DERIVATION OF LOWER ERROR BOUNDS FOR THE BILINEAR FINITE ELEMENT METHOD WITH A WEIGHT FOR THE ONE-DIMENSIONAL WAVE EQUATION

We study a three-level in time bilinear finite element method with weight for an initial-boundary value problem for the one-dimensional wave equation. We derive lower error estimates of orders (h + τ)2λ/3, 0 ⩽ λ ⩽ 3 in the L1 and W1,1 h norms. In them, each of the two initial functions or the free term in the equation belongs to Holder-type spaces of the corresponding orders of smoothness. They substantiate the accuracy in ¨ order of the corresponding known error estimates (from above) of the finite element method with a weight of the second-order approximation for second-order hyperbolic equations, as well as the impossibility of improving them with the maximum weakening of the degree of summability in the error norms and its maximum strengthening in the data norms. The derivation is based on the Fourier method.

wave equation, finite element method, lower error estimates on data spaces, Fourier method