arXiv subjects

Loic Cappanera

Publications and source records attributed to Loic Cappanera.

5 recordsLinked to original sources

arXiv2026

A Mountain-Pass Algorithm for Nonlocal Problems with Super-quadratic Nonlinearities

In this paper we consider a nonlinear equation $-\mathcal{L} u(x) = f(x, u(x))$ with a super-quadratic nonlinearity, $f$, and a nonlocal operator, $\mathcal{L}$, generated by a special class of radially symmetric $L^1$ convolution kernels with finite second moments. The assumptions on this operator are mild and allow for a variety of kernels used in biological and physical applications, including kernels with algebraic decay and sign changing kernels. Using the strong nonlinearities present in the equation, we prove the existence of nontrivial solutions using the classical Mountain Pass Theorem, a central result in minimax theory that equates solutions of our equation to critical points of a corresponding energy functional. This existence result holds with both homogeneous nonlocal Dirichlet and nonlocal Neumann boundary conditions. We supplement these theoretical results with numerical simulations for various nonlinearities with odd maximal degree in the unknown $u$. The numerical scheme exploits the resulting energy landscape which allows one to adapt a gradient descent algorithm.

math.AP↗

arXiv2025

The role of boundary constraints in simulating a nonlocal Gray-Scott model

We present a second-order algorithm for approximating solutions to nonlocal diffusive processes in reaction-diffusion equations. The numerical scheme relies on a quadrature method for the spatial discretization and a second-order Adams-Bashford method for the time marching. This algorithm is then used to simulate a nonlocal Gray-Scott model, known for generating interesting structures including periodic patterns, traveling waves, pulse and multi-pulse solutions. Our main goal is to study the impact of boundary constraints on the formation of stationary pulse solutions. We consider nonlocal Dirichlet and Neumann boundary constraints, as well as what we refer to as `free' boundary conditions. In addition, we investigate the effects of using different convolution kernels, fat- or thin-tailed, on the formation of these localized solutions. Our numerical results show that when the spread of the kernel is large, i.e. when the model is nonlocal, both the type of kernel and the type of boundary constraint used have a strong impact on the solution's profile.

math.NA↗

arXiv2024

Analysis and Simulations of a Nonlocal Gray-Scott Model

The Gray-Scott model is a set of reaction-diffusion equations that describes chemical systems far from equilibrium. Interest in this model stems from its ability to generate spatio-temporal structures, including pulses, spots, stripes, and self-replicating patterns. We consider an extension of this model in which the spread of the different chemicals is assumed to be nonlocal, and can thus be represented by an integral operator. In particular, we focus on the case of strictly positive, symmetric, $L^1$ convolution kernels that have a finite second moment. Modeling the equations on a finite interval, we prove the existence of small-time weak solutions in the case of nonlocal Dirichlet and Neumann boundary constraints. We then use this result to develop a finite element numerical scheme that helps us explore the effects of nonlocal diffusion on the formation of pulse solutions.

math.NA↗

arXiv2022

Existence and convergence of a discontinuous Galerkin method for the incompressible three-phase flow problem in porous media

This paper presents and analyzes a discontinuous Galerkin method for the incompressible three-phase flow problem in porous media. We use a first order time extrapolation which allows us to solve the equations implicitly and sequentially. We show that the discrete problem is well-posed, and obtain a priori error estimates. Our numerical results validate the theoretical results, i.e. the algorithm converges with first order.

math.NA↗

arXiv2020

Numerical Methods for a Diffusive Class Nonlocal Operators

In this paper we develop a numerical scheme based on quadratures to approximate solutions of integro-differential equations involving convolution kernels, $ν$, of diffusive type. In particular, we assume $ν$ is symmetric and exponentially decaying at infinity. We consider problems posed in bounded domains and in $\R$. In the case of bounded domains with nonlocal Dirichlet boundary conditions, we show the convergence of the scheme for kernels that have positive tails, but that can take on negative values. When the equations are posed on all of $\R$, we show that our scheme converges for nonnegative kernels. Since nonlocal Neumann boundary conditions lead to an equivalent formulation as in the unbounded case, we show that these last results also apply to the Neumann problem.

math.NA↗