Finite element Galerkin approximations to a class of nonlinear and nonlocal parabolic problems

Abstract

In this article, a finite element Galerkin method is applied to a general class of nonlinear and nonlocal parabolic problems. Based on an exponential weight function, new a priori bounds which are valid for uniform in time are derived. As a result, existence of an attractor is proved for the problem with nonhomogeneous right hand side which is independent of time. In particular, when the forcing function is zero or decays exponentially, it is shown that solution has exponential decay property which improves even earlier results in one dimensional problems. For the semidiscrete method, global existence of a unique discrete solution is derived and it is shown that the discrete problem has an attractor. Moreover, optimal error estimates are derived in both L 2 ( H 0 1 ( Ω ) ) and L ∞ ( H 0 1 ( Ω ) ) ‐norms with later estimate is a new result in this context. For completely discrete scheme, backward Euler method with its linearized version is discussed and existence of a unique discrete solution is established. Further, optimal estimates in ℓ 2 ( H 0 1 ( Ω ) ) ‐norm are proved for fully discrete schemes. Finally, several numerical experiments are conducted to confirm our theoretical findings. © 2016 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 32: 1232–1264, 2016

Convergence analysis of expanded mixed virtual element methods for nonlocal problems

“…Adak et al [3], extended the technique to the framework of the virtual element method (VEM). In [4], the Galerkin approximation has been proposed, and a priori error estimations are derived for nonlinear nonlocal coupled Maxwell system in L 2 ( H 1 0 (Ω)…”

“…Adak et al [3], extended the technique to the framework of the virtual element method (VEM). In [4], the Galerkin approximation has been proposed, and a priori error estimations are derived for nonlinear nonlocal coupled Maxwell system in $$$ {L}^2\left({H}_0^1\left(\Omega \right)\right) $$$ and $$$ {L}^{\infty}\left({H}_0^1\left(\Omega \right)\right) $$$ norms for semi‐discrete schemes.…”

Optimal error estimates of a linearized second-order BDF scheme for a nonlocal parabolic problem

“…Simsen and Ferreira [4] have discussed not only the existence and uniqueness of solutions for this problem but also continuity with respect to initial values, the exponential stability of weak solutions and important results on the existence of a global attractor. The numerical methods for the nonlocal problems have been investigated by many authors as like in Refs [5,6] and the references therein. However, they are restricted to nonlocal reaction terms or nonlocal boundary conditions.…”

Uniqueness of solution and fully discrete scheme to nonlinear integro-differential averaged model with source terms

“…In those works main attention is paid to one-dimensional cases. Finite element analogues and Galerkin method algorithm as well as settling of semi-discrete and finite difference schemes for (1.1) type one-dimensional integro-differential models are studied in [2], [7], [8], [9], [15], [18], [22], [23], [24] and in the other works as well for the linear case of diffusion coefficient, i.e. a.S / D 1 C S .…”