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Nonlinear partial differential equations (PDEs) are used to model dynamical processes in a large number of scientific fields, ranging from finance to biology. In many applications standard local models are not sufficient to accurately…
Motivated by the mathematics literature on the algebraic properties of so-called polynomial vector flows, we propose a technique for approximating nonlinear differential equations by linear differential equations. Although the idea of…
A local weighted discontinuous Galerkin gradient discretization method for solving elliptic equations is introduced. The local scheme is based on a coarse grid and successively improves the solution solving a sequence of local elliptic…
In this paper we study the randomized non-autonomous complete linear differential equation. The diffusion coefficient and the source term in the differential equation are assumed to be stochastic processes and the initial condition is…
Critical points of energy functionals, which are of broad interest, for instance, in physics and chemistry, in solid and quantum mechanics, in material science, or in general diffusion-reaction models arise as solutions to the associated…
This paper is concerned with finite element approximations of $W^{2,p}$ strong solutions of second-order linear elliptic partial differential equations (PDEs) in non-divergence form with continuous coefficients. A nonstandard (primal)…
Nonlocally related partial differential equation (PDE) systems are useful in the analysis of a given PDE system. It is known that each local conservation law of a given PDE system systematically yields a nonlocally related system. In this…
This paper is concerned with the approximations of random dispersal operators/equations by nonlocal dispersal operators/equations. It first proves that the solutions of properly rescaled nonlocal dispersal initial-boundary value problems…
As a counterpoint to classical stochastic particle methods for linear diffusion equations, we develop a deterministic particle method for the weighted porous medium equation (WPME) and prove its convergence on bounded time intervals. This…
We develop a new method to solve the Fokker-Planck or Kolmogorov's forward equation that governs the time evolution of the joint probability density function of a continuous-time stochastic nonlinear system. Numerical solution of this…
We introduce an accurate and efficient method for a class of nonlocal potential evaluations with free boundary condition, including the 3D/2D Coulomb, 2D Poisson and 3D dipolar potentials. Our method is based on a Gaussian-sum approximation…
We investigate a class of systems of partial differential equations with nonlinear cross-diffusion and nonlocal interactions, which are of interest in several contexts in social sciences, finance, biology, and real world applications.…
Numerically solving high-dimensional random parametric PDEs poses a challenging computational problem. It is well-known that numerical methods can greatly benefit from adaptive refinement algorithms, in particular when functional…
This article introduces a simple weak Galerkin (WG) finite element method for solving convection-diffusion-reaction equation. The proposed method offers significant flexibility by supporting discontinuous approximating functions on general…
This paper considers efficient spectral solutions for weakly singular nonlocal diffusion equations with Dirichlet-type volume constraints. The equation we consider contains an integral operator that typically has a singularity at the…
We present a continuous and a discontinuous linear Finite Element method based on a predictor-corrector scheme for the numerical approximation of the Ericksen-Leslie equations, a model for nematic liquid crystal flow including a non-convex…
We deal with the problem of approximating a scalar conservation law by a conservation law with nonlocal flux. As convolution kernel in the nonlocal flux, we consider an exponential-type approximation of the Dirac distribution. This enables…
We obtain the comparison principle for discontinuous viscosity sub- and supersolutions of nonlocal Hamilton-Jacobi equations, with superlinear and coercive gradient terms. The nonlocal terms are integro-differential operators in L\'evy…
We propose a predictor-corrector adaptive method for the simulation of hyperbolic partial differential equations (PDEs) on networks under general uncertainty in parameters, initial conditions, or boundary conditions. The approach is based…
We propose a novel algorithm based on inexact GMRES methods for linear response calculations in density functional theory. Such calculations require iteratively solving a nested linear problem $\mathcal{E} \delta\rho = b$ to obtain the…