Related papers: High order solution of Poisson problems with piece…
In this paper we present a method to treat interface jump conditions for constant coefficients Poisson problems that allows the use of standard "black box" solvers, without compromising accuracy. The basic idea of the new approach is…
This paper presents an efficient high-order sharp-interface method for solving the three-dimensional (3D) Poisson equation with Dirichlet boundary conditions on a nonuniform Cartesian grid with irregular domain boundaries. The new approach…
In this paper a novel method to solve the constant coefficient wave equation, subject to interface jump conditions, is presented. In general, such problems pose issues for standard finite difference solvers, as the inherent discontinuity in…
We introduce a technique that simplifies the problem of imposing jump conditions on interfaces that are not aligned with a computational grid in the context of the Correction Function Method (CFM). The CFM offers a general framework to…
We solve Poisson's equation using new multigrid algorithms that converge rapidly. The novel feature of the 2D and 3D algorithms are the use of extra diagonal grids in the multigrid hierarchy for a much richer and effective communication…
In multi-phase fluid flow, fluid-structure interaction, and other applications, partial differential equations (PDEs) often arise with discontinuous coefficients and singular sources (e.g., Dirac delta functions). These complexities arise…
Computations of incompressible flows with velocity boundary conditions require solution of a Poisson equation for pressure with all Neumann boundary conditions. Discretization of such a Poisson equation results in a rank-deficient matrix of…
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…
This work presents a multigrid preconditioned high order immersed finite difference solver to accurately and efficiently solve the Poisson equation on complex 2D and 3D domains. The solver employs a low order Shortley-Weller multigrid…
We present a new multigrid scheme for solving the Poisson equation with Dirichlet boundary conditions on a Cartesian grid with irregular domain boundaries. This scheme was developed in the context of the Adaptive Mesh Refinement (AMR)…
We propose a high-order FDTD scheme based on the correction function method (CFM) to treat interfaces with complex geometry without increasing the complexity of the numerical approach for constant coefficients. Correction functions are…
We propose high-order FDTD schemes based on the Correction Function Method (CFM) for Maxwell's interface problems with discontinuous coefficients and complex interfaces. The key idea of the CFM is to model the correction function near an…
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…
We present an efficient numerical method, inspired by transformation optics, for solving the Poisson equation in complex and arbitrarily shaped geometries. The approach operates by mapping the physical domain to a uniform computational…
We present a simple numerical algorithm for solving elliptic equations where the diffusion coefficient, the source term, the solution and its flux are discontinuous across an irregular interface. The algorithm produces second-order accurate…
A new and efficient neural-network and finite-difference hybrid method is developed for solving Poisson equation in a regular domain with jump discontinuities on embedded irregular interfaces. Since the solution has low regularity across…
The Fast Multipole Method (FMM) provides a highly efficient computational tool for solving constant coefficient partial differential equations (e.g. the Poisson equation) on infinite domains. The solution to such an equation is given as the…
We present a direct Poisson solver for massively parallel simulations on three-dimensional Cartesian grids with non-uniform spacing. The method uses a tensor-based formulation in which the operator is diagonalized numerically along two…
We propose the Compact Coupling Interface Method (CCIM), a finite difference method capable of obtaining second-order accurate approximations of not only solution values but their gradients, for elliptic complex interface problems with…
Multigrid solvers are among the most efficient methods for solving the Poisson equation, which is ubiquitous in computational physics. For example, in the context of incompressible flows, it is typically the costliest operation. The present…