Related papers: Hybrid Finite Difference Schemes for Elliptic Inte…
In this paper I uncover and explain---using contour integrals and residues---a connection between cubic splines and a popular compact finite difference formula. The connection is that on a uniform mesh the simplest Pad\'e scheme for…
In the past decades, the finite difference methods for space fractional operators develop rapidly; to the best of our knowledge, all the existing finite difference schemes, including the first and high order ones, just work on uniform…
This work focuses on a class of elliptic boundary value problems with diffusive, advective and reactive terms, motivated by the study of three-dimensional heterogeneous physical systems composed of two or more media separated by a selective…
This paper proposes a deep unfitted Nitsche method for computing elliptic interface problems with high contrasts in high dimensions. To capture discontinuities of the solution caused by interfaces, we reformulate the problem as an energy…
Harmonic average method has been widely utilized to deal with heterogeneous coefficients in solving differential equations. One remarkable advantage of the harmonic averaging method is that no derivative of the coefficient is needed.…
In this work, a new compact sixth order accurate finite difference scheme for the two and three-dimensional Helmholtz equation is presented. The main significance of the proposed scheme is that its sixth order leading truncation error term…
Let $\Gamma$ be a smooth curve inside a two-dimensional rectangular region $\Omega$. In this paper, we consider the Poisson interface problem $-\nabla^2 u=f$ in $\Omega\setminus \Gamma$ with Dirichlet boundary condition such that $f$ is…
In this paper, we propose a mesh-free method to solve interface problems using the deep learning approach. Two interface problems are considered. The first one is an elliptic PDE with a discontinuous and high-contrast coefficient. While the…
Elastic materials are ubiquitous in nature and indispensable components in man-made devices and equipments. When a device or equipment involves composite or multiple elastic materials, elasticity interface problems come into play. The…
In this paper, we propose a novel high order unfitted finite element method on Cartesian meshes for solving the acoustic wave equation with discontinuous coefficients having complex interface geometry. The unfitted finite element method…
This work introduces a novel, fully robust and highly-scalable, $h$-adaptive aggregated unfitted finite element method for large-scale interface elliptic problems. The new method is based on a recent distributed-memory implementation of the…
We introduce a novel class of finite difference approximations, termed zigzag schemes, that employ a hybrid stencil that is neither symmetrical, nor fully one-sided. These zigzag schemes often enjoy more permissive stability constraints and…
This paper presents a high-order method for solving an interface problem for the Poisson equation on embedded meshes through a coupled finite element and integral equation approach. The method is capable of handling homogeneous or…
An expandable local and parallel two-grid finite element scheme based on superposition principle for elliptic problems is proposed and analyzed in this paper by taking example of Poisson equation. Compared with the usual local and parallel…
We propose and analyze a two-level method for mimetic finite difference approximations of second order elliptic boundary value problems. We prove that the two-level algorithm is uniformly convergent, i.e., the number of iterations needed to…
In this article we consider two-grid finite element methods for solving semilinear interface problems in d space dimensions, for d=2 or d=3. We first describe in some detail the target problem class with discontinuous diffusion…
We present an arbitrary order discontinuous Galerkin finite element method for solving the biharmonic interface problem on the unfitted mesh. The approximation space is constructed by a patch reconstruction process with at most one degree…
The solution $u$ of an elliptic interface problem in a domain $\Omega$ is often smooth away from the interface $\Gamma\subset \Omega$, but its gradient is discontinuous across $\Gamma$, resulting in low regularity; in particular, $u \notin…
The paper proposes a novel hybrid method for solving equilibrium problems and fixed point problems. By constructing specially cutting-halfspaces, in this algorithm, only an optimization program is solved at each iteration without the…
We propose a $k^{\rm th}$-order unfitted finite element method ($2\le k\le 4$) to solve the moving interface problem of the Oseen equations. Thorough error estimates for the discrete solutions are presented by considering errors from…