Related papers: Improving the accuracy of meshless methods via res…
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…
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,…
In this paper, we address a way to reduce the total computational cost of meshless approximation by reducing the required stencil size through spatial variation of computational node regularity. Rather than covering the entire domain with…
Finding accurate solutions to partial differential equations (PDEs) is a crucial task in all scientific and engineering disciplines. It has recently been shown that machine learning methods can improve the solution accuracy by correcting…
Building on the successes of local kernel methods for approximating the solutions to partial differential equations (PDE) and the evaluation of definite integrals (quadrature/cubature), a local estimate of the error in such approximations…
We are interested in mesh-free formulas based on the Monte-Carlo methodology for the approximation of multi-dimensional integrals, and we investigate their accuracy when the functions belong to a reproducing-kernel space. A kernel typically…
This paper introduces a multilevel kernel-based approximation method to estimate efficiently solutions to elliptic partial differential equations (PDEs) with periodic random coefficients. Building upon the work of Kaarnioja, Kazashi, Kuo,…
We present a detailed comparison between two adaptive numerical approaches to solve partial differential equations (PDEs), adaptive multiresolution (MR) and adaptive mesh refinement (AMR). Both discretizations are based on finite volumes in…
Eulerian smoothed particle hydrodynamics (Eulerian SPH) is considered as a potential meshless alternative to a traditional Eulerian mesh-based method, i.e. finite volume method (FVM), in computational fluid dynamics (CFD). While researchers…
Meshless methods approximate operators in a specific node as a weighted sum of values in its neighbours. Higher order approximations of derivatives provide more accurate solutions with better convergence characteristics, but they come at…
We develop a new spatial semidiscrete multiscale method based upon the edge multiscale methods to solve semilinear parabolic problems with heterogeneous coefficients and smooth initial data. This method allows for a cheap spatial…
The computational complexity of classical numerical methods for solving Partial Differential Equations (PDE) scales significantly as the resolution increases. As an important example, climate predictions require fine spatio-temporal…
Mesh generation is essential for accurate and efficient computational fluid dynamics simulations. To resolve critical features in the flow, adaptive mesh refinement (AMR) is routinely employed in certain regions of the computational domain,…
Partial differential equations (PDEs) on surfaces are fundamental to scientific computing and geometry processing. A popular approach to solving PDEs on surfaces is the finite element method (FEM), where the surface is divided into discrete…
(Partial) differential equations (PDEs) are fundamental tools for describing natural phenomena, making their solution crucial in science and engineering. While traditional methods, such as the finite element method, provide reliable…
We present an exponentially convergent semi-implicit meshless algorithm for the solution of Navier-Stokes equations in complex domains. The algorithm discretizes partial derivatives at scattered points using radial basis functions as…
We present new convergence analyses for parallel subspace correction methods for unconstrained semicoercive and nearly semicoercive convex optimization problems, generalizing the theory of singular and nearly singular linear problems to a…
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…
Physics-informed Neural Networks (PINNs) have recently emerged as a principled way to include prior physical knowledge in form of partial differential equations (PDEs) into neural networks. Although PINNs are generally viewed as mesh-free,…
Machine learning based partial differential equations (PDEs) solvers have received great attention in recent years. Most progress in this area has been driven by deep neural networks such as physics-informed neural networks (PINNs) and…