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

Abstract

© 2016 Wiley Periodicals, Inc. 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 2 (H 0 1 (Ω)) -norm are proved for fully discrete schemes. Finally, several numerical experiments are conducted to confirm our theoretical findings. Numer Methods Partial Differential Eq 32: 1232-1264, 2016.