Buyang Li

The Hong Kong Polytechnic University | PolyU · Department of Applied Mathematics

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-Ni...

In general polygons and polyhedra, possibly nonconvex, the analyticity of the finite element heat semigroup in the Lq-norm, 1 ≤ q ≤ ∞, and the maximal Lp-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...

A proof of convergence is given for semi- and full discretizations of mean curvature flow of closed two-dimensional surfaces. The numerical method proposed and studied here combines evolving finite elements, whose nodes determine the discrete surface like in Dziuk’s method, and linearly implicit backward difference formulae for time integration. Th...

We construct and analyze a class of extrapolated and linearized Runge-Kutta (RK) methods, which can be of arbitrarily high order, for the time discretization of the Allen-Cahn and Cahn-Hilliard phase field equations, based on the scalar auxiliary variable (SAV) formulation. We prove that the proposed q-stage RK-SAV methods have qth-order convergenc...

The parabolic Dirichlet boundary control problem and its finite element discretization are considered in convex polygonal and polyhedral domains. We improve the existing results on the regularity of the solutions by establishing and utilizing the maximal L^p-regularity of parabolic equations under inho-mogeneous Dirichlet boundary conditions. Based...

A high-frequency recovered fully discrete low-regularity integrator is constructed to approximate rough and possibly discontinuous solutions of the semilinear wave equation. The proposed method, with high-frequency recovery techniques, can capture the discontinuities of the solutions correctly without spurious oscillations and approximate rough and...

Numerical analysis for the stochastic Stokes equations is still challenging even though it has been well done for the corresponding deterministic equations. In particular, the pre-existing error estimates of finite element methods for the stochastic Stokes equations in the L ∞ ( 0 , T ; L 2 ( Ω ; L 2 ) ) L^\infty (0, T; L^2(\Omega ; L^2)) norm all...

In contrast with the diffusion equation which smoothens the initial data to C ∞ C^\infty for t > 0 t>0 (away from the corners/edges of the domain), the subdiffusion equation only exhibits limited spatial regularity. As a result, one generally cannot expect high-order accuracy in space in solving the subdiffusion equation with nonsmooth initial data...

In the perfect conductivity problem (i.e., the conductivity problem with perfectly conducting inclusions), the gradient of the electric field is often very large in a narrow region between two inclusions and blows up as the distance between the inclusions tends to zero. The rigorous error analysis for the computation of such perfect conductivity pr...

As a specific type of shape gradient descent algorithm, shape gradient flow is widely used for shape optimization problems constrained by partial differential equations. In this approach, the constraint partial differential equations could be solved by finite element methods on a domain with a solution-driven evolving boundary. Rigorous analysis fo...

Parametric finite element methods have achieved great success in approximating the evolution of surfaces under various different geometric flows, including mean curvature flow, Willmore flow, surface diffusion, and so on. However, the convergence of Dziuk’s parametric finite element method, as well as many other widely used parametric finite elemen...

A new spectral method is constructed for the linear and semilinear subdiffusion equations with possibly discontinuous rough initial data. The new method effectively combines several computational techniques, including the contour integral representation of the solutions, the quadrature approximation of contour integrals, the exponential integrator...

The weak maximum principle of the isoparametric finite element method is proved for the Poisson equation under the Dirichlet boundary condition in a (possibly concave) curvilinear polyhedral domain with edge openings smaller than π \pi , which include smooth domains and smooth deformations of convex polyhedra. The proof relies on the analysis of a...

A class of stochastic Besov spaces \({B^p}{L^2}({\rm{\Omega}};{\dot{H}^\alpha}({\cal O}))\), 1 ⩽ p ⩽ ∞ and α ∈ [−2, 2], is introduced to characterize the regularity of the noise in the semilinear stochastic heat equation under the following conditions for some α ∈ (0,1] The conditions above are shown to be satisfied by both trace-class noises (with...

