Related papers: Improved Stencil Selection for Meshless Finite Dif…
We introduce generalised finite difference methods for solving fully nonlinear elliptic partial differential equations. Methods are based on piecewise Cartesian meshes augmented by additional points along the boundary. This allows for…
Meshless methods are an active and modern branch of numerical analysis with many intriguing benefits. One of the main open research questions related to local meshless methods is how to select the best possible stencil - a collection of…
We present a method for generating higher-order finite volume discretizations for Poisson's equation on Cartesian cut cell grids in two and three dimensions. The discretization is in flux-divergence form, and stencils for the flux are…
We introduce a numerical method for approximating arbitrary differential operators on vector fields in the weak form given point cloud data sampled randomly from a $d$ dimensional manifold embedded in $\mathbb{R}^n$. This method generalizes…
In this work, we propose efficient and accurate numerical algorithms based on Difference Potentials Method for numerical solution of chemotaxis systems and related models in 3D. The developed algorithms handle 3D irregular geometry with the…
In this paper, we address a way to reduce the total computational cost of meshless approximation by reducing the required stencil size through spatially varying computational node regularity. Rather than covering the entire domain with…
We consider a sketched implementation of the finite element method for elliptic partial differential equations on high-dimensional models. Motivated by applications in real-time simulation and prediction we propose an algorithm that…
A grid-overlay finite difference method is proposed for the numerical approximation of the fractional Laplacian on arbitrary bounded domains. The method uses an unstructured simplicial mesh and an overlay uniform grid for the underlying…
We design a monotone meshfree finite difference method for linear elliptic equations in the non-divergence form on point clouds via a nonlocal relaxation method. The key idea is a novel combination of a nonlocal integral relaxation of the…
The study of generalising the central difference for integer order Laplacian to fractional order is discussed in this paper. Analysis shows that, in contrary to the conclusion of a previous study, difference stencils evaluated through fast…
In this paper a fourth order finite difference ghost point method for the Poisson equation on regular Cartesian mesh is presented. The method can be considered the high order extension of the second ghost method introduced earlier by the…
Meshfree finite difference methods for the Poisson equation approximate the Laplace operator on a point cloud. Desirable are positive stencils, i.e. all neighbor entries are of the same sign. Classical least squares approaches yield large…
Mesh-free numerical methods provide flexible discretisations for complex geometries; however, classical meshless discrete differential operators typically trade low computational cost for limited accuracy or high accuracy for substantial…
A numerical scheme is presented for approximating fractional order Poisson problems in two and three dimensions. The scheme is based on reformulating the original problem posed over $\Omega$ on the extruded domain…
An extension of the restricted Delaunay-refinement algorithm for surface mesh generation is described, where a new point-placement scheme is introduced to improve element quality in the presence of mesh size constraints. Specifically, it is…
Shape optimization involves the minimization of a cost function defined over a set of shapes, often governed by a partial differential equation (PDE). In the absence of closed-form solutions, one relies on numerical methods to approximate…
We study finite element approximations of the nonhomogeneous Dirichlet problem for the fractional Laplacian. Our approach is based on weak imposition of the Dirichlet condition and incorporating a nonlocal analogous of the normal derivative…
In this paper, we propose a novel mesh-free numerical method for solving the elliptic interface problems based on deep learning. We approximate the solution by the neural networks and, since the solution may change dramatically across the…
A so-called grid-overlay finite difference method (GoFD) was proposed recently for the numerical solution of homogeneous Dirichlet boundary value problems of the fractional Laplacian on arbitrary bounded domains. It was shown to have…
A new approach to building explicit time-marching stencil computation schemes for the transient 2D acoustic wave equation is implemented. It is based on using Poisson's formula and its three time level modification combined with polynomial…