This paper is an experimental continuation of , where we presented one realization of a locally-conservative Eulerian-Lagrangian finite element method (LCELM) for a semilinear
parabolic equation and proved an optimal convergence rate. In this paper, we present two higher-
order extensions of the method of , along with one lower-order procedure. We show some
numerical results to illustrate the accuracy and efficiency of the LCELM procedures. Optimal
convergence rates for each method will be presented.