High Order Finite Difference Methods for Interface Problems with Singularities
Creators: Feng, Qiwei
Publisher: name: University of Alberta Library
Year: 2022

<p>Interface problems arise in many applications such as modeling of underground waste disposal, oil reservoirs, composite materials, and many others. The coefficient $a$, the source term $f$, the solution $u$ and the flux $a\nabla u\cdot \vec{n}$ are possibly discontinuous across the interface curve $\Gamma$ in such problems. In realistic problems, the coefficient $a$ may have large jumps across the interface curve, or it can be highly oscillatory across the whole domain.  This leads to accuracy deterioration and huge condition numbers of resulting linear systems. In order to obtain reasonable numerical solutions, higher order numerical schemes are desirable.</p>

<p>In Chapter 2 we propose a sixth order compact 9-point finite difference method (FDM) on uniform Cartesian grids, for Poisson interface problems with singular sources in a rectangular domain. The matrix $A$ in the resulting linear system $Ax=b$, following from the proposed compact 9-point scheme, is independent of any source terms $f$, jump conditions, and interface curves $\Gamma$. We prove the sixth order convergence rate for the proposed compact 9-point scheme using the discrete maximum principle. Our numerical experiments confirm the sixth order of accuracy of the proposed compact 9-point scheme. This chapter has been published in \emph{Computers and Mathematics with Applications} in 2021.</p>

<p>In Chapter 3, elliptic interface problems with discontinuous and high-contrast piecewise smooth coefficients in a rectangle are considered. We propose a high order compact 9-point FDM and a high order local calculation for approximation of the solution $u$ and the gradient $\nabla u$ respectively. The scheme is developed on uniform Cartesian grids, avoiding the transformation into local coordinates. We also numerically verify the sign conditions of our proposed compact 9-point scheme and prove the fourth order convergence rate by the discrete maximum principle. Our numerical experiments confirm the fourth order accuracy for the numerically approximated solution $u$ in both $l<em>2$ and $l</em>{\infty}$ norms, and the fourth/third order accuracy for the numerically approximated gradient $\big((u<em>h)</em>x,(u<em>h)</em>y\big)$ in the $l<em>2$/$l</em>{\infty}$ norm. This chapter has been published in \emph{Applied Mathematics and Computation} in 2022.</p>

<p>In Chapter 4, we propose an efficient and flexible way to achieve the implementation of a hybrid FDM in uniform Cartesian meshes for elliptic interface problems with discontinuous and high-contrast piecewise smooth coefficients in a rectangular domain.  The scheme utilizes a 9-point compact stencil with a sixth order accuracy for interior regular points and 13-point stencil with a fifth order accuracy for interior irregular points. Near the boundary, the stencil is reduced to six points and near the domain corners - to four points, and the corresponding discretization has a sixth order of accuracy on uniform Cartesian meshes, for various boundary conditions (Dirichlet, Neumann and Robin). Our numerical experiments confirm the flexibility and the accuracy order in $l<em>2$ and $l</em>{\infty}$ norms.</p>

<p>In Chapter 5, we present a sixth order compact FDM on uniform Cartesian meshes for the Helmholtz equation with singular sources, and any possible combination of boundary conditions (Dirichlet, Neumann, and impedance) in a rectangular domain. To reduce the pollution effect, we propose a new pollution minimization strategy that is based on the average truncation error of plane waves. Our numerical experiments demonstrate the superiority of the proposed compact finite difference scheme with reduced pollution effect, as compared to several state-of-the-art finite difference schemes in the literature, particularly in the critical pre-asymptotic region where $\textsf{k}h$ is near $1$ with $\textsf{k}$ being the wavenumber and $h$ the mesh size. This chapter has been submitted in \emph{SIAM Journal on Scientific Computing}.</p>

<p>In Chapter 6, we propose a sixth order compact 9-point FDM on uniform Cartesian meshes for elliptic interface problems with particular intersecting interfaces and four discontinuous constant coefficients in a square domain, where the solution is smooth enough, and interface curves are horizontal and vertical straight lines. The formulas of proposed sixth order compact 9-point finite difference scheme are constructed explicitly for all grid points (regular points, interface points, and the intersection point). We prove the order $6$ convergence of our proposed compact 9-point scheme by the discrete maximum principle. Our numerical experiments confirm the flexibility and the sixth order accuracy in $l<em>2$ and $l</em>{\infty}$ norms of our proposed compact 9-point scheme.</p>