In this talk I will discuss the newly developed symmetric direct discontinuous Galerkin (DDG) method for the elliptic interface problems associated with solution jump and flux jump interface conditions. We focus on the case with mesh partition aligning with the curved interface. Numerical fluxes on the edges of curved triangular elements that overlap with the interface are carefully designed. Both the solution jump and flux jump conditions are incorporated into numerical flux definitions and enforced in the weak sense and a stable and high order method is obtained. Optimal $(k+1)th$ order $L^2$ norm error estimate is proved for polygonal interface. A sequence of numerical examples are carried out to verify the optimal convergence of the symmetric DDG method with high order $P_2$, $P_3$ and $P_4$ approximations. Uniform convergence orders that are independent of the diffusion coefficient ratio inside and outside of the interface are obtained. The symmetric DDG method is shown to be capable to handle interface problems with complicated geometries.

The immersed finite element method (IFEM) is a class of numerical methods for solving PDE interface problems with unfitted meshes. In this talk, we will introduce a new class of IFEM in the nonconforming framework, such as the Crouzeix-Raviart element on triangular meshes or the rotated Q1 element on rectangular meshes. For second-order elliptic interface problems, the nonconforming IFEM is proved to converge optimally in standard Galerkin formulation without extra stabilization terms required by conforming IFEMs. In particular, our error estimates in the discrete H1 and L2-norms are proved to be optimal sans the usual | log h| factor, which reflects the coefficient discontinuity. Moreover, we will discuss the application of the nonconforming IFEM to Stokes interface problems. The lowest order nonconforming P1-P0 immersed finite element is a stable pair. Numerical results are provided to demonstrate the features of these nonconforming IFEMs.

Since their introduction in the late 1950’s, the Cahn-Hilliard equation has played an important role in understanding phase transition phenomena that is observed in materials. In particular, the Cahn-Hilliard equation describes the process of phase separation in which a mixture of two materials separate or fuse to form pure material domains. In this talk, we discuss possible schemes to obtain stable and robust Newton iterations when solving the Cahn-Hilliard Equation with nonlinear potential function. In particular, we focus on formulating suitable preconditioners to solve the resultant linear system based on multigrid approach.

In this article, we reanalyze the fully discrete PPIFE method presented in our previous work. We are able to derive optimal a-priori error bounds for this PPIFE method not only in the energy norm but also in L2 norm under the standard piecewise H2 regularity assumption in the space variable of the exact solution, rather than the excessive piecewise H3 regularity. Numerical simulations for standing and traveling waves are presented.

This work presents a novel two-dimensional interface-fitted adaptive mesh method to solve elliptic problems of jump conditions across the interface, and its application in free interface problems with surface tension. The interface-fitted mesh is achieved by two operations: (i) the projection of mesh nodes onto the interface and (ii) the insertion of mesh nodes right on the interface. The interface-fitting technique is combined with an existing adaptive mesh approach which uses addition/subtraction and displacement of mesh nodes. We develop a simple piecewise linear finite element method built on this interface-fitted mesh and prove its almost optimal convergence for elliptic problems with jump conditions across the interface. Applications to two free interface problems, a sheared drop in Stokes flow and the growth of a solid tumor, are presented.

Interface inverse problems widely appear in many engineering applications. These problems in general involve multiple materials coupled through interface and the interface itself is unknown needed to be recovered. In this talk, I will describe a parameterized shape optimization algorithm based on immersed finite element (IFE) methods to solve a group of interface inverse problems including electrical impedance tomography (EIT) and elastography. A set of numerical experiments are shown to demonstrate the strength and versatility of the proposed method.

In this talk, we present a high order immersed finite element (IFE) method for solving parabolic interface problems. This method allows the solution mesh to be independent of the interface. Hence, there is no need to regenerate meshes for problems with a moving interface. Standard time marching schemes including Backward-Euler and Crank-Nicolson methods are implemented to fully discretize the system. Optimal convergence rates in both H1 and L2 norms are observed. Numerical examples are provided to test the performance of our numerical schemes.

A systematic approach is introduced to implement high order central difference schemes for solving Poisson's equation via the fast Fourier transform (FFT). FFT-based high order central difference schemes have never been developed for Poisson problems, because with long stencils, central differences require artificial nodes information outside the boundary, which poses a challenge to integrate boundary conditions in FFT computations. To overcome this difficulty, several layers of exterior grid lines are introduced to convert the problem to an immersed boundary problem with zero-padding solutions beyond the original cubic domain. Over the boundary of the enlarged cubic domain, the anti-symmetric property is naturally satisfied so that the FFT fast inversion is feasible, while the immersed boundary problem can be efficiently solved by the proposed augmented matched interface and boundary (AMIB) method. As the first fast Poisson solver based on high order central differences, the AMIB method can be easily implemented in any high dimension, due to the tensor product nature of the discretization. As a systematical approach, the AMIB method can be made to arbitrarily high order in principle, and can handle the Dirichlet, Neumann, Robin or any combination of boundary conditions.