A new coupled perfectly matched layer (PML) method is proposed for the Helmholtz equation in the whole space with inhomogeneity concentrated on a nonconvex domain. Rigorous analysis is presented for the stability and convergence of the proposed coupled PML method, which shows that the PML solution converges to the solution of the original Helmholtz...

This article is concerned with the question of whether it is possible to construct a time discretization for the one-dimensional cubic nonlinear Schrödinger equation with second-order convergence for initial data with regularity strictly below $H^2$. We address this question with a positive answer by constructing a new second-order low-regularity i...

The numerical approximation of nonsmooth solutions of the semilinear Klein--Gordon equation in the $d$-dimensional space, with $d=1,2,3$, is studied based on the discovery of a new cancellation structure in the equation. This cancellation structure allows us to construct a low-regularity correction of the Lie splitting method (i.e., exponential Eul...

A new harmonic analysis technique by using the Littlewood-Paley dyadic decomposition is developed for constructing low-regularity integrators for the one-dimensional cubic nonlinear Schrödinger equation in a bounded domain under the Neumann boundary condition, when the frequency analysis based on the Fourier series cannot be used. In particular, a...

Maximal parabolic $L^p$-regularity of linear parabolic equations on an evolving surface is shown by pulling back the problem to the initial surface and studying the maximal $L^p$-regularity on a fixed surface. By freezing the coefficients in the parabolic equations at a fixed time and utilizing a perturbation argument around the freezed time, it is...

An artificial tangential velocity is introduced into the evolving finite element methods for mean curvature flow and Willmore flow proposed by Kovács et al. (Numer Math 143(4), 797-853, 2019, Numer Math 149, 595-643, 2021) in order to improve the mesh quality in the computation. The artificial tangential velocity is constructed by considering a lim...

First-order convergence in time and space is proved for a fully discrete semi-implicit finite element method for the two-dimensional Navier-Stokes equations with L^2 initial data in convex polygonal domains, without extra regularity assumptions or grid-ratio conditions. The proof utilises the smoothing properties of the Navier-Stokes equations in t...

This paper is concerned with an optimal control problem in a bounded-domain Ω0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varOmega _0$$\end{document} under the con...

This article is concerned with the construction and analysis of new time discretizations for the KdV equation on a torus for low-regularity solutions below $H^1$. New harmonic analysis tools, including new averaging approximations to the exponential phase functions, new frequency decomposition techniques, and new trilinear estimates of the KdV oper...

A new type of low-regularity integrator is proposed for the Navier-Stokes equations. Unlike the other low-regularity integrators for nonlinear dispersive equations, which are all fully explicit in time, the proposed method is a semi-implicit exponential method in time in order to preserve the energy-decay structure of the Navier-Stokes equations. F...

An error estimate is presented for the Newton iterative Crank–Nicolson finite element method for the nonlinear Schrödinger equation, fully discretized by quadrature, without restriction on the grid ratio between temporal step size and spatial mesh size. It is shown that the Newton iterative solution converges double exponentially with respect to th...

We propose a linearized semi-implicit and decoupled finite element method for the incompressible Navier–Stokes equations with variable density. Our method is fully discrete and shown to be unconditionally stable. The velocity equation is solved by an H1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \us...

A linearized fully discrete arbitrary Lagrangian–Eulerian finite element method is proposed for solving the two-phase Navier–Stokes flow system and to preserve the energy-diminishing structure of the system at the discrete level, by taking account of the kinetic, potential and surface energy. Two benchmark problems of rising bubbles in fluids in bo...

A fully discrete surface finite element method is proposed for solving the viscous shallow water equations in a bounded Lipschitz domain on the sphere based on a general triangular mesh. The method consists of a modified Crank–Nicolson method in time and a Galerkin surface finite element method in space for the fluid thickness H and the fluid veloc...

The Galerkin finite element solution of the Possion equation under the Neumann boundary condition in a possibly nonconvex polygon, with a graded mesh locally refined at the corners of the domain, is shown to have the maximum-norm stability. As a result, the best approximation result in the maximum norm is proved.

