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 {ut−a1(l1(u),l2(v))Δu+λ1|u|p−2u=f1(x,t)in  Ω×]0,T]vt−a2(l1(u),l2(v))Δv+λ2|v|p−2v=f2(x,t)in  Ω×]0,T]u(x,t)=v(x,t)=0on  ∂Ω×]0,T]u(x,0)=u0(x),v(x,0)=v0(x)in  Ω, where a1a1 and a2a2 are Lipschitz-continuous positive functions, l1l1 and l2l2 are continuous linear forms, λ1,λ2≥0λ1,λ2≥0 and p≥2p≥2.We prove the convergence of a linearized Euler–Galerkin finite element method and obtain the order of convergence in the L2L2 norm. Finally we implement and simulate the presented method in Matlab’s environment.