Practical Error Analysis for the Three-Level Bilinear Fem and Finite-Difference Scheme for the 1d Wave Equation With Non-Smooth Data

Abstract

Abstract. We deal with the standard three-level bilinear FEM and finite-difference scheme with a weight to solve the initial-boundary value problem for the 1D wave equation. We consider the rich collection of initial data and the free term which are the Dirac δ-functions, discontinuous, continuous but with discontinuous derivatives and from the Sobolev spaces, accomplish the practical error analysis in the L 2 , L 1 , energy and uniform norms as the mesh refines and compare results with known theoretical error bounds.

On Compact 4th Order Finite-Difference Schemes for the Wave Equation

“…according to [19]; recall that the middle error order is derived from two other ones. These orders also have recently been confirmed practically in [16].…”

“…The cases of the delta-shaped, discontinuous or with discontinuous derivatives data are covered. The higher-order practical error behavior is shown compared to standard 2nd approximation order schemes [16,19] thus confirming the essential advantages of 4th order schemes over them in the non-smooth case as well. Second, we present numerical results in the case of non-uniform spatial meshes with various node distribution functions (for the smooth data).…”

“…For more visibility, here we include the error norms r The main observation is the nice agreement between γ pr and γ th for all three norms in all Examples E α , thus the sensitive dependence of γ pr on the data smoothness order α becomes quite clear. This agreement is mainly better for the first and second norms (5.1) (similarly to [16]). Notice that γ for N = 200 and N max especially as α grows.…”

“…The properties of u in Example E α have been described in [16] or are similar. Recall that, for example, u is piecewise-constant and discontinuous onQ for α = 1 2 , or u is piecewise-linear with discontinuous piecewise-constant derivatives onQ for α = 3 2 , etc.…”

“…. , 11/2, of non-smooth data supplementing the study in [16]. The initial functions u 0 = P [α] and u 1 = c 1 P [α]−1 are piecewise-polynomial functions of the degree [α] and [α] − 1 respectively, with a unique singularity point x = 0, excluding the case [α] = 0 for u 1 , where u 1 (x) = c 1 δ(x).…”