We study two numerical approximations of solutions of nonlocal diffusion evolution problems which are inspired in algorithms for computing the bilateral denoising filtering of an image, and which are based on functional rearrangements and on the Fourier transform. Apart from the usual time-space discretization, these algorithms also use the discretization of the range of the solution (quantization). We show that the discrete approximations converge to the continuous solution in suitable functional spaces, and provide error estimates. Finally, we present some numerical experiments illustrating the performance of the algorithms, specially focusing in the execution time.
Time discretizations of phase-field systems have been studied. For example, a time discretization and an error estimate for a parabolic-parabolic phase-field system have been studied by Colli--K. [Commun. Pure Appl. Anal. 18 (2019)]. Also, a time discretization and an error estimate for a simultaneous abstract evolution equation applying parabolic-hyperbolic phase field systems and the linearized equations of coupled sound and heat flow have been studied (see K. [ESAIM Math. Model. Numer. Anal.54 (2020), Electron. J. Differential Equations 2020, Paper No. 96]). On the other hand, although exis…
In this paper we design efficient quadrature rules for finite element discretizations of nonlocal diffusion problems with compactly supported kernel functions. Two of the main challenges in nonlocal modeling and simulations are the prohibitive computational cost and the nontrivial implementation of discretization schemes, especially in three-dimensional settings. In this work we circumvent both challenges by introducing a parametrized mollifying function that improves the regularity of the integrand, utilizing an adaptive integration technique, and exploiting parallelization. We first show tha…
We propose a domain decomposition method for the efficient simulation of nonlocal problems. Our approach is based on a multi-domain formulation of a nonlocal diffusion problem where the subdomains share "nonlocal" interfaces of the size of the nonlocal horizon. This system of nonlocal equations is first rewritten in terms of minimization of a nonlocal energy, then discretized with a meshfree approximation and finally solved via a Lagrange multiplier approach in a way that resembles the finite element tearing and interconnect method. Specifically, we propose a distributed projected gradient alg…
A nonlocal coupled system: analysis and discretization
Universidad de Buenos Aires（布宜诺斯艾利斯大学）；IMAS CONICET（阿根廷国家科学研究委员会数学应用与模拟研究所）；Universidad Técnica Federico Santa María（费德里科·圣玛丽亚理工大学）；Universidad de Los Lagos（洛斯拉戈斯大学）
In this paper, we propose and analyze a finite element discretization for the computation of fractional minimal graphs of order~$s \in (0,1/2)$ on a bounded domain $Ω$. Such a Plateau problem of order $s$ can be reinterpreted as a Dirichlet problem for a nonlocal, nonlinear, degenerate operator of order $s + 1/2$. We prove that our numerical scheme converges in $W^{2r}_1(Ω)$ for all $r<s$, where $W^{2s}_1(Ω)$ is closely related to the natural energy space. Moreover, we introduce a geometric notion of error that, for any pair of $H^1$ functions, in the limit $s \to 1/2$ recovers a weighted $L^2…
In this paper, we propose a hybrid collocation method based on finite difference and Haar wavelets to solve nonlocal hyperbolic partial differential equations. Developing an efficient and accurate numerical method to solve such problem is a difficult task due to the presence of nonlocal boundary condition. The speciality of the proposed method is to handle integral boundary condition efficiently using the given data. Due to various attractive properties of Haar wavelets such as closed form expression, compact support and orthonormality, Haar wavelets are efficiently used for spatial discretiza…
This paper considers numerical discretization of a nonlocal conservation law modeling vehicular traffic flows involving nonlocal inter-vehicle interactions. The nonlocal model involves an integral over the range measured by a horizon parameter and it recovers the local Lighthill-Richards-Whitham model as the nonlocal horizon parameter goes to zero. Good numerical schemes for simulating these parameterized nonlocal traffic flow models should be robust with respect to the change of the model parameters but this has not been systematically investigated in the literature. We fill this gap through …
This work presents a probabilistic scheme for solving semilinear nonlocal diffusion equations with volume constraints and integrable kernels. The nonlocal model of interest is defined by a time-dependent semilinear partial integro-differential equation (PIDE), in which the integro-differential operator consists of both local convection-diffusion and nonlocal diffusion operators. Our numerical scheme is based on the direct approximation of the nonlinear Feynman-Kac formula that establishes a link between nonlinear PIDEs and stochastic differential equations. The exploitation of the Feynman-Kac …
