A POSTERIORI ERROR ESTIMATION WITH FINITE-ELEMENT METHODS OF LINES FOR ONE-DIMENSIONAL PARABOLIC-SYSTEMS

Consider the solution of one-dimensional linear initial-boundary value problems by a finite element method of lines using a piecewise p(th)-degree polynomial basis. A posteriori estimates of the discretization error are obtained as the solutions of either local parabolic or local elliptic finite element problems using piecewise polynomial corrections of degree p + 1 that vanish at element ends. Error estimates computed in this manner are shown to converge in energy under mesh refinement to the exact finite element discretization error. Computational results indicate that the error estimates are robust over a wide range of mesh spacings and polynomial degrees and are, furthermore, applicable in situations that are not supported by the analysis.