An exponential type of convolution quadrature is proposed as a time-stepping method for the nonlinear subdiffusion equation with bounded measurable initial data. The method combines contour integral representation of the solution, quadrature approximation of contour integrals, multi-step exponential integrators for ordinary differential equations,...

An optimal-order error estimate is presented for the arbitrary Lagrangian-Eulerian (ALE) finite element method for a parabolic equation in an evolving domain, using high-order iso-parametric finite elements with flat simplices in the interior of the domain. The mesh velocity can be a linear approximation of a given bulk velocity field or a numerica...

A proof of convergence is given for a novel evolving surface finite element semi-discretization of Willmore flow of closed two-dimensional sur- faces, and also of surface diffusion flow. The numerical method proposed and studied here discretizes fourth-order evolution equations for the normal vector and mean curvature, reformulated as a system of s...

A linearly implicit renormalized lumped mass finite element method is considered for solving the equations describing heat flow of harmonic maps, of which the exact solution naturally satisfies the pointwise constraint |m| = 1. At every time level, the method first computes an auxiliary numerical solution by a linearly implicit lumped mass method a...

A fully discrete and fully explicit low-regularity integrator is constructed for the one-dimensional periodic cubic nonlinear Schrödinger equation. The method can be implemented by using fast Fourier transform with \(O(N\ln N)\) operations at every time level, and is proved to have an \(L^2\)-norm error bound of \(O(\tau \sqrt{\ln (1/\tau )}+N^{-1}...

A fully discrete finite element method, based on a new weak formulation and a new time-stepping scheme, is proposed for the surface diffusion flow of closed curves in the two-dimensional plane. It is proved that the proposed method can preserve two geometric structures simultaneously in the discrete level, i.e., the perimeter of the curve decreases...

This article concerns the numerical approximation of the two-dimensional nonstationary Navier-Stokes equations with H 1 initial data. By utilizing special locally refined temporal stepsizes, we prove that the linearly extrapolated Crank-Nicolson scheme, with the usual stabilized Taylor-Hood finite element method in space, can achieve second-order c...

A variable stepsize exponential multistep integrator, with contour integral approximation of the operator-valued exponential functions, is proposed for solving semilinear parabolic equations with nonsmooth initial data. By this approach, the exponential k-step method would have kth-order convergence in approximating a mild solution, possibly nonsmo...

Dziuk's surface finite element method for mean curvature flow has had significant impact on the development of parametric and evolving surface finite element methods for surface evolution equations and curvature flows. However, the convergence of Dziuk's surface finite element method for mean curvature flow of closed surfaces still remains open sin...

A family of arbitrarily high-order fully discrete space-time finite element methods are proposed for the nonlinear Schrödinger equation based on the scalar auxiliary variable formulation, which consists of a Gauss collocation temporal discretization and the finite element spatial discretization. The proposed methods are proved to be well-posed and...

This article extends the semidiscrete maximal L p-regularity results in [27] to mul-tistep fully discrete finite element methods for parabolic equations with more general diffusion coefficients in W^{1,d+β}, where d is the dimension of space and β > 0. The maximal angles of R-boundedness are characterized for the analytic semigroup e^{zA_h} and the...

An error estimate is presented for a fully discrete, linearized and stabilized finite element method for solving the coupled system of nonlinear hyperbolic and parabolic equations describing incompressible flow with variable density in a two-dimensional convex polygon. In particular, the error of the numerical solution is split into the temporal an...

We prove well-posedness and regularity of solutions to a fractional diffusion porous media equation with a variable order that may depend on the unknown solution. We present a linearly implicit time-stepping method to linearize and discretize the equation in time, and present rigorous analysis for the convergence of numerical solutions based on pro...

We prove that for a given smooth initial value, if the finite element solution of the three-dimensional Navier-Stokes equations is bounded in a certain norm with a relatively small mesh size, then the solution of the Navier-Stokes equations with this given initial value must be smooth and unique, and is successfully approximated by the numerical so...

The evolution of a closed two-dimensional surface driven by both mean curvature flow and a reaction–diffusion process on the surface is formulated as a system that couples the velocity law not only to the surface partial differential equation but also to the evolution equations for the normal vector and the mean curvature on the surface. Two algori...