In this paper, we consider a class of convection-diffusion equations with memory effects. These equations arise as a result of homogenization or upscaling of linear transport equations in heterogeneous media and play an important role in many applications. First, we present a dememorization technique for these equations. We show that the convection-diffusion equations with memory effects can be written as a system of standard convection-diffusion-reaction equations. This allows removing the memory term and simplifying the computations. We consider a relation between dememorized equations and m…
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 midpoint of the integral domain, and the approximation of the integral operator is one of the essential difficulties in solving nonlocal equations. To overcome this problem, two-sided Jacobi spectral quadrature rules are proposed to develop a Jacobi spectral collocation method for nonlocal diffusion equations. A rigorous convergence analysis of the proposed method…
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 collocat…
We propose a second order, fully semi-Lagrangian method for the numerical solution of systems of advection-diffusion-reaction equations, which employs a semi-Lagrangian approach to approximate in time both the advective and the diffusive terms. Standard interpolation procedures are used for the space discretization on structured and unstructured meshes. The proposed method allows for large time steps, while avoiding the solution of large linear systems, which would be required by an implicit time discretization technique. Numerical experiments demonstrate the effectiveness of the proposed appr…
We present recent finite element numerical results on a model convection-diffusion problem in the singular perturbed case when the convection term dominates the problem. We compare the standard Galerkin discretization using the linear element with a saddle point least square discretization that uses quadratic test functions, trying to control and explain the non-physical oscillations of the discrete solutions. We also relate the up-winding Petrov-Galerkin method and the stream-line diffusion discretization method, by emphasizing the resulting linear systems and by comparing appropriate error n…
In this work, we propose a new quasi-optimal test norm for a discontinuous Petrov-Galerkin (DPG) discretization of the ultra-weak formulation of the convection-diffusion equation. We prove theoretically that the proposed test norm leads to bounds between the target norm and the energy norm induced by the test norm which are robust with respect to the diffusion parameter in the solution and gradient components and have favorable scalings in the trace components. We conclude with numerical experiments to confirm our theoretical results.
We consider the reaction diffusion problem and present efficient ways to discretize and precondition in the singular perturbed case when the reaction term dominates the equation. Using the concepts of optimal test norm and saddle point reformulation, we provide efficient discretization processes for uniform and non-uniform meshes. We present a preconditioning strategy that works for a large range of the perturbation parameter. Numerical examples to illustrate the efficiency of the method are included for a problem on the unit square.
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 on the test space as induced by the trial space norm, i.e., the optimal test norm, so that the inf-sup condition can be satisfied uniformly independent of the problem. We show the well-posedness of a class of nonlocal convection-dominated diffusion problems under the optimal test norm with general assumptions on the nonlocal diffusion and convection kernels. Second, following the framework of Cohen et al.~(2012), we …
The balancing domain decomposition methods (BDDC) are originally introduced for symmetric positive definite systems and have been extended to the nonsymmetric positive definite system from the linear finite element discretization of advection-diffusion equations. In this paper, the convergence of the GMRES method is analyzed for the BDDC preconditioned linear system from advection-diffusion equations with the hybridizable discontinuous Galerkin (HDG) discretization. Compared to the finite element discretizations, several additional norms for the numerical trace have to be used and the equivale…
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, convergence, and stability of such schemes. Employing an FE method also facilitates meshing of complicated domain geometries and coupling with FE methods for local problems. However, nonlocal weak problems involve the computation of a double-integral, which is computationally expensive and presents several challenges. In particular, the inner integral of the variational form associated with the stiffness matrix is defi…
Anisotropic diffusion is a well recognized tool in digital image processing, including edge detection and denoising. We present here a particular nonlinear time-dependent operator together with an appropriate high-order discretization for the space variable. The iterative procedure emphasizes the contour lines encircling the objects, paving the way to accurate reconstructions at a very low cost. One of the main features of such an approach is the possibility of relying on a rather large set of invariant discontinuous images, whose edges can be determined without introducing any approximation.
