The Euler-Galerkin finite element method for a nonlocal coupled system of reaction-diffusion type

In this work, we study a system of parabolic equations with nonlocal nonlinearity of the following type {u(t) - a(1) (l(1) (u), l(2)(v)) Delta u + lambda(i)vertical bar u vertical bar(p-2)u = f(1)(x, t) in Omega x]0, T] v(t) - a(2)(l(1)(u), l(2)(v)Delta v + lambda(2)vertical bar v vertical bar(p-2)v = f(2)(x, t) in Omega x]0, T] u(x, t) = v(x, t) = 0 on partial derivative Omega x]0, T] u(x, 0) = u(0)(x), v(x, 0) = v(0)(x) in Omega where a(1) and a(2) are Lipschitz-continuous positive functions, l(1) and l(2) are continuous linear forms, lambda(1), lambda(2) >= 0 and p >= 2. We prove the convergence of a linearized Euler-Galerkin finite element method and obtain the order of convergence in the L-2 norm. Finally we implement and simulate the presented method in Matlab's environment. (C) 2015 Elsevier B.V. All rights reserved.