Research interests

Mathematical modeling and simulation: fluid flow through poroelastic media, ocean and atmosphere dynamics, sea ice floe dynamics.

Scientific computing and numerical analysis: parallel computing, preconditioners, postprocessing, PDE numerical solvers such as FDM, FVM, FEM, IGA, DG, HHO, Runge–Kutta methods, and generalizedalpha methods, operator splitting schemes, dispersion and spectral analysis, a priori and a posteriori error analysis.

Uncertainty quantification and data assimilation: stochastic models, Kalman filters, OrnsteinUhlenbeck process.

Artificial intelligence and machine learning: particle swarm optimization, physicsinformed neural networks, supervised learning.
Publications
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A talked given at ANU MSI MACS:
Several research lines
Development of FEM approximations
(1) A FEM approximated solution lacks the local conservation property on its fluxes. We propose a simple and efficient postprocessing technique to recover the locally conservative fluxes on control volumes (FEM dual mesh elements). The technqiue solves an elemental Neumannboundary value problem and it can be naturally implemented in a parallel environment. See below figures for control volumes for rectangular and triangular elements. See this paper for details.


(2) Quadratic and higherorder FEMs suffer from high stiffness (large condition numbers) in their discretised systems. We propose to reduce the stiffness of the problem by subtracting a leastsquares penalty on the gradient jumps across the mesh interfaces from the standard stiffness bilinear form. The two key advantages of softFEM over the standard Galerkin FEM are to improve the approximation of the eigenvalues in the upper part of the discrete spectrum and to reduce the condition number of the stiffness matrix. The resulting approximation technique is called softFEM since it reduces the stiffness of the problem. See below a figure which compares the quadratic softFEM with FEM on spectral and eigenstate errors. See this paper for details.

Spectral approximation with finite and isogeometric elements
(1) The spectral approximation by isogoemetric analysis has outliers (large eigenvalue errors) in the highfrequency region. We propose a boundary penalty technique to remove the outliers and consequently reduce the stiffness (condition numbers) of the discretised system; See below a figure where the large spectral errors have been significantly reduced. See this paper for details.

(2) In general, when the domain of the model problem is irregular, the discontinuous Galerkin (DG) methods perform better than the standard FEMs. We develop hybrid highorder (HHO) DG method to approximate the spectra of the secondorder elliptic operator on irregular domains. See below the HHO approximated eigenmodes on a circular and an Lshaped domain. See this paper for details.




Multiphase flow simulations for flow in porous or poroelastic media
It is a challenging task to simulate the fluid flow through porous or poroelastic media. The main challenges are (1) complicated coupling between the Darcy fluid flow and the transport and (2) local conservative fluxes are required to maintain physical saturation (bounded from 0 to 1). We develop a simulation tool based on FEMs with our local conservation postprocessing technique. See below the left column for twophase flow through porous media while the right column for flow through poroelastic media. See this paper for porous media while this paper for poroelastic media.




Superfloe parameterisation with data assimilation for sea ice dynamics
The discrete element method (DEM) is providing a new modeling approach for describing sea ice dynamics. It exploits particlebased methods to characterize the physical quantities of each sea ice floe along its trajectory under Lagrangian coordinates. One major challenge in applying the DEM models is the heavy computational cost when the number of the floes becomes large. We develop an efficient Lagrangian parameterization algorithm to reduce the computational cost of simulating the DEM models while preserving the key features of the sea ice. The new parameterization takes advantage of a small number of artificial ice floes, named the superfloes, to effectively approximate a considerable number of the floes, where the parameterization scheme satisfies several important physics constraints. See below figures on the superfloe parameterisation. See this paper for details.


See below a presentation on this topic.
Isogeometric analysis of a quantum threebody problem
The quantum threebody problem has been wellknown to be difficult to solve. We initiate the numerical study of this problem by isogeometric analysis. We represent the wavefunctions by linear combinations of Bspline basis functions and solve the problem as a matrix eigenvalue problem. The eigenvalue gives the eigenstate energy while the eigenvector gives the coefficients of the Bsplines that lead to the eigenstate. The major difficulty of isogeometric or other finiteelementmethodbased analyses lies in the lack of boundary conditions and a large number of degrees of freedom required for accuracy. For a typical manybody problem with attractive interaction, there are bound and scattering states where bound states have negative eigenvalues. We focus on bound states and start with the analysis for a twobody problem. See below the figure on the first few eigenstates. See this paper for details.

Spectra of an unconditionally stable explicit time integrator
We propose a new strategy for solving stiff ordinary differential equations (ODEs). We stabilize explicit schemes in a constructive and repeatable manner. The key insight is to modify an explicit time marching scheme using Newmark’s ideas and to use analysis to discretely correct the algorithm to deliver stability. For example, we adopt the update rules from the Newmark method and apply an estimate (a constant) of the stiffness to discretely correct the forwardEuler scheme to obtain an unconditionallystable method. The new scheme solves simultaneously for the ODE unknown and its first timederivative. In practice, we eliminate the time derivative and solve for the variable at the present time using the variable and its time derivate from the prior time. We then update the first timederivative using the update rule from the Newmark method. The scheme inherits the auxiliary parameter $\beta$ from the Newmark$\beta$ method and introduces an extra auxiliary scaling parameter $\eta$. We establish that the scheme is unconditionallystable when $\beta>1$ and $\eta > (2\beta  1)/\beta^2$. We also discuss a special case that delivers Lstability. The prototypical scheme is firstorder accurate in time for both the ODE unknown and its first timederivative. Using the same construction, we further correct the local truncation error of the forward Euler method to achieve a secondorder scheme. We analyze the stability and accuracy of this explicit secondorder scheme. We demonstrate its accuracy and stability. Both methods have the overall cost of Heun’s method (twostage RungeKutta). See below the figure for spectra of the scheme.


Research visits, talks, and other activities
 See CV here.