Shengfeng Zhu

Inhomogeneous Dirichlet Boundary-Value Problems of Space-Fractional Diffusion Equations and their Finite Element Approximations

We prove the wellposedness of the Galerkin weak formulation and Petrov--Galerkin weak formulation for inhomogeneous Dirichlet boundary-value problems of constant- or variable-coefficient conservative Caputo space-fractional diffusion equations. We also show that the weak solutions to their Riemann--Liouville analogues do not exist, in general. In addition, we develop an indirect finite element method for the Dirichlet boundary-value problems of Caputo fractional differential equations, which reduces the computational work for the numerical solution of variable-coefficient fractional diffusion equations from

New Variational Formulations for Level Set Evolution Without Reinitialization with Applications to Image Segmentation

O(N^3)

Variational piecewise constant level set methods for shape optimization of a two-density drum

to

Numerical solution of the Falkner-Skan equation based on quasilinearization

O(N)

An adaptive algorithm for the Thomas---Fermi equation

and the memory requirement from

A variational binary level-set method for elliptic shape optimization problems

O(N^2)

Accuracy of Finite Element Methods for Boundary-Value Problems of Steady-State Fractional Diffusion Equations

to

Isogeometric analysis and proper orthogonal decomposition for parabolic problems

O(N)

A level set method for shape optimization in semilinear elliptic problems

on any quasiuniform space partition. We further prove a nearly sharp error estimate for the method, which is expressed in terms of the smoothness of the prescribed data of the problem only. We carry out numerical experiments to investigate the performance of the method in comparison with the Galerkin finite element method.

A multi-mesh finite element method for phase-field based photonic band structure optimization

Interface evolution problems are often solved elegantly by the level set method, which generally requires the time-consuming reinitialization process. In order to avoid reinitialization, we reformulate the variational model as a constrained optimization problem. Then we present an augmented Lagrangian method and a projection Lagrangian method to solve the constrained model and propose two gradient-type algorithms. For the augmented Lagrangian method, we employ the Uzawa scheme to update the Lagrange multiplier. For the projection Lagrangian method, we use the variable splitting technique and get an explicit expression for the Lagrange multiplier. We apply the two approaches to the Chan-Vese model and obtain two efficient alternating iterative algorithms based on the semi-implicit additive operator splitting scheme. Numerical results on various synthetic and real images are provided to compare our methods with two others, which demonstrate effectiveness and efficiency of our algorithms.