An implicit energy-decaying modified Crank-Nicolson time-stepping method is constructed for the viscous rotating shallow water equation on the plane. Existence, uniqueness and convergence of semidiscrete solutions are proved by using Schaefer's fixed point theorem and H 2 estimates of the discretized hyperbolic-parabolic system. For practical compu...

A new class of high-order maximum principle preserving numerical methods is proposed for solving parabolic equations, with application to the semilinear Allen-Cahn equation. The proposed method consists of a kth-order multistep exponential integrator in time, and a lumped mass finite element method in space with piecewise rth-order polynomials and...

This article concerns second-order time discretization of subdiffu-sion equations with time-dependent diffusion coefficients. High-order differentiability and regularity estimates are established for subdiffusion equations with time-dependent coefficients. Using these regularity results and a perturbation argument of freezing the diffusion coeffici...

For a class of compatible profiles of initial data describing bulk phase regions separated by transition zones, we approximate the Cauchy problem of the Allen-Cahn phase field equation by an initial-boundary value problem in a bounded domain with the Dirichlet boundary condition. The initial-boundary value problem is discretized in time by the back...

A new time discretization method for strongly nonlinear parabolic systems is constructed by combining the fully explicit two-step backward difference formula and a second-order stabilization of wave type. The proposed method linearizes and decouples a nonlinear parabolic system at every time level, with second-order consistency error. The convergen...

In a general polygonal domain, possibly nonconvex and multi-connected (with holes), the time-dependent Ginzburg-Landau equation is reformulated into a new system of equations. The magnetic field B := ∇×A is introduced as an unknown solution in the new system, while the magnetic potential A is solved implicitly through its Hodge decomposition into d...

Convergence of Dziuk's fully discrete linearly implicit parametric finite element method for curve shortening flow on the plane remains still open since it was proposed in 1990, though the corresponding semidiscrete method with piecewise linear finite elements has been proved to be convergent in 1994, while the error analysis for the semidiscrete m...

We prove that the Galerkin finite element solution u h u_h of the Laplace equation in a convex polyhedron Ω \varOmega , with a quasi-uniform tetrahedral partition of the domain and with finite elements of polynomial degree r ⩾ 1 r\geqslant 1 , satisfies the following weak maximum principle: ‖ u h ‖ L ∞ ( Ω ) ⩽ C ‖ u h ‖ L ∞ ( ∂ Ω ) , \begin{align*}...

The implicit Euler scheme for nonlinear partial differential equations of gradient flows is linearized by Newton's method, discretized in space by the finite element method. With two Newton iterations at each time level, almost optimal order convergence of the numerical solutions is established in both the L q (Ω) and W 1,q (Ω) norms. The proof is...

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...

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(u)=γdmI+|u|(αTI+(αL-αT)u⊗u|u|2).Previous works on optimal-order L ∞ (0 , T; L ² ) -norm error estimate required the regularity assumption ∇ x ∂ t D(u(x, t)) ∈ L...

A feasible approach to study tempered anomalous dynamics is to analyze its functional distribution, which is governed by the tempered fractional Feynman-Kac equation. The main challenges of numerically solving the equation come from the time-space coupled nonlocal operators and the complex parameters involved. In this work, we introduce an efficien...

In this work, a complete error analysis is presented for fully discrete solutions of the sub-diffusion 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 quad-rature in time. The regularity of the solutions of t...

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...

This paper is concerned with time-harmonic electromagnetic scattering from a cavity embedded in an impedance ground plane. The fillings (which may be inhomogeneous) do not protrude the cavity and the space above the ground plane is empty. This problem is obviously different from those considered in previous work where either perfectly conducting bo...

The stochastic time-fractional equation $\partial_t \psi -\Delta\partial_t^{1-\alpha} \psi = f + \dot W$ with space-time white noise $\dot W$ is discretized in time by a backward-Euler convolution quadrature for which the sharp-order error estimate \[ {\mathbb E}\|\psi(\cdot,t_n)-\psi_n\|_{L^2(\mathcal{O})}^2=O(\tau^{1-\alpha d/2}) \] is establishe...

