Shu Ma

• Doctor of Philosophy
• Research Fellow at National University of Singapore

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...

A fully discrete finite element method with a Gauss collocation in time is proposed for solving the nonlinear Schrödinger equation with a wave operator in the d-dimensional torus, \(d\in \{1,2,3\}\). Based on Gauss collocation method in time and the scalar auxiliary variable technique, the proposed method preserves both mass and energy conservation...

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 optimal error estimate that depending only on the polynomial degree of $ \varepsilon^{-1}$ is established for the temporal semi-discrete scheme of the Cahn-Hilliard equation, which is based on the scalar auxiliary variable (SAV) formulation. The key to our analysis is to convert the structure of the SAV time-stepping scheme back to a form compa...

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...

Numerical analysis for the stochastic Stokes/Navier-Stokes equations is still challenging even though it has been well done for the corresponding deterministic equations. In particular, the existing error estimates of finite element methods for the stochastic equations all suffer from the order reduction with respect to the spatial discretizations....

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 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,...

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

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...

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...

An error estimate is presented 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 CFL conditions, by utilizing the smoothing properties of the Navier--Stokes equations, an appropriate duality argument,...

This paper is concerned with fully discrete finite element approximations of a stochastic nonlinear Schrödinger (sNLS) equation with linear multiplicative noise of the Stratonovich type. The goal of studying the sNLS equation is to understand the role played by the noises for a possible delay or prevention of the collapsing and/or blow-up of the so...

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 $k$th-order convergence in approximating a mild solution, possibly nons...

A family of arbitrarily high-order fully discrete space-time finite element methods are proposed for the nonlinear Schr\"odinger 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 an...

In this paper, we propose a family of time-stepping schemes for approximating general nonlinear Schr\"odinger equations. The proposed schemes all satisfy both mass conservation and energy conservation. Truncation and dispersion error analyses are provided for each proposed scheme. Efficient fixed-point iterative solvers are also constructed to solv...