Derivation of Lower Error Bounds for the Bilinear Element Method with a Weight for the One-Dimensional Wave Equation

The three-level in time bilinear finite element method with a weight is studied for an initial-boundary value problem for the one-dimensional wave equation. A derivation of lower error bounds of the orders (h + tau)(2 lambda/3), 0 <= lambda <= 3, in the L-1 and W-h(1,1) norms is given. 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 upper error bounds for a second-order finite element method with a weight for second-order hyperbolic equations as well as the impossibility of improving them under the maximal weakening of the summability order in the error norms and its maximal strengthening in the data norms. The derivation is based on the Fourier method.