Asymptotically compatible reproducing kernel collocation and meshfree integration for nonlocal diffusion

07/28/2019
by   Yu Leng, et al.
0

Reproducing kernel (RK) approximations are meshfree methods that construct shape functions from sets of scattered data. We present an asymptotically compatible (AC) RK collocation method for nonlocal diffusion models with Dirichlet boundary condition. The scheme is shown to be convergent to both nonlocal diffusion and its corresponding local limit as nonlocal interaction vanishes. The analysis is carried out on a special family of rectilinear Cartesian grids for linear RK method with designed kernel support. The key idea for the stability of the RK collocation scheme is to compare the collocation scheme with the standard Galerkin scheme which is stable. In addition, there is a large computational cost for assembling the stiffness matrix of the nonlocal problem because high order Gaussian quadrature is usually needed to evaluate the integral. We thus provide a remedy to the problem by introducing a quasi-discrete nonlocal diffusion operator for which no numerical quadrature is further needed after applying the RK collocation scheme. The quasi-discrete nonlocal diffusion operator combined with RK collocation is shown to be convergent to the correct local diffusion problem by taking the limits of nonlocal interaction and spatial resolution simultaneously. The theoretical results are then validated with numerical experiments. We additionally illustrate a connection between the proposed technique and an existing optimization based approach based on generalized moving least squares (GMLS).

READ FULL TEXT

page 1

page 2

page 3

page 4

research
01/02/2020

Asymptotically compatible reproducing kernel collocation and meshfree integration for the peridynamic Navier equation

In this work, we study the reproducing kernel (RK) collocation method fo...
research
11/03/2022

A Scharfetter-Gummerl stabilization scheme for HDG approximations of convection-diffusion problems

We present a Scharfetter-Gummel (SG) stabilization scheme for high-order...
research
12/18/2019

Superconvergence of local discontinuous Galerkin methods with generalized alternating fluxes for 1D linear convection-diffusion equations

This paper investigates superconvergence properties of the local discont...
research
10/25/2022

Non-Oscillatory Limited-Time Integration for Conservation Laws and Convection-Diffusion Equations

In this study we consider unconditionally non-oscillatory, high order im...
research
11/27/2022

OBMeshfree: An optimization-based meshfree solver for nonlocal diffusion and peridynamics models

We present OBMeshfree, an Optimization-Based Meshfree solver for compact...
research
10/18/2022

Analyses of the contour integral method for time fractional subdiffusion-normal transport equation

In this work, we theoretically and numerically discuss the time fraction...
research
08/02/2019

A Kernel Based High Order "Explicit" Unconditionally Stable Constrained Transport Method for Ideal Magnetohydrodynamics

The ideal Magnetohydrodynamics (MHD) equations are challenging because o...

Please sign up or login with your details

Forgot password? Click here to reset