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 $\alpha\in(0,2)$, $\alpha\neq 1$, in time. These schemes include convolution quadratures generated by backward Euler method and second- order backward difference formula, the...

For characterizing the Brownian motion in a bounded domain: $\Omega$, 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 L\'evy flights or tempered L\'evy flights in a bounded domain, it involves the information...

We analyze a parallel, noniterative, multiphysics domain decomposition method for decoupling the Stokes–Darcy model with multistep backward differentiation schemes for the time discretization and finite elements for the spatial discretization. Based on a rigorous analysis of the Ritz projection error shown in this article, we prove almost optimal L...

A stable and convergent second-order fully discrete finite difference scheme with efficient approximation of the exact absorbing boundary conditions is proposed to solve the Cauchy problem of the one-dimensional Schrödinger equation. Our approximation is based on the Padé expansion of the square root function in the complex plane. By introducing a...

A second-order Crank-Nicolson finite difference method, integrating a fast approximation of an exact discrete absorbing boundary condition, is proposed for solving the one-dimensional Schrödinger equation in the whole space. The fast approximation is based on Gaussian quadrature approximation of the convolution coefficients in the discrete absorbin...

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...

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 c...

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 wea...

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 $\alpha\in(0,1)$ in time. The fully discrete scheme is obtained by applying the conforming linear Galerkin finite element method in space, L1 s...

In this paper, we establish the unconditional stability and optimal error estimates of a linearized backward Euler–Galerkin finite element method (FEM) for the time-dependent nonlinear thermistor equations in a two-dimensional nonconvex polygon. Due to the nonlinearity of the equations and the non-smoothness of the solution in a nonconvex polygon,...

We present a general framework for the rigorous numerical analysis of time-fractional nonlinear parabolic partial differential equations, with a fractional derivative of order $\alpha\in(0,1)$ in time. The framework relies on three technical tools: a fractional version of the discrete Gr\"onwall-type inequality, discrete maximal regularity, and reg...

A second-order leapfrog finite difference scheme in time is proposed and developed for solving the first-order necessary optimality system of the distributed parabolic optimal control problems. Different from available approaches, the proposed leapfrog scheme for the two-point boundary optimality system is shown to be unconditionally stable and pro...

In this work, we analyse a Crank-Nicolson type time-stepping scheme for the subdiffusion equation, which involves a Caputo fractional derivative of order α ∈ (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...

We establish optimal order a priori error estimates for implicit–explicit backward difference formula (BDF) methods for abstract semilinear parabolic equations with time-dependent operators in a complex Banach space setting, under a sharp condition on the non-self-adjointness of the linear operator. Our approach relies on the discrete maximal parab...

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...

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 th...

We establish the maximal lp-regularity for fully discrete finite element solutions of parabolic equations with time-dependent Lipschitz continuous coefficients. The analysis is based on a discrete lp(W¹,q) estimate together with a duality argument and a perturbation method. Optimalorder error estimates of fully discrete finite element solutions in...

As a model of the second order elliptic equation with non-trivial boundary conditions, we consider the Laplace equation with mixed Dirichlet and Neumann boundary conditions on convex polygonal domains. Our goal is to establish that finite element discrete harmonic functions with mixed Dirichlet and Neumann boundary conditions satisfy a weak (Agmon-...

In this paper, we develop a new central finite difference scheme in terms of both time and space for solving the first-order necessary optimality systems that characterize the optimal control of wave equations. The obtained new scheme is proved to be unconditionally convergent with a second-order accuracy, without the requirement of the Courant-Fri...

A second-order leapfrog finite difference scheme in time is proposed to solve the first-order necessary optimality systems arising from parabolic optimal control problems. Different from classical approximation, the proposed leapfrog scheme appears to be unconditionally stable. More importantly, the developed leapfrog scheme provides a well-structu...