In this paper, we study the Crank-Nicolson method for temporal dimension and the piecewise quadratic polynomial collocation method for spatial dimensions of time-dependent nonlocal problems. The new theoretical results of such discretization are that the proposed numerical method is unconditionally stable and its global truncation error is of $\mathcal{O}\left(τ^2+h^{4-γ}\right)$ with $0<γ<1$, where $τ$ and $h$ are the discretization sizes in the temporal and spatial dimensions respectively. Also we develop the conjugate gradient squared method to solving the resulting discretized nonsymmetric…
In a recent paper (see [7]), a quasi-nonlocal coupling method was introduced to seamlessly bridge a nonlocal diffusion model with the classical local diffusion counterpart in a one-dimensional space. The proposed coupling framework removes inconsistency on the interface, preserves the balance of fluxes, and satisfies the maximum principle of diffusion problem. However, the numerical scheme proposed in that paper does not maintain all of these properties on a discrete level. In this paper we resolve these issues by proposing a new finite difference scheme that ensures the balance of fluxes and …
An implicit Euler finite-volume scheme for a nonlocal cross-diffusion system on the one-dimensional torus, arising in population dynamics, is proposed and analyzed. The kernels are assumed to be in detailed balance and satisfy a weak cross-diffusion condition. The latter condition allows for negative off-diagonal coefficients and for kernels defined by an indicator function. The scheme preserves the nonnegativity of the densities, conservation of mass, and production of the Boltzmann and Rao entropies. The key idea is to ``translate'' the entropy calculations for the continuous equations to th…
We study a model of crowd motion following a gradient vector field, with possibly additional interaction terms such as attraction/repulsion, and we present a numerical scheme for its solution through a Lagrangian discretization. The density constraint of the resulting particles is enforced by means of a partial optimal transport problem at each time step. We prove the convergence of the discrete measures to a solution of the continuous PDE describing the crowd motion in dimension one. In a second part, we show how a similar approach can be used to construct a Lagrangian discretization of a lin…
In this article, we propose a linearized fully-discrete scheme for solving a time fractional nonlocal diffusion-wave equation of Kirchhoff type. The scheme is established by using the finite element method in space and the $L1$ scheme in time. We derive the $α$-robust \textit{a priori} bound and \textit{a priori} error estimate for the fully-discrete solution in $L^{\infty}\big(H^1_0(Ω)\big)$ norm, where $α\in (1,2)$ is the order of time fractional derivative. Finally, we perform some numerical experiments to verify the theoretical results.
We analyze the well-posedness of an anisotropic, nonlocal diffusion equation. Establishing an equivalence between weighted and unweighted anisotropic nonlocal diffusion operators in the vein of unified nonlocal vector calculus, we apply our analysis to a class of fractional-order operators and present rigorous estimates for the solution of the corresponding anisotropic anomalous diffusion equation. Furthermore, we extend our analysis to the anisotropic diffusion-advection equation and prove well-posedness for fractional orders s in [0.5,1). We also present an application of the advection-diffu…
In this work we present the mathematical foundation of an assembly code for finite element approximations of nonlocal models with compactly supported, weakly singular kernels. We demonstrate the code on a nonlocal diffusion model in various configurations and on a two-dimensional bond-based peridynamics model. The code nlfem is published under the MIT License and can be freely downloaded.
In this paper we consider 2D nonlocal diffusion models with a finite nonlocal horizon parameter $δ$ characterizing the range of nonlocal interactions, and consider the treatment of Neumann-like boundary conditions that have proven challenging for discretizations of nonlocal models. While existing 2D nonlocal flux boundary conditions have been shown to exhibit at most first order convergence to the local counter part as $δ\rightarrow 0$, we present a new generalization of classical local Neumann conditions that recovers the local case as $O(δ^2)$ in the $L^{\infty}(Ω)$ norm. This convergence ra…
For some spatially nonlocal diffusion models with a finite range of nonlocal interactions measured by a positive parameter $δ$, we review their formulation defined on a bounded domain subject to various conditions that correspond to some inhomogeneous data. We consider their consistency to similar inhomogeneous boundary value problems of classical partial differential equation (PDE) models as the nonlocal interaction kernel gets localized in the local $δ\to 0$ limit, and at the same time, for rescaled fractional type kernels, to corresponding inhomogeneous nonlocal boundary value problems of f…
The solution of time fractional partial differential equations in general exhibit a weak singularity near the initial time. In this article we propose a method for solving time fractional diffusion equation with nonlocal diffusion term. The proposed method comprises L1 scheme on graded mesh, finite element method and Newton's method. We discuss the well-posedness of the weak formulation at discrete level and derive \emph{a priori} error estimates for fully-discrete formulation in $L^2(Ω)$ and $H^1(Ω)$ norms. Finally, some numerical experiments are conducted to validate the theoretical findings…
Stochastic evolution equations with compensated Poisson noise are considered in the variational approach with monotone and coercive coefficients. Here the Poisson noise is assumed to be time-homogeneous with $σ$-finite intensity measure on a metric space. By using finite element methods and Galerkin approximations, some explicit and implicit discretizations for this equation are presented and their convergence is proved. Polynomial growth condition and linear growth condition are assumed on the drift operator, respectively for the implicit and explicit schemes.
We present OBMeshfree, an Optimization-Based Meshfree solver for compactly supported nonlocal integro-differential equations (IDEs) that can describe material heterogeneity and brittle fractures. OBMeshfree is developed based on a quadrature rule calculated via an equality constrained least square problem to reproduce exact integrals for polynomials. As such, a meshfree discretization method is obtained, whose solution possesses the asymptotically compatible convergence to the corresponding local solution. Moreover, when fracture occurs, this meshfree formulation automatically provides a sharp…
