Cascading Multilevel Finite-Element Analysis for Local and Nonlocal Parabolic Problems

In this article, several cascading multilevel finite-element algorithms are considered to discretize nonlinear parabolic problems, of which the nonlinearity has either local or nonlocal form. Algorithm I solves only a stationary linear system of equations at each level of P1 finite element spaces, while Algorithm II works on the coupling of a stationary linear system of equations with a linear parabolic equation. The convergence orders of Algorithms I and II are both O(hJ) in the energy norm; in Algorithm I the estimation depends on the number of grids, while Algorithm II does not. Algorithm III is based on Picard linearization techniques and Algorithm IV on Newton iteration. Both algorithms have convergence orderO(hJ).