u})$ being Lipschitz continuous with respect to ${\bf u}$.

Buyang Li, Zhimin Zhang

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.

Buyang Li, Zhimin Zhang

We introduce a new approach for finite element simulations of the time-dependent Ginzburg-Landau equations (TDGL) in a general curved polygon, possibly with reentrant corners. Specifically, we reformulate the TDGL into an equivalent system of equations by decomposing the magnetic potential to the sum of its divergence-free and curl-free parts, respectively. Numerical simulations of vortex dynamics show that, in a domain with reentrant corners, the new approach is much more stable and accurate than the old approaches of solving the TDGL directly (under either the temporal gauge or the Lorentz gauge); in a convex domain, the new approach gives comparably accurate solutions as the old approaches.