Related papers: Mathematical and Numerical Studies on Meshless Met…
The method of fundamental solutions (MFS) is a numerical method for solving boundary value problems involving linear partial differential equations. It is well known that it can be very effective assuming regularity of the domain and…
The method of the fundamental solutions (MFS) is used to construct an approximate solution for a partial differential equation in a bounded domain. It is demonstrated by combining the fundamental solutions shifted to the points outside the…
The Method of Fundamental Solutions (MFS) is a popular tool to solve Laplace and Helmholtz boundary value problems. Its main drawback is that it often leads to ill-conditioned systems of equations. In this paper we investigate for the…
An efficient numerical method is proposed for computing the Dirichlet-to-Neumann (DtN) map associated with the exterior Dirichlet problem for the two-dimensional Helmholtz equation with an inhomogeneous term. The exterior solution is…
The method of fundamental solutions (MFS) is known to be effective for solving 3D Laplace and Stokes Dirichlet boundary value problems in the exterior of a large collection of simple smooth objects. Here we present new scalable MFS…
The method of fundamental solutions (MFS), also known as the method of auxiliary sources (MAS), is a well-known computational method for the solution of boundary-value problems. The final solution ("MAS solution") is obtained once we have…
This paper further develops the Method of Matched Sections (MMS), a robust numerical framework for the solution of boundary value problems governed by partial differential equations. It demonstrates its unique applicability to the…
Meshless solution to differential equations using radial basis functions (RBF) is an alternative to grid based methods commonly used. Since the meshless method does not need an underlying connectivity in the form of control volumes or…
In this paper, using the approximate particular solutions of Helmholtz equations, we solve the boundary value problems of Helmholtz equations by combining the methods of fundamental solutions (MFS) with the methods of particular solutions…
Meshfree simulation methods are emerging as compelling alternatives to conventional mesh-based approaches, particularly in the fields of Computational Fluid Dynamics (CFD) and continuum mechanics. In this publication, we provide a…
Nonnegative matrix factorization (NMF) seeks a low-rank approximation $X \approx UV^T$ with nonnegative factors and is commonly solved using interior methods that enforce feasibility throughout optimization. We show that such…
Numerical solution of equations governing time domain simulations in computational electromagnetics, is usually based on grid methods in space and on explicit schemes for the time evolution. A predefined grid in the problem domain and a…
A major obstacle to the application of the standard Radial Basis Function-generated Finite Difference (RBF-FD) meshless method is constituted by its inability to accurately and consistently solve boundary value problems involving Neumann…
There are many application papers that solve elliptic boundary value problems by meshless methods, and they use various forms of generalized stiffness matrices that approximate derivatives of functions from values at scattered nodes…
We suggest a method for simultaneously generating high order quadrature weights for integrals over Lipschitz domains and their boundaries that requires neither meshing nor moment computation. The weights are determined on pre-defined…
This paper introduces a novel meshfree methodology based on Radial Basis Function-Finite Difference (RBF-FD) approximations for the numerical solution of partial differential equations (PDEs) on surfaces of codimension 1 embedded in…
We present a new finite element method, called $\phi$-FEM, to solve numerically elliptic partial differential equations with natural (Neumann or Robin) boundary conditions using simple computational grids, not fitted to the boundary of the…
The paper describes a numerical method for solving acoustic multibody scattering problems in two and three dimensions. The idea is to compute a highly accurate approximation to the scattering operator for each body through a local…
This paper considers flow problems in multiscale heterogeneous porous media. The multiscale nature of the modeled process significantly complicates numerical simulations due to the need to compute huge and ill-conditioned sparse matrices,…
This paper presents a matrix-free approach for implementing the shifted boundary method (SBM) in finite element analysis. The SBM is a versatile technique for solving partial differential equations on complex geometries by shifting boundary…