209 citations · 348 across the 22 of their papers we have counts for
13 papers · 1 filter
Numerical analysis of a class of penalty discontinuous Galerkin methods for nonlocal diffusion problems
Qiang Du, Lili Ju, Jianfang Lu +1
In this paper, we consider a class of discontinuous Galerkin (DG) methods for one-dimensional nonlocal diffusion (ND) problems. The nonlocal models, which are integral equations, a…
Asymptotically compatible schemes for nonlinear variational models via Gamma-convergence and applications to nonlocal problems
Qiang Du, James M. Scott, Xiaochuan Tian
We present a study on asymptotically compatible Galerkin discretizations for a class of parametrized nonlinear variational problems. The abstract analytical framework is based on v…
Monotone meshfree methods for linear elliptic equations in non-divergence form via nonlocal relaxation
Qihao Ye, Xiaochuan Tian
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…
Efficient optimization-based quadrature for variational discretization of nonlocal problems
Marco Pasetto, Zhaoxiang Shen, Marta D'Elia +3
Casting nonlocal problems in variational form and discretizing them with the finite element (FE) method facilitates the use of nonlocal vector calculus to prove well-posedeness, co…
A Petrov-Galerkin method for nonlocal convection-dominated diffusion problems
Yu Leng, Xiaochuan Tian, Leszek Demkowicz +2
We present a Petrov-Gelerkin (PG) method for a class of nonlocal convection-dominated diffusion problems. There are two main ingredients in our approach. First, we define the norm…
An asymptotically compatible probabilistic collocation method for randomly heterogeneous nonlocal problems
Yiming Fan, Xiaochuan Tian, Xiu Yang +3
In this paper we present an asymptotically compatible meshfree method for solving nonlocal equations with random coefficients, describing diffusion in heterogeneous media. In parti…
Numerical methods for nonlocal and fractional models
Marta D'Elia, Qiang Du, Christian Glusa +3
Partial differential equations (PDEs) are used, with huge success, to model phenomena arising across all scientific and engineering disciplines. However, across an equally wide swa…
Asymptotically compatible reproducing kernel collocation and meshfree integration for the peridynamic Navier equation
Yu Leng, Xiaochuan Tian, Nathaniel A. Trask +1
In this work, we study the reproducing kernel (RK) collocation method for the peridynamic Navier equation. We first apply a linear RK approximation on both displacements and dilata…
Multiscale Modeling, Homogenization and Nonlocal Effects: Mathematical and Computational Issues
Qiang Du, Bjorn Engquist, Xiaochuan Tian
In this work, we review the connection between the subjects of homogenization and nonlocal modeling and discuss the relevant computational issues. By further exploring this connect…
Asymptotically compatible reproducing kernel collocation and meshfree integration for nonlocal diffusion
Yu Leng, Xiaochuan Tian, Nathaniel Trask +1
Reproducing kernel (RK) approximations are meshfree methods that construct shape functions from sets of scattered data. We present an asymptotically compatible (AC) RK collocation…
Fast algorithm for computing nonlocal operators with finite interaction distance
Xiaochuan Tian, Bjorn Engquist
Developments of nonlocal operators for modeling processes that traditionally have been described by local differential operators have been increasingly active during the last few y…
Mathematics of Smoothed Particle Hydrodynamics: a Study via Nonlocal Stokes Equations
Qiang Du, Xiaochuan Tian
Smoothed Particle Hydrodynamics (SPH) is a popular numerical technique developed for simulating complex fluid flows. Among its key ingredients is the use of nonlocal integral relax…
A quasinonlocal coupling method for nonlocal and local diffusion models
Qiang Du, Xingjie Helen Li, Jianfeng Lu +1
In this paper, we extend the idea of "geometric reconstruction" to couple a nonlocal diffusion model directly with the classical local diffusion in one dimensional space. This new…