README.md

Compact Coupling Interface Method (CCIM) for Elliptic Interface Problem

Elliptic Interface Problem

The repository demonstrates the Compact Coupling Interface Method (CCIM) [1] for the following Elliptic Interface Problem in 3D.

$$ \begin{cases} -\nabla \cdot(\epsilon \nabla u) + a u = f & \text{ in }\Omega \setminus \Gamma \\ \left[u\right] = \tau,\quad \left[\epsilon\nabla u\cdot \mathbf{n} \right] = \sigma & \text{on }\Gamma \\ u = g & \text{on }\partial \Omega. \end{cases} $$

Here $\Gamma$ is an interface that separates $\Omega \subset R^d$ into an inside region $\Omega^-$ and an outside region $\Omega^+$. $\Gamma$ is represented implicitly by the zero level set of a function $\phi$ so that $\Gamma ={ x| \phi(x) = 0}$. $\mathbf{n}$ is the outward unit normal vector of the interface. g is the Dirichlet boundary condition. $\epsilon$, $f$, $a$: $\Omega \to R$ are given functions that might be discontinuous across $\Gamma$. The notation $[v]$ stands for jump of $v$ across the interface.

CCIM

CCIM obtains second order accurate solution and second order accurate gradient at the interface for complex interface in 3D

Linear System Solver

CCIM results in an asymmetric sparse linear system. The code includes self-contained implementation of BICGSTAB (with ILU preconditioner) and Hypre[4] (currently without MPI).

Level Set Method

The code also includes examples of a moving interface using the level set method $$\phi_t + v_n |\nabla \phi| = 0$$ where the normal velocity of the interface is $v_n = [\nabla u \cdot \mathbb{n}]$, and $u$ is the solution of the elliptic interface problem. We use the forward Euler method for time stepping, Godunov scheme for the Hamiltonian, WENO scheme, and the Fast marching Method to extend $v_n$.

Other Finite Difference Method

Additionally, the repository also includes implementation of the Coupling Interface Method (CIM)and the Improved Coupling Interface Method (ICIM).

Usage

Reference

[1] Zhang, Z., Cheng, L.-T., 2021. A Compact Coupling Interface Method with Accurate Gradient Approximation for Elliptic Interface Problems. https://doi.org/10.48550/arXiv.2110.12414

[2] Chern, I.-L., Shu, Y.-C., 2007. A coupling interface method for elliptic interface problems. Journal of Computational Physics 225, 2138–2174. https://doi.org/10.1016/j.jcp.2007.03.012

[3] Shu, Y.-C., Chern, I.-L., Chang, C.C., 2014. Accurate gradient approximation for complex interface problems in 3D by an improved coupling interface method. Journal of Computational Physics 275, 642–661. https://doi.org/10.1016/j.jcp.2014.07.017

[4] Falgout, R.D., Yang, U.M., 2002. hypre: A Library of High Performance Preconditioners, in: Sloot, P.M.A., Hoekstra, A.G., Tan, C.J.K., Dongarra, J.J. (Eds.), Computational Science — ICCS 2002, Lecture Notes in Computer Science. Springer, Berlin, Heidelberg, pp. 632–641. https://doi.org/10.1007/3-540-47